src/HOL/Groups.thy
author Christian Sternagel
Thu Aug 30 15:44:03 2012 +0900 (2012-08-30)
changeset 49093 fdc301f592c4
parent 48891 c0eafbd55de3
child 49388 1ffd5a055acf
permissions -rw-r--r--
forgot to add lemmas
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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 proof
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qed (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 (advanced) {*
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  let
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    fun tr' c = (c, fn ctxt => fn T => fn ts =>
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      if not (null ts) orelse T = dummyT
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        orelse not (Config.get ctxt show_types) andalso can Term.dest_Type T
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      then raise Match
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      else
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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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  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 proof
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qed (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 proof
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qed (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 proof
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qed (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 proof
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qed (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 proof
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qed (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 proof
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qed (insert add_0, simp add: ac_simps)
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subclass (in comm_monoid_add) monoid_add proof
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qed (fact add.left_neutral add.right_neutral)+
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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 proof
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qed (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 proof
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qed (insert mult_1, simp add: ac_simps)
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subclass (in comm_monoid_mult) monoid_mult proof
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qed (fact mult.left_neutral mult.right_neutral)+
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class cancel_semigroup_add = semigroup_add +
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  assumes add_left_imp_eq: "a + b = a + c \<Longrightarrow> b = c"
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  assumes add_right_imp_eq: "b + a = c + a \<Longrightarrow> b = c"
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begin
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lemma add_left_cancel [simp]:
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  "a + b = a + c \<longleftrightarrow> b = c"
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by (blast dest: add_left_imp_eq)
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lemma add_right_cancel [simp]:
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  "b + a = c + a \<longleftrightarrow> b = c"
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by (blast dest: add_right_imp_eq)
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end
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class cancel_ab_semigroup_add = ab_semigroup_add +
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  assumes add_imp_eq: "a + b = a + c \<Longrightarrow> b = c"
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begin
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subclass cancel_semigroup_add
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proof
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  fix a b c :: 'a
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  assume "a + b = a + c" 
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  then show "b = c" by (rule add_imp_eq)
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next
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  fix a b c :: 'a
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  assume "b + a = c + a"
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  then have "a + b = a + c" by (simp only: add_commute)
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  then show "b = c" by (rule add_imp_eq)
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qed
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end
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class cancel_comm_monoid_add = cancel_ab_semigroup_add + comm_monoid_add
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subsection {* Groups *}
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class group_add = minus + uminus + monoid_add +
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  assumes left_minus [simp]: "- a + a = 0"
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  assumes diff_minus: "a - b = a + (- b)"
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begin
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lemma minus_unique:
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  assumes "a + b = 0" shows "- a = b"
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proof -
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  have "- a = - a + (a + b)" using assms by simp
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  also have "\<dots> = b" by (simp add: add_assoc [symmetric])
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  finally show ?thesis .
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qed
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lemmas equals_zero_I = minus_unique (* legacy name *)
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lemma minus_zero [simp]: "- 0 = 0"
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proof -
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  have "0 + 0 = 0" by (rule add_0_right)
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  thus "- 0 = 0" by (rule minus_unique)
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qed
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lemma minus_minus [simp]: "- (- a) = a"
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proof -
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  have "- a + a = 0" by (rule left_minus)
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  thus "- (- a) = a" by (rule minus_unique)
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qed
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lemma right_minus [simp]: "a + - a = 0"
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proof -
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  have "a + - a = - (- a) + - a" by simp
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  also have "\<dots> = 0" by (rule left_minus)
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  finally show ?thesis .
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qed
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subclass cancel_semigroup_add
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proof
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  fix a b c :: 'a
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  assume "a + b = a + c"
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  then have "- a + a + b = - a + a + c"
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    unfolding add_assoc by simp
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  then show "b = c" by simp
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next
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  fix a b c :: 'a
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  