src/HOL/Groups.thy
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fact generalization and name consolidation
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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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1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25230
diff changeset
   301
begin
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   302
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25230
diff changeset
   303
subclass cancel_semigroup_add
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28262
diff changeset
   304
proof
22390
378f34b1e380 now using "class"
haftmann
parents: 21382
diff changeset
   305
  fix a b c :: 'a
378f34b1e380 now using "class"
haftmann
parents: 21382
diff changeset
   306
  assume "a + b = a + c" 
378f34b1e380 now using "class"
haftmann
parents: 21382
diff changeset
   307
  then show "b = c" by (rule add_imp_eq)
378f34b1e380 now using "class"
haftmann
parents: 21382
diff changeset
   308
next
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   309
  fix a b c :: 'a
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   310
  assume "b + a = c + a"
22390
378f34b1e380 now using "class"
haftmann
parents: 21382
diff changeset
   311
  then have "a + b = a + c" by (simp only: add_commute)
378f34b1e380 now using "class"
haftmann
parents: 21382
diff changeset
   312
  then show "b = c" by (rule add_imp_eq)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   313
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   314
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25230
diff changeset
   315
end
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25230
diff changeset
   316
29904
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29886
diff changeset
   317
class cancel_comm_monoid_add = cancel_ab_semigroup_add + comm_monoid_add
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29886
diff changeset
   318
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29886
diff changeset
   319
23085
fd30d75a6614 Introduced new classes monoid_add and group_add
nipkow
parents: 22997
diff changeset
   320
subsection {* Groups *}
fd30d75a6614 Introduced new classes monoid_add and group_add
nipkow
parents: 22997
diff changeset
   321
25762
c03e9d04b3e4 splitted class uminus from class minus
haftmann
parents: 25613
diff changeset
   322
class group_add = minus + uminus + monoid_add +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   323
  assumes left_minus [simp]: "- a + a = 0"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   324
  assumes add_uminus_conv_diff [simp]: "a + (- b) = a - b"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   325
begin
23085
fd30d75a6614 Introduced new classes monoid_add and group_add
nipkow
parents: 22997
diff changeset
   326
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   327
lemma diff_conv_add_uminus:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   328
  "a - b = a + (- b)"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   329
  by simp
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   330
34147
319616f4eecf generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents: 34146
diff changeset
   331
lemma minus_unique:
319616f4eecf generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents: 34146
diff changeset
   332
  assumes "a + b = 0" shows "- a = b"
319616f4eecf generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents: 34146
diff changeset
   333
proof -
319616f4eecf generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents: 34146
diff changeset
   334
  have "- a = - a + (a + b)" using assms by simp
319616f4eecf generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents: 34146
diff changeset
   335
  also have "\<dots> = b" by (simp add: add_assoc [symmetric])
319616f4eecf generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents: 34146
diff changeset
   336
  finally show ?thesis .
319616f4eecf generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents: 34146
diff changeset
   337
qed
319616f4eecf generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents: 34146
diff changeset
   338
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   339
lemma minus_zero [simp]: "- 0 = 0"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   340
proof -
34147
319616f4eecf generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents: 34146
diff changeset
   341
  have "0 + 0 = 0" by (rule add_0_right)
319616f4eecf generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents: 34146
diff changeset
   342
  thus "- 0 = 0" by (rule minus_unique)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   343
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   344
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   345
lemma minus_minus [simp]: "- (- a) = a"
23085
fd30d75a6614 Introduced new classes monoid_add and group_add
nipkow
parents: 22997
diff changeset
   346
proof -
34147
319616f4eecf generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents: 34146
diff changeset
   347
  have "- a + a = 0" by (rule left_minus)
319616f4eecf generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents: 34146
diff changeset
   348
  thus "- (- a) = a" by (rule minus_unique)
23085
fd30d75a6614 Introduced new classes monoid_add and group_add
nipkow
parents: 22997
diff changeset
   349
qed
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   350
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   351
lemma right_minus: "a + - a = 0"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   352
proof -
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   353
  have "a + - a = - (- a) + - a" by simp
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   354
  also have "\<dots> = 0" by (rule left_minus)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   355
  finally show ?thesis .
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   356
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   357
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   358
lemma diff_self [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   359
  "a - a = 0"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   360
  using right_minus [of a] by simp
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   361
40368
47c186c8577d added class relation group_add < cancel_semigroup_add
haftmann
parents: 39134
diff changeset
   362
subclass cancel_semigroup_add
47c186c8577d added class relation group_add < cancel_semigroup_add
haftmann
parents: 39134
diff changeset
   363
proof
47c186c8577d added class relation group_add < cancel_semigroup_add
haftmann
parents: 39134
diff changeset
   364
  fix a b c :: 'a
47c186c8577d added class relation group_add < cancel_semigroup_add
haftmann
parents: 39134
diff changeset
   365
  assume "a + b = a + c"
47c186c8577d added class relation group_add < cancel_semigroup_add
haftmann
parents: 39134
diff changeset
   366
  then have "- a + a + b = - a + a + c"
47c186c8577d added class relation group_add < cancel_semigroup_add
haftmann
parents: 39134
diff changeset
   367
    unfolding add_assoc by simp
47c186c8577d added class relation group_add < cancel_semigroup_add
haftmann
parents: 39134
diff changeset
   368
  then show "b = c" by simp
47c186c8577d added class relation group_add < cancel_semigroup_add
haftmann
parents: 39134
diff changeset
   369
next
47c186c8577d added class relation group_add < cancel_semigroup_add
haftmann
parents: 39134
diff changeset
   370
  fix a b c :: 'a
47c186c8577d added class relation group_add < cancel_semigroup_add
haftmann
parents: 39134
diff changeset
   371
  assume "b + a = c + a"
47c186c8577d added class relation group_add < cancel_semigroup_add
haftmann
parents: 39134
diff changeset
   372
  then have "b + a + - a = c + a  + - a" by simp
47c186c8577d added class relation group_add < cancel_semigroup_add
haftmann
parents: 39134
diff changeset
   373
  then show "b = c" unfolding add_assoc by simp
47c186c8577d added class relation group_add < cancel_semigroup_add
haftmann
parents: 39134
diff changeset
   374
qed
47c186c8577d added class relation group_add < cancel_semigroup_add
haftmann
parents: 39134
diff changeset
   375
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   376
lemma minus_add_cancel [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   377
  "- a + (a + b) = b"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   378
  by (simp add: add_assoc [symmetric])
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   379
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   380
lemma add_minus_cancel [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   381
  "a + (- a + b) = b"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   382
  by (simp add: add_assoc [symmetric])
34147
319616f4eecf generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents: 34146
diff changeset
   383
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   384
lemma diff_add_cancel [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   385
  "a - b + b = a"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   386
  by (simp only: diff_conv_add_uminus add_assoc) simp
34147
319616f4eecf generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents: 34146
diff changeset
   387
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   388
lemma add_diff_cancel [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   389
  "a + b - b = a"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   390
  by (simp only: diff_conv_add_uminus add_assoc) simp
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   391
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   392
lemma minus_add:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   393
  "- (a + b) = - b + - a"
34147
319616f4eecf generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents: 34146
diff changeset
   394
proof -
319616f4eecf generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents: 34146
diff changeset
   395
  have "(a + b) + (- b + - a) = 0"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   396
    by (simp only: add_assoc add_minus_cancel) simp
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   397
  then show "- (a + b) = - b + - a"
34147
319616f4eecf generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents: 34146
diff changeset
   398
    by (rule minus_unique)
319616f4eecf generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents: 34146
diff changeset
   399
qed
319616f4eecf generalize lemma add_minus_cancel, add lemma minus_add, simplify some proofs
huffman
parents: 34146
diff changeset
   400
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   401
lemma right_minus_eq [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   402
  "a - b = 0 \<longleftrightarrow> a = b"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   403
proof
23085
fd30d75a6614 Introduced new classes monoid_add and group_add
nipkow
parents: 22997
diff changeset
   404
  assume "a - b = 0"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   405
  have "a = (a - b) + b" by (simp add: add_assoc)
23085
fd30d75a6614 Introduced new classes monoid_add and group_add
nipkow
parents: 22997
diff changeset
   406
  also have "\<dots> = b" using `a - b = 0` by simp
fd30d75a6614 Introduced new classes monoid_add and group_add
nipkow
parents: 22997
diff changeset
   407
  finally show "a = b" .
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   408
next
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   409
  assume "a = b" thus "a - b = 0" by simp
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   410
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   411
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   412
lemma eq_iff_diff_eq_0:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   413
  "a = b \<longleftrightarrow> a - b = 0"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   414
  by (fact right_minus_eq [symmetric])
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   415
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   416
lemma diff_0 [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   417
  "0 - a = - a"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   418
  by (simp only: diff_conv_add_uminus add_0_left)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   419
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   420
lemma diff_0_right [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   421
  "a - 0 = a" 
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   422
  by (simp only: diff_conv_add_uminus minus_zero add_0_right)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   423
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   424
lemma diff_minus_eq_add [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   425
  "a - - b = a + b"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   426
  by (simp only: diff_conv_add_uminus minus_minus)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   427
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   428
lemma neg_equal_iff_equal [simp]:
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   429
  "- a = - b \<longleftrightarrow> a = b" 
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   430
proof 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   431
  assume "- a = - b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   432
  hence "- (- a) = - (- b)" by simp
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   433
  thus "a = b" by simp
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   434
next
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   435
  assume "a = b"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   436
  thus "- a = - b" by simp
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   437
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   438
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   439
lemma neg_equal_0_iff_equal [simp]:
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   440
  "- a = 0 \<longleftrightarrow> a = 0"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   441
  by (subst neg_equal_iff_equal [symmetric]) simp
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   442
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   443
lemma neg_0_equal_iff_equal [simp]:
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   444
  "0 = - a \<longleftrightarrow> 0 = a"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   445
  by (subst neg_equal_iff_equal [symmetric]) simp
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   446
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   447
text{*The next two equations can make the simplifier loop!*}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   448
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   449
lemma equation_minus_iff:
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   450
  "a = - b \<longleftrightarrow> b = - a"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   451
proof -
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   452
  have "- (- a) = - b \<longleftrightarrow> - a = b" by (rule neg_equal_iff_equal)
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   453
  thus ?thesis by (simp add: eq_commute)
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   454
qed
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   455
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   456
lemma minus_equation_iff:
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   457
  "- a = b \<longleftrightarrow> - b = a"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   458
proof -
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   459
  have "- a = - (- b) \<longleftrightarrow> a = -b" by (rule neg_equal_iff_equal)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   460
  thus ?thesis by (simp add: eq_commute)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   461
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   462
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   463
lemma eq_neg_iff_add_eq_0:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   464
  "a = - b \<longleftrightarrow> a + b = 0"
29914
c9ced4f54e82 generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents: 29904
diff changeset
   465
proof
c9ced4f54e82 generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents: 29904
diff changeset
   466
  assume "a = - b" then show "a + b = 0" by simp
c9ced4f54e82 generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents: 29904
diff changeset
   467
next
c9ced4f54e82 generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents: 29904
diff changeset
   468
  assume "a + b = 0"
c9ced4f54e82 generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents: 29904
diff changeset
   469
  moreover have "a + (b + - b) = (a + b) + - b"
c9ced4f54e82 generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents: 29904
diff changeset
   470
    by (simp only: add_assoc)
c9ced4f54e82 generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents: 29904
diff changeset
   471
  ultimately show "a = - b" by simp
c9ced4f54e82 generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents: 29904
diff changeset
   472
qed
c9ced4f54e82 generalize lemma eq_neg_iff_add_eq_0, and move to OrderedGroup
huffman
parents: 29904
diff changeset
   473
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   474
lemma add_eq_0_iff2:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   475
  "a + b = 0 \<longleftrightarrow> a = - b"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   476
  by (fact eq_neg_iff_add_eq_0 [symmetric])
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   477
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   478
lemma neg_eq_iff_add_eq_0:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   479
  "- a = b \<longleftrightarrow> a + b = 0"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   480
  by (auto simp add: add_eq_0_iff2)
44348
40101794c52f move lemma add_eq_0_iff to Groups.thy
huffman
parents: 42248
diff changeset
   481
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   482
lemma add_eq_0_iff:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   483
  "a + b = 0 \<longleftrightarrow> b = - a"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   484
  by (auto simp add: neg_eq_iff_add_eq_0 [symmetric])
45548
3e2722d66169 Groups.thy: generalize several lemmas from class ab_group_add to class group_add
huffman
parents: 45294
diff changeset
   485
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   486
lemma minus_diff_eq [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   487
  "- (a - b) = b - a"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   488
  by (simp only: neg_eq_iff_add_eq_0 diff_conv_add_uminus add_assoc minus_add_cancel) simp
45548
3e2722d66169 Groups.thy: generalize several lemmas from class ab_group_add to class group_add
huffman
parents: 45294
diff changeset
   489
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   490
lemma add_diff_eq [algebra_simps, field_simps]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   491
  "a + (b - c) = (a + b) - c"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   492
  by (simp only: diff_conv_add_uminus add_assoc)
45548
3e2722d66169 Groups.thy: generalize several lemmas from class ab_group_add to class group_add
huffman
parents: 45294
diff changeset
   493
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   494
lemma diff_add_eq_diff_diff_swap:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   495
  "a - (b + c) = a - c - b"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   496
  by (simp only: diff_conv_add_uminus add_assoc minus_add)
45548
3e2722d66169 Groups.thy: generalize several lemmas from class ab_group_add to class group_add
huffman
parents: 45294
diff changeset
   497
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   498
lemma diff_eq_eq [algebra_simps, field_simps]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   499
  "a - b = c \<longleftrightarrow> a = c + b"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   500
  by auto
45548
3e2722d66169 Groups.thy: generalize several lemmas from class ab_group_add to class group_add
huffman
parents: 45294
diff changeset
   501
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   502
lemma eq_diff_eq [algebra_simps, field_simps]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   503
  "a = c - b \<longleftrightarrow> a + b = c"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   504
  by auto
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   505
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   506
lemma diff_diff_eq2 [algebra_simps, field_simps]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   507
  "a - (b - c) = (a + c) - b"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   508
  by (simp only: diff_conv_add_uminus add_assoc) simp
45548
3e2722d66169 Groups.thy: generalize several lemmas from class ab_group_add to class group_add
huffman
parents: 45294
diff changeset
   509
3e2722d66169 Groups.thy: generalize several lemmas from class ab_group_add to class group_add
huffman
parents: 45294
diff changeset
   510
lemma diff_eq_diff_eq:
3e2722d66169 Groups.thy: generalize several lemmas from class ab_group_add to class group_add
huffman
parents: 45294
diff changeset
   511
  "a - b = c - d \<Longrightarrow> a = b \<longleftrightarrow> c = d"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   512
  by (simp only: eq_iff_diff_eq_0 [of a b] eq_iff_diff_eq_0 [of c d])
45548
3e2722d66169 Groups.thy: generalize several lemmas from class ab_group_add to class group_add
huffman
parents: 45294
diff changeset
   513
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   514
end
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   515
25762
c03e9d04b3e4 splitted class uminus from class minus
haftmann
parents: 25613
diff changeset
   516
class ab_group_add = minus + uminus + comm_monoid_add +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   517
  assumes ab_left_minus: "- a + a = 0"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   518
  assumes ab_add_uminus_conv_diff: "a - b = a + (- b)"
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25230
diff changeset
   519
begin
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   520
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25230
diff changeset
   521
subclass group_add
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   522
  proof qed (simp_all add: ab_left_minus ab_add_uminus_conv_diff)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   523
29904
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29886
diff changeset
   524
subclass cancel_comm_monoid_add
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28262
diff changeset
   525
proof
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   526
  fix a b c :: 'a
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   527
  assume "a + b = a + c"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   528
  then have "- a + a + b = - a + a + c"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   529
    by (simp only: add_assoc)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   530
  then show "b = c" by simp
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   531
qed
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   532
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   533
lemma uminus_add_conv_diff [simp]:
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   534
  "- a + b = b - a"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   535
  by (simp add: add_commute)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   536
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   537
lemma minus_add_distrib [simp]:
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   538
  "- (a + b) = - a + - b"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   539
  by (simp add: algebra_simps)
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   540
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   541
lemma diff_add_eq [algebra_simps, field_simps]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   542
  "(a - b) + c = (a + c) - b"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   543
  by (simp add: algebra_simps)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   544
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   545
lemma diff_diff_eq [algebra_simps, field_simps]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   546
  "(a - b) - c = a - (b + c)"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   547
  by (simp add: algebra_simps)
30629
5cd9b19edef3 move diff_eq_0_iff_eq into class locale context
huffman
parents: 29914
diff changeset
   548
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   549
lemma diff_add_eq_diff_diff:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   550
  "a - (b + c) = a - b - c"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   551
  using diff_add_eq_diff_diff_swap [of a c b] by (simp add: add.commute)
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   552
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   553
lemma add_diff_cancel_left [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   554
  "(c + a) - (c + b) = a - b"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   555
  by (simp add: algebra_simps)
48556
62a3fbf9d35b replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents: 45548
diff changeset
   556
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   557
end
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   558
37884
314a88278715 discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents: 36977
diff changeset
   559
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   560
subsection {* (Partially) Ordered Groups *} 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   561
35301
90e42f9ba4d1 distributed theory Algebras to theories Groups and Lattices
haftmann
parents: 35267
diff changeset
   562
text {*
90e42f9ba4d1 distributed theory Algebras to theories Groups and Lattices
haftmann
parents: 35267
diff changeset
   563
  The theory of partially ordered groups is taken from the books:
90e42f9ba4d1 distributed theory Algebras to theories Groups and Lattices
haftmann
parents: 35267
diff changeset
   564
  \begin{itemize}
90e42f9ba4d1 distributed theory Algebras to theories Groups and Lattices
haftmann
parents: 35267
diff changeset
   565
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
90e42f9ba4d1 distributed theory Algebras to theories Groups and Lattices
haftmann
parents: 35267
diff changeset
   566
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
90e42f9ba4d1 distributed theory Algebras to theories Groups and Lattices
haftmann
parents: 35267
diff changeset
   567
  \end{itemize}
90e42f9ba4d1 distributed theory Algebras to theories Groups and Lattices
haftmann
parents: 35267
diff changeset
   568
  Most of the used notions can also be looked up in 
90e42f9ba4d1 distributed theory Algebras to theories Groups and Lattices
haftmann
parents: 35267
diff changeset
   569
  \begin{itemize}
90e42f9ba4d1 distributed theory Algebras to theories Groups and Lattices
haftmann
parents: 35267
diff changeset
   570
  \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
90e42f9ba4d1 distributed theory Algebras to theories Groups and Lattices
haftmann
parents: 35267
diff changeset
   571
  \item \emph{Algebra I} by van der Waerden, Springer.
90e42f9ba4d1 distributed theory Algebras to theories Groups and Lattices
haftmann
parents: 35267
diff changeset
   572
  \end{itemize}
90e42f9ba4d1 distributed theory Algebras to theories Groups and Lattices
haftmann
parents: 35267
diff changeset
   573
*}
90e42f9ba4d1 distributed theory Algebras to theories Groups and Lattices
haftmann
parents: 35267
diff changeset
   574
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   575
class ordered_ab_semigroup_add = order + ab_semigroup_add +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   576
  assumes add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   577
begin
24380
c215e256beca moved ordered_ab_semigroup_add to OrderedGroup.thy
haftmann
parents: 24286
diff changeset
   578
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   579
lemma add_right_mono:
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   580
  "a \<le> b \<Longrightarrow> a + c \<le> b + c"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   581
by (simp add: add_commute [of _ c] add_left_mono)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   582
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   583
text {* non-strict, in both arguments *}
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   584
lemma add_mono:
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   585
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   586
  apply (erule add_right_mono [THEN order_trans])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   587
  apply (simp add: add_commute add_left_mono)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   588
  done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   589
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   590
end
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   591
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   592
class ordered_cancel_ab_semigroup_add =
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   593
  ordered_ab_semigroup_add + cancel_ab_semigroup_add
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   594
begin
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   595
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   596
lemma add_strict_left_mono:
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   597
  "a < b \<Longrightarrow> c + a < c + b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   598
by (auto simp add: less_le add_left_mono)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   599
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   600
lemma add_strict_right_mono:
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   601
  "a < b \<Longrightarrow> a + c < b + c"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   602
by (simp add: add_commute [of _ c] add_strict_left_mono)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   603
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   604
text{*Strict monotonicity in both arguments*}
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   605
lemma add_strict_mono:
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   606
  "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   607
apply (erule add_strict_right_mono [THEN less_trans])
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   608
apply (erule add_strict_left_mono)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   609
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   610
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   611
lemma add_less_le_mono:
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   612
  "a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c < b + d"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   613
apply (erule add_strict_right_mono [THEN less_le_trans])
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   614
apply (erule add_left_mono)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   615
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   616
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   617
lemma add_le_less_mono:
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   618
  "a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   619
apply (erule add_right_mono [THEN le_less_trans])
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   620
apply (erule add_strict_left_mono) 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   621
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   622
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   623
end
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   624
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   625
class ordered_ab_semigroup_add_imp_le =
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   626
  ordered_cancel_ab_semigroup_add +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   627
  assumes add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   628
begin
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   629
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   630
lemma add_less_imp_less_left:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   631
  assumes less: "c + a < c + b" shows "a < b"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   632
proof -
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   633
  from less have le: "c + a <= c + b" by (simp add: order_le_less)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   634
  have "a <= b" 
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   635
    apply (insert le)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   636
    apply (drule add_le_imp_le_left)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   637
    by (insert le, drule add_le_imp_le_left, assumption)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   638
  moreover have "a \<noteq> b"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   639
  proof (rule ccontr)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   640
    assume "~(a \<noteq> b)"
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   641
    then have "a = b" by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   642
    then have "c + a = c + b" by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   643
    with less show "False"by simp
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   644
  qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   645
  ultimately show "a < b" by (simp add: order_le_less)
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   646
qed
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   647
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   648
lemma add_less_imp_less_right:
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   649
  "a + c < b + c \<Longrightarrow> a < b"
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   650
apply (rule add_less_imp_less_left [of c])
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   651
apply (simp add: add_commute)  
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   652
done
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   653
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   654
lemma add_less_cancel_left [simp]:
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   655
  "c + a < c + b \<longleftrightarrow> a < b"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   656
  by (blast intro: add_less_imp_less_left add_strict_left_mono) 
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   657
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   658
lemma add_less_cancel_right [simp]:
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   659
  "a + c < b + c \<longleftrightarrow> a < b"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   660
  by (blast intro: add_less_imp_less_right add_strict_right_mono)
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   661
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   662
lemma add_le_cancel_left [simp]:
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   663
  "c + a \<le> c + b \<longleftrightarrow> a \<le> b"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   664
  by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono) 
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   665
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   666
lemma add_le_cancel_right [simp]:
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   667
  "a + c \<le> b + c \<longleftrightarrow> a \<le> b"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   668
  by (simp add: add_commute [of a c] add_commute [of b c])
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   669
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
   670
lemma add_le_imp_le_right:
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   671
  "a + c \<le> b + c \<Longrightarrow> a \<le> b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   672
by simp
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   673
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   674
lemma max_add_distrib_left:
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   675
  "max x y + z = max (x + z) (y + z)"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   676
  unfolding max_def by auto
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   677
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   678
lemma min_add_distrib_left:
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   679
  "min x y + z = min (x + z) (y + z)"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   680
  unfolding min_def by auto
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   681
44848
f4d0b060c7ca remove lemmas nat_add_min_{left,right} in favor of generic lemmas min_add_distrib_{left,right}
huffman
parents: 44433
diff changeset
   682
lemma max_add_distrib_right:
f4d0b060c7ca remove lemmas nat_add_min_{left,right} in favor of generic lemmas min_add_distrib_{left,right}
huffman
parents: 44433
diff changeset
   683
  "x + max y z = max (x + y) (x + z)"
f4d0b060c7ca remove lemmas nat_add_min_{left,right} in favor of generic lemmas min_add_distrib_{left,right}
huffman
parents: 44433
diff changeset
   684
  unfolding max_def by auto
f4d0b060c7ca remove lemmas nat_add_min_{left,right} in favor of generic lemmas min_add_distrib_{left,right}
huffman
parents: 44433
diff changeset
   685
f4d0b060c7ca remove lemmas nat_add_min_{left,right} in favor of generic lemmas min_add_distrib_{left,right}
huffman
parents: 44433
diff changeset
   686
lemma min_add_distrib_right:
f4d0b060c7ca remove lemmas nat_add_min_{left,right} in favor of generic lemmas min_add_distrib_{left,right}
huffman
parents: 44433
diff changeset
   687
  "x + min y z = min (x + y) (x + z)"
f4d0b060c7ca remove lemmas nat_add_min_{left,right} in favor of generic lemmas min_add_distrib_{left,right}
huffman
parents: 44433
diff changeset
   688
  unfolding min_def by auto
f4d0b060c7ca remove lemmas nat_add_min_{left,right} in favor of generic lemmas min_add_distrib_{left,right}
huffman
parents: 44433
diff changeset
   689
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   690
end
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   691
52289
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   692
class ordered_cancel_comm_monoid_diff = comm_monoid_diff + ordered_ab_semigroup_add_imp_le +
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   693
  assumes le_iff_add: "a \<le> b \<longleftrightarrow> (\<exists>c. b = a + c)"
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   694
begin
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   695
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   696
context
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   697
  fixes a b
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   698
  assumes "a \<le> b"
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   699
begin
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   700
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   701
lemma add_diff_inverse:
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   702
  "a + (b - a) = b"
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   703
  using `a \<le> b` by (auto simp add: le_iff_add)
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   704
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   705
lemma add_diff_assoc:
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   706
  "c + (b - a) = c + b - a"
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   707
  using `a \<le> b` by (auto simp add: le_iff_add add_left_commute [of c])
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   708
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   709
lemma add_diff_assoc2:
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   710
  "b - a + c = b + c - a"
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   711
  using `a \<le> b` by (auto simp add: le_iff_add add_assoc)
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   712
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   713
lemma diff_add_assoc:
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   714
  "c + b - a = c + (b - a)"
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   715
  using `a \<le> b` by (simp add: add_commute add_diff_assoc)
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   716
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   717
lemma diff_add_assoc2:
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   718
  "b + c - a = b - a + c"
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   719
  using `a \<le> b`by (simp add: add_commute add_diff_assoc)
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   720
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   721
lemma diff_diff_right:
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   722
  "c - (b - a) = c + a - b"
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   723
  by (simp add: add_diff_inverse add_diff_cancel_left [of a c "b - a", symmetric] add_commute)
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   724
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   725
lemma diff_add:
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   726
  "b - a + a = b"
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   727
  by (simp add: add_commute add_diff_inverse)
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   728
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   729
lemma le_add_diff:
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   730
  "c \<le> b + c - a"
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   731
  by (auto simp add: add_commute diff_add_assoc2 le_iff_add)
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   732
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   733
lemma le_imp_diff_is_add:
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   734
  "a \<le> b \<Longrightarrow> b - a = c \<longleftrightarrow> b = c + a"
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   735
  by (auto simp add: add_commute add_diff_inverse)
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   736
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   737
lemma le_diff_conv2:
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   738
  "c \<le> b - a \<longleftrightarrow> c + a \<le> b" (is "?P \<longleftrightarrow> ?Q")
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   739
proof
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   740
  assume ?P
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   741
  then have "c + a \<le> b - a + a" by (rule add_right_mono)
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   742
  then show ?Q by (simp add: add_diff_inverse add_commute)
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   743
next
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   744
  assume ?Q
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   745
  then have "a + c \<le> a + (b - a)" by (simp add: add_diff_inverse add_commute)
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   746
  then show ?P by simp
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   747
qed
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   748
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   749
end
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   750
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   751
end
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   752
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 52210
diff changeset
   753
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   754
subsection {* Support for reasoning about signs *}
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   755
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   756
class ordered_comm_monoid_add =
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   757
  ordered_cancel_ab_semigroup_add + comm_monoid_add
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   758
begin
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   759
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   760
lemma add_pos_nonneg:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   761
  assumes "0 < a" and "0 \<le> b" shows "0 < a + b"
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   762
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   763
  have "0 + 0 < a + b" 
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   764
    using assms by (rule add_less_le_mono)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   765
  then show ?thesis by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   766
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   767
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   768
lemma add_pos_pos:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   769
  assumes "0 < a" and "0 < b" shows "0 < a + b"
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   770
by (rule add_pos_nonneg) (insert assms, auto)
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   771
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   772
lemma add_nonneg_pos:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   773
  assumes "0 \<le> a" and "0 < b" shows "0 < a + b"
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   774
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   775
  have "0 + 0 < a + b" 
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   776
    using assms by (rule add_le_less_mono)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   777
  then show ?thesis by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   778
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   779
36977
71c8973a604b declare add_nonneg_nonneg [simp]; remove now-redundant lemmas realpow_two_le_order(2)
huffman
parents: 36348
diff changeset
   780
lemma add_nonneg_nonneg [simp]:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   781
  assumes "0 \<le> a" and "0 \<le> b" shows "0 \<le> a + b"
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   782
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   783
  have "0 + 0 \<le> a + b" 
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   784
    using assms by (rule add_mono)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   785
  then show ?thesis by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   786
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   787
30691
0047f57f6669 lemmas add_sign_intros
huffman
parents: 30629
diff changeset
   788
lemma add_neg_nonpos:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   789
  assumes "a < 0" and "b \<le> 0" shows "a + b < 0"
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   790
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   791
  have "a + b < 0 + 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   792
    using assms by (rule add_less_le_mono)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   793
  then show ?thesis by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   794
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   795
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   796
lemma add_neg_neg: 
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   797
  assumes "a < 0" and "b < 0" shows "a + b < 0"
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   798
by (rule add_neg_nonpos) (insert assms, auto)
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   799
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   800
lemma add_nonpos_neg:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   801
  assumes "a \<le> 0" and "b < 0" shows "a + b < 0"
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   802
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   803
  have "a + b < 0 + 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   804
    using assms by (rule add_le_less_mono)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   805
  then show ?thesis by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   806
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   807
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   808
lemma add_nonpos_nonpos:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   809
  assumes "a \<le> 0" and "b \<le> 0" shows "a + b \<le> 0"
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   810
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   811
  have "a + b \<le> 0 + 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   812
    using assms by (rule add_mono)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   813
  then show ?thesis by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   814
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   815
30691
0047f57f6669 lemmas add_sign_intros
huffman
parents: 30629
diff changeset
   816
lemmas add_sign_intros =
0047f57f6669 lemmas add_sign_intros
huffman
parents: 30629
diff changeset
   817
  add_pos_nonneg add_pos_pos add_nonneg_pos add_nonneg_nonneg
0047f57f6669 lemmas add_sign_intros
huffman
parents: 30629
diff changeset
   818
  add_neg_nonpos add_neg_neg add_nonpos_neg add_nonpos_nonpos
0047f57f6669 lemmas add_sign_intros
huffman
parents: 30629
diff changeset
   819
29886
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   820
lemma add_nonneg_eq_0_iff:
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   821
  assumes x: "0 \<le> x" and y: "0 \<le> y"
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   822
  shows "x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   823
proof (intro iffI conjI)
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   824
  have "x = x + 0" by simp
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   825
  also have "x + 0 \<le> x + y" using y by (rule add_left_mono)
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   826
  also assume "x + y = 0"
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   827
  also have "0 \<le> x" using x .
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   828
  finally show "x = 0" .
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   829
next
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   830
  have "y = 0 + y" by simp
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   831
  also have "0 + y \<le> x + y" using x by (rule add_right_mono)
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   832
  also assume "x + y = 0"
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   833
  also have "0 \<le> y" using y .
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   834
  finally show "y = 0" .
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   835
next
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   836
  assume "x = 0 \<and> y = 0"
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   837
  then show "x + y = 0" by simp
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   838
qed
b8a6b9c56fdd add lemma add_nonneg_eq_0_iff
huffman
parents: 29833
diff changeset
   839
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   840
lemma add_increasing:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   841
  "0 \<le> a \<Longrightarrow> b \<le> c \<Longrightarrow> b \<le> a + c"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   842
  by (insert add_mono [of 0 a b c], simp)
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   843
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   844
lemma add_increasing2:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   845
  "0 \<le> c \<Longrightarrow> b \<le> a \<Longrightarrow> b \<le> a + c"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   846
  by (simp add: add_increasing add_commute [of a])
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   847
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   848
lemma add_strict_increasing:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   849
  "0 < a \<Longrightarrow> b \<le> c \<Longrightarrow> b < a + c"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   850
  by (insert add_less_le_mono [of 0 a b c], simp)
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   851
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   852
lemma add_strict_increasing2:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   853
  "0 \<le> a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   854
  by (insert add_le_less_mono [of 0 a b c], simp)
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   855
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   856
end
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   857
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   858
class ordered_ab_group_add =
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   859
  ab_group_add + ordered_ab_semigroup_add
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   860
begin
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   861
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   862
subclass ordered_cancel_ab_semigroup_add ..
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   863
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   864
subclass ordered_ab_semigroup_add_imp_le
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28262
diff changeset
   865
proof
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   866
  fix a b c :: 'a
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   867
  assume "c + a \<le> c + b"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   868
  hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono)
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   869
  hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc)
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   870
  thus "a \<le> b" by simp
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   871
qed
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
   872
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
   873
subclass ordered_comm_monoid_add ..
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
   874
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   875
lemma add_less_same_cancel1 [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   876
  "b + a < b \<longleftrightarrow> a < 0"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   877
  using add_less_cancel_left [of _ _ 0] by simp
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   878
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   879
lemma add_less_same_cancel2 [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   880
  "a + b < b \<longleftrightarrow> a < 0"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   881
  using add_less_cancel_right [of _ _ 0] by simp
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   882
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   883
lemma less_add_same_cancel1 [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   884
  "a < a + b \<longleftrightarrow> 0 < b"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   885
  using add_less_cancel_left [of _ 0] by simp
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   886
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   887
lemma less_add_same_cancel2 [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   888
  "a < b + a \<longleftrightarrow> 0 < b"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   889
  using add_less_cancel_right [of 0] by simp
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   890
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   891
lemma add_le_same_cancel1 [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   892
  "b + a \<le> b \<longleftrightarrow> a \<le> 0"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   893
  using add_le_cancel_left [of _ _ 0] by simp
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   894
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   895
lemma add_le_same_cancel2 [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   896
  "a + b \<le> b \<longleftrightarrow> a \<le> 0"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   897
  using add_le_cancel_right [of _ _ 0] by simp
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   898
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   899
lemma le_add_same_cancel1 [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   900
  "a \<le> a + b \<longleftrightarrow> 0 \<le> b"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   901
  using add_le_cancel_left [of _ 0] by simp
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   902
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   903
lemma le_add_same_cancel2 [simp]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   904
  "a \<le> b + a \<longleftrightarrow> 0 \<le> b"
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   905
  using add_le_cancel_right [of 0] by simp
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   906
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   907
lemma max_diff_distrib_left:
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   908
  shows "max x y - z = max (x - z) (y - z)"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   909
  using max_add_distrib_left [of x y "- z"] by simp
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   910
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   911
lemma min_diff_distrib_left:
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   912
  shows "min x y - z = min (x - z) (y - z)"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   913
  using min_add_distrib_left [of x y "- z"] by simp
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   914
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   915
lemma le_imp_neg_le:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   916
  assumes "a \<le> b" shows "-b \<le> -a"
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   917
proof -
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   918
  have "-a+a \<le> -a+b" using `a \<le> b` by (rule add_left_mono) 
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   919
  then have "0 \<le> -a+b" by simp
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   920
  then have "0 + (-b) \<le> (-a + b) + (-b)" by (rule add_right_mono) 
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   921
  then show ?thesis by (simp add: algebra_simps)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   922
qed
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   923
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   924
lemma neg_le_iff_le [simp]: "- b \<le> - a \<longleftrightarrow> a \<le> b"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   925
proof 
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   926
  assume "- b \<le> - a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   927
  hence "- (- a) \<le> - (- b)" by (rule le_imp_neg_le)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   928
  thus "a\<le>b" by simp
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   929
next
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   930
  assume "a\<le>b"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   931
  thus "-b \<le> -a" by (rule le_imp_neg_le)
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   932
qed
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   933
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   934
lemma neg_le_0_iff_le [simp]: "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   935
by (subst neg_le_iff_le [symmetric], simp)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   936
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   937
lemma neg_0_le_iff_le [simp]: "0 \<le> - a \<longleftrightarrow> a \<le> 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   938
by (subst neg_le_iff_le [symmetric], simp)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   939
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   940
lemma neg_less_iff_less [simp]: "- b < - a \<longleftrightarrow> a < b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   941
by (force simp add: less_le) 
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   942
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   943
lemma neg_less_0_iff_less [simp]: "- a < 0 \<longleftrightarrow> 0 < a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   944
by (subst neg_less_iff_less [symmetric], simp)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   945
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   946
lemma neg_0_less_iff_less [simp]: "0 < - a \<longleftrightarrow> a < 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   947
by (subst neg_less_iff_less [symmetric], simp)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   948
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   949
text{*The next several equations can make the simplifier loop!*}
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   950
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   951
lemma less_minus_iff: "a < - b \<longleftrightarrow> b < - a"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   952
proof -
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   953
  have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   954
  thus ?thesis by simp
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   955
qed
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   956
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   957
lemma minus_less_iff: "- a < b \<longleftrightarrow> - b < a"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   958
proof -
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   959
  have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   960
  thus ?thesis by simp
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   961
qed
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   962
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   963
lemma le_minus_iff: "a \<le> - b \<longleftrightarrow> b \<le> - a"
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   964
proof -
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   965
  have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff)
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   966
  have "(- (- a) <= -b) = (b <= - a)" 
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   967
    apply (auto simp only: le_less)
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   968
    apply (drule mm)
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   969
    apply (simp_all)
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   970
    apply (drule mm[simplified], assumption)
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   971
    done
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   972
  then show ?thesis by simp
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   973
qed
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   974
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   975
lemma minus_le_iff: "- a \<le> b \<longleftrightarrow> - b \<le> a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
   976
by (auto simp add: le_less minus_less_iff)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   977
54148
c8cc5ab4a863 killed more "no_atp"s
blanchet
parents: 54147
diff changeset
   978
lemma diff_less_0_iff_less [simp]:
37884
314a88278715 discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents: 36977
diff changeset
   979
  "a - b < 0 \<longleftrightarrow> a < b"
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   980
proof -
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   981
  have "a - b < 0 \<longleftrightarrow> a + (- b) < b + (- b)" by simp
37884
314a88278715 discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents: 36977
diff changeset
   982
  also have "... \<longleftrightarrow> a < b" by (simp only: add_less_cancel_right)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   983
  finally show ?thesis .
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   984
qed
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   985
37884
314a88278715 discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents: 36977
diff changeset
   986
lemmas less_iff_diff_less_0 = diff_less_0_iff_less [symmetric]
314a88278715 discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents: 36977
diff changeset
   987
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   988
lemma diff_less_eq [algebra_simps, field_simps]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   989
  "a - b < c \<longleftrightarrow> a < c + b"
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   990
apply (subst less_iff_diff_less_0 [of a])
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   991
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   992
apply (simp add: algebra_simps)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   993
done
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   994
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   995
lemma less_diff_eq[algebra_simps, field_simps]:
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   996
  "a < c - b \<longleftrightarrow> a + b < c"
36302
4e7f5b22dd7d more localization; tuned proofs
haftmann
parents: 36176
diff changeset
   997
apply (subst less_iff_diff_less_0 [of "a + b"])
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
   998
apply (subst less_iff_diff_less_0 [of a])
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
   999
apply (simp add: algebra_simps)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1000
done
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1001
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
  1002
lemma diff_le_eq[algebra_simps, field_simps]: "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
  1003
by (auto simp add: le_less diff_less_eq )
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1004
36348
89c54f51f55a dropped group_simps, ring_simps, field_eq_simps; classes division_ring_inverse_zero, field_inverse_zero, linordered_field_inverse_zero
haftmann
parents: 36343
diff changeset
  1005
lemma le_diff_eq[algebra_simps, field_simps]: "a \<le> c - b \<longleftrightarrow> a + b \<le> c"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 54148
diff changeset
  1006
by (auto simp add: le_less less_diff_eq)
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1007
54148
c8cc5ab4a863 killed more "no_atp"s
blanchet
parents: 54147
diff changeset
  1008
lemma diff_le_0_iff_le [simp]:
37884
314a88278715 discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents: 36977
diff changeset
  1009
  "a - b \<le> 0 \<longleftrightarrow> a \<le> b"
314a88278715 discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents: 36977
diff changeset
  1010
  by (simp add: algebra_simps)
314a88278715 discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents: 36977
diff changeset
  1011
314a88278715 discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents: 36977
diff changeset
  1012
lemmas le_iff_diff_le_0 = diff_le_0_iff_le [symmetric]
314a88278715 discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents: 36977
diff changeset
  1013
314a88278715 discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents: 36977
diff changeset
  1014
lemma diff_eq_diff_less:
314a88278715 discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents: 36977
diff changeset
  1015
  "a - b = c - d \<Longrightarrow> a < b \<longleftrightarrow> c < d"
314a88278715 discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents: 36977
diff changeset
  1016
  by (auto simp only: less_iff_diff_less_0 [of a b] less_iff_diff_less_0 [of c d])
314a88278715 discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents: 36977
diff changeset
  1017
37889
0d8058e0c270 keep explicit diff_def as legacy theorem; modernized abel_cancel simproc setup
haftmann
parents: 37884
diff changeset
  1018
lemma diff_eq_diff_less_eq:
0d8058e0c270 keep explicit diff_def as legacy theorem; modernized abel_cancel simproc setup
haftmann
parents: 37884
diff changeset
  1019
  "a - b = c - d \<Longrightarrow> a \<le> b \<longleftrightarrow> c \<le> d"
0d8058e0c270 keep explicit diff_def as legacy theorem; modernized abel_cancel simproc setup
haftmann
parents: 37884
diff changeset
  1020
  by (auto simp only: le_iff_diff_le_0 [of a b] le_iff_diff_le_0 [of c d])
25077
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1021
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1022
end
c2ec5e589d78 continued localization
haftmann
parents: 25062
diff changeset
  1023
48891
c0eafbd55de3 prefer ML_file over old uses;
wenzelm
parents: 48556
diff changeset
  1024
ML_file "Tools/group_cancel.ML"
48556
62a3fbf9d35b replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents: 45548
diff changeset
  1025
62a3fbf9d35b replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents: 45548
diff changeset
  1026
simproc_setup group_cancel_add ("a + b::'a::ab_group_add") =
62a3fbf9d35b replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents: 45548
diff changeset
  1027
  {* fn phi => fn ss => try Group_Cancel.cancel_add_conv *}
62a3fbf9d35b replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents: 45548
diff changeset
  1028
62a3fbf9d35b replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents: 45548
diff changeset
  1029
simproc_setup group_cancel_diff ("a - b::'a::ab_group_add") =
62a3fbf9d35b replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents: 45548
diff changeset
  1030
  {* fn phi => fn ss => try Group_Cancel.cancel_diff_conv *}
37884
314a88278715 discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents: 36977
diff changeset
  1031
48556
62a3fbf9d35b replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents: 45548
diff changeset
  1032
simproc_setup group_cancel_eq ("a = (b::'a::ab_group_add)") =
62a3fbf9d35b replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents: 45548
diff changeset
  1033
  {* fn phi => fn ss => try Group_Cancel.cancel_eq_conv *}
37889
0d8058e0c270 keep explicit diff_def as legacy theorem; modernized abel_cancel simproc setup
haftmann
parents: 37884
diff changeset
  1034
48556
62a3fbf9d35b replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents: 45548
diff changeset
  1035
simproc_setup group_cancel_le ("a \<le> (b::'a::ordered_ab_group_add)") =
62a3fbf9d35b replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents: 45548
diff changeset
  1036
  {* fn phi => fn ss => try Group_Cancel.cancel_le_conv *}
62a3fbf9d35b replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents: 45548
diff changeset
  1037
62a3fbf9d35b replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents: 45548
diff changeset
  1038
simproc_setup group_cancel_less ("a < (b::'a::ordered_ab_group_add)") =
62a3fbf9d35b replace abel_cancel simprocs with functionally equivalent, but simpler and faster ones
huffman
parents: 45548
diff changeset
  1039
  {* fn phi => fn ss => try Group_Cancel.cancel_less_conv *}
37884
314a88278715 discontinued pretending that abel_cancel is logic-independent; cleaned up junk
haftmann
parents: 36977
diff changeset
  1040
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
  1041
class linordered_ab_semigroup_add =
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
  1042
  linorder + ordered_ab_semigroup_add
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1043
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
  1044
class linordered_cancel_ab_semigroup_add =
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
  1045
  linorder + ordered_cancel_ab_semigroup_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25230
diff changeset
  1046
begin
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1047
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
  1048
subclass linordered_ab_semigroup_add ..
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1049
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
  1050
subclass ordered_ab_semigroup_add_imp_le
28823
dcbef866c9e2 tuned unfold_locales invocation
haftmann
parents: 28262
diff changeset
  1051
proof
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1052
  fix a b c :: 'a
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1053
  assume le: "c + a <= c + b"  
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1054
  show "a <= b"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1055
  proof (rule ccontr)
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1056
    assume w: "~ a \<le> b"
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1057
    hence "b <= a" by (simp add: linorder_not_le)
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1058
    hence le2: "c + b <= c + a" by (rule add_left_mono)
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1059
    have "a = b" 
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1060
      apply (insert le)
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1061
      apply (insert le2)
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1062
      apply (drule antisym, simp_all)
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1063
      done
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1064
    with w show False 
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1065
      by (simp add: linorder_not_le [symmetric])
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1066
  qed
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1067
qed
af5ef0d4d655 global class syntax
haftmann
parents: 24748
diff changeset
  1068
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25230
diff changeset
  1069
end
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25230
diff changeset
  1070
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
  1071
class linordered_ab_group_add = linorder + ordered_ab_group_add
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25230
diff changeset
  1072
begin
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1073
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
  1074
subclass linordered_cancel_ab_semigroup_add ..
25230
022029099a83 continued localization
haftmann
parents: 25194
diff changeset
  1075
35036
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1076
lemma equal_neg_zero [simp]:
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1077
  "a = - a \<longleftrightarrow> a = 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1078
proof
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1079
  assume "a = 0" then show "a = - a" by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1080
next
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1081
  assume A: "a = - a" show "a = 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1082
  proof (cases "0 \<le> a")
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1083
    case True with A have "0 \<le> - a" by auto
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1084
    with le_minus_iff have "a \<le> 0" by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1085
    with True show ?thesis by (auto intro: order_trans)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1086
  next
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1087
    case False then have B: "a \<le> 0" by auto
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1088
    with A have "- a \<le> 0" by auto
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1089
    with B show ?thesis by (auto intro: order_trans)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1090
  qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1091
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1092
35036
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1093
lemma neg_equal_zero [simp]:
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1094
  "- a = a \<longleftrightarrow> a = 0"
35036
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1095
  by (auto dest: sym)
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1096
54250
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1097
lemma neg_less_eq_nonneg [simp]:
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1098
  "- a \<le> a \<longleftrightarrow> 0 \<le> a"
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1099
proof
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1100
  assume A: "- a \<le> a" show "0 \<le> a"
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1101
  proof (rule classical)
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1102
    assume "\<not> 0 \<le> a"
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1103
    then have "a < 0" by auto
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1104
    with A have "- a < 0" by (rule le_less_trans)
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1105
    then show ?thesis by auto
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1106
  qed
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1107
next
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1108
  assume A: "0 \<le> a" show "- a \<le> a"
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1109
  proof (rule order_trans)
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1110
    show "- a \<le> 0" using A by (simp add: minus_le_iff)
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1111
  next
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1112
    show "0 \<le> a" using A .
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1113
  qed
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1114
qed
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1115
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1116
lemma neg_less_pos [simp]:
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1117
  "- a < a \<longleftrightarrow> 0 < a"
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1118
  by (auto simp add: less_le)
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1119
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1120
lemma less_eq_neg_nonpos [simp]:
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1121
  "a \<le> - a \<longleftrightarrow> a \<le> 0"
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1122
  using neg_less_eq_nonneg [of "- a"] by simp
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1123
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1124
lemma less_neg_neg [simp]:
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1125
  "a < - a \<longleftrightarrow> a < 0"
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1126
  using neg_less_pos [of "- a"] by simp
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1127
35036
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1128
lemma double_zero [simp]:
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1129
  "a + a = 0 \<longleftrightarrow> a = 0"
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1130
proof
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1131
  assume assm: "a + a = 0"
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1132
  then have a: "- a = a" by (rule minus_unique)
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 35092
diff changeset
  1133
  then show "a = 0" by (simp only: neg_equal_zero)
35036
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1134
qed simp
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1135
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1136
lemma double_zero_sym [simp]:
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1137
  "0 = a + a \<longleftrightarrow> a = 0"
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1138
  by (rule, drule sym) simp_all
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1139
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1140
lemma zero_less_double_add_iff_zero_less_single_add [simp]:
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1141
  "0 < a + a \<longleftrightarrow> 0 < a"
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1142
proof
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1143
  assume "0 < a + a"
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1144
  then have "0 - a < a" by (simp only: diff_less_eq)
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1145
  then have "- a < a" by simp
54250
7d2544dd3988 fact generalization and name consolidation
haftmann
parents: 54230
diff changeset
  1146
  then show "0 < a" by simp
35036
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1147
next
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1148
  assume "0 < a"
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1149
  with this have "0 + 0 < a + a"
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1150
    by (rule add_strict_mono)
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1151
  then show "0 < a + a" by simp
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1152
qed
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1153
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1154
lemma zero_le_double_add_iff_zero_le_single_add [simp]:
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1155
  "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1156
  by (auto simp add: le_less)
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1157
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1158
lemma double_add_less_zero_iff_single_add_less_zero [simp]:
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1159
  "a + a < 0 \<longleftrightarrow> a < 0"
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1160
proof -
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1161
  have "\<not> a + a < 0 \<longleftrightarrow> \<not> a < 0"
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1162
    by (simp add: not_less)
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1163
  then show ?thesis by simp
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1164
qed
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1165
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1166
lemma double_add_le_zero_iff_single_add_le_zero [simp]:
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1167
  "a + a \<le> 0 \<longleftrightarrow> a \<le> 0" 
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1168
proof -
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1169
  have "\<not> a + a \<le> 0 \<longleftrightarrow> \<not> a \<le> 0"
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1170
    by (simp add: not_le)
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1171
  then show ?thesis by simp
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1172
qed
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1173
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1174
lemma minus_max_eq_min:
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1175
  "- max x y = min (-x) (-y)"
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1176
  by (auto simp add: max_def min_def)
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1177
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1178
lemma minus_min_eq_max:
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1179
  "- min x y = max (-x) (-y)"
b8c8d01cc20d separate library theory for type classes combining lattices with various algebraic structures; more simp rules
haftmann
parents: 35028
diff changeset
  1180
  by (auto simp add: max_def min_def)
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1181
25267
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25230
diff changeset
  1182
end
1f745c599b5c proper reinitialisation after subclass
haftmann
parents: 25230
diff changeset
  1183
35092
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1184
class abs =
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1185
  fixes abs :: "'a \<Rightarrow> 'a"
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1186
begin
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1187
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1188
notation (xsymbols)
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1189
  abs  ("\<bar>_\<bar>")
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1190
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1191
notation (HTML output)
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1192
  abs  ("\<bar>_\<bar>")
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1193
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1194
end
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1195
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1196
class sgn =
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1197
  fixes sgn :: "'a \<Rightarrow> 'a"
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1198
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1199
class abs_if = minus + uminus + ord + zero + abs +
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1200
  assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1201
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1202
class sgn_if = minus + uminus + zero + one + ord + sgn +
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1203
  assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1204
begin
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1205
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1206
lemma sgn0 [simp]: "sgn 0 = 0"
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1207
  by (simp add:sgn_if)
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1208
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35050
diff changeset
  1209
end
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents:
diff changeset
  1210
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34973
diff changeset
  1211
class ordered_ab_group_add_abs = ordered_ab_group_add + abs +
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1212
  assumes abs_ge_zero [simp]: "\<bar>a\<bar> \<ge> 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1213
    and abs_ge_self: "a \<le> \<bar>a\<bar>"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1214
    and abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1215
    and abs_minus_cancel [simp]: "\<bar>-a\<bar> = \<bar>a\<bar>"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1216
    and abs_triangle_ineq: "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1217
begin
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1218
25307
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1219
lemma abs_minus_le_zero: "- \<bar>a\<bar> \<le> 0"
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1220
  unfolding neg_le_0_iff_le by simp
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1221
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1222
lemma abs_of_nonneg [simp]:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1223
  assumes nonneg: "0 \<le> a" shows "\<bar>a\<bar> = a"
25307
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1224
proof (rule antisym)
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1225
  from nonneg le_imp_neg_le have "- a \<le> 0" by simp
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1226
  from this nonneg have "- a \<le> a" by (rule order_trans)
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1227
  then show "\<bar>a\<bar> \<le> a" by (auto intro: abs_leI)
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1228
qed (rule abs_ge_self)
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1229
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1230
lemma abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1231
by (rule antisym)
36302
4e7f5b22dd7d more localization; tuned proofs
haftmann
parents: 36176
diff changeset
  1232
   (auto intro!: abs_ge_self abs_leI order_trans [of "- \<bar>a\<bar>" 0 "\<bar>a\<bar>"])
25307
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1233
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1234
lemma abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1235
proof -
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1236
  have "\<bar>a\<bar> = 0 \<Longrightarrow> a = 0"
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1237
  proof (rule antisym)
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1238
    assume zero: "\<bar>a\<bar> = 0"
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1239
    with abs_ge_self show "a \<le> 0" by auto
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1240
    from zero have "\<bar>-a\<bar> = 0" by simp
36302
4e7f5b22dd7d more localization; tuned proofs
haftmann
parents: 36176
diff changeset
  1241
    with abs_ge_self [of "- a"] have "- a \<le> 0" by auto
25307
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1242
    with neg_le_0_iff_le show "0 \<le> a" by auto
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1243
  qed
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1244
  then show ?thesis by auto
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1245
qed
389902f0a0c8 simplified specification of *_abs class
haftmann
parents: 25303
diff changeset
  1246
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1247
lemma abs_zero [simp]: "\<bar>0\<bar> = 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1248
by simp
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
  1249
54148
c8cc5ab4a863 killed more "no_atp"s
blanchet
parents: 54147
diff changeset
  1250
lemma abs_0_eq [simp]: "0 = \<bar>a\<bar> \<longleftrightarrow> a = 0"
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1251
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1252
  have "0 = \<bar>a\<bar> \<longleftrightarrow> \<bar>a\<bar> = 0" by (simp only: eq_ac)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1253
  thus ?thesis by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1254
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1255
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1256
lemma abs_le_zero_iff [simp]: "\<bar>a\<bar> \<le> 0 \<longleftrightarrow> a = 0" 
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1257
proof
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1258
  assume "\<bar>a\<bar> \<le> 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1259
  then have "\<bar>a\<bar> = 0" by (rule antisym) simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1260
  thus "a = 0" by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1261
next
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1262
  assume "a = 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1263
  thus "\<bar>a\<bar> \<le> 0" by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1264
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1265
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1266
lemma zero_less_abs_iff [simp]: "0 < \<bar>a\<bar> \<longleftrightarrow> a \<noteq> 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1267
by (simp add: less_le)
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1268
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1269
lemma abs_not_less_zero [simp]: "\<not> \<bar>a\<bar> < 0"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1270
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1271
  have a: "\<And>x y. x \<le> y \<Longrightarrow> \<not> y < x" by auto
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1272
  show ?thesis by (simp add: a)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1273
qed
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
  1274
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1275
lemma abs_ge_minus_self: "- a \<le> \<bar>a\<bar>"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1276
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1277
  have "- a \<le> \<bar>-a\<bar>" by (rule abs_ge_self)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1278
  then show ?thesis by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1279
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1280
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1281
lemma abs_minus_commute: 
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1282
  "\<bar>a - b\<bar> = \<bar>b - a\<bar>"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1283
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1284
  have "\<bar>a - b\<bar> = \<bar>- (a - b)\<bar>" by (simp only: abs_minus_cancel)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1285
  also have "... = \<bar>b - a\<bar>" by simp
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1286
  finally show ?thesis .
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1287
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1288
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1289
lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1290
by (rule abs_of_nonneg, rule less_imp_le)
16775
c1b87ef4a1c3 added lemmas to OrderedGroup.thy (reasoning about signs, absolute value, triangle inequalities)
avigad
parents: 16417
diff changeset
  1291
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1292
lemma abs_of_nonpos [simp]:
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1293
  assumes "a \<le> 0" shows "\<bar>a\<bar> = - a"
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1294
proof -
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1295
  let ?b = "- a"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1296
  have "- ?b \<le> 0 \<Longrightarrow> \<bar>- ?b\<bar> = - (- ?b)"
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1297
  unfolding abs_minus_cancel [of "?b"]
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1298
  unfolding neg_le_0_iff_le [of "?b"]
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1299
  unfolding minus_minus by (erule abs_of_nonneg)
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1300
  then show ?thesis using assms by auto
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1301
qed
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1302
  
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1303
lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1304
by (rule abs_of_nonpos, rule less_imp_le)
25303
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1305
0699e20feabd renamed lordered_*_* to lordered_*_add_*; further localization
haftmann
parents: 25267
diff changeset
  1306
lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29269
diff changeset
  1307
by (insert abs_ge_self, blast intro: order_trans)
25303
0699e20feabd renamed lo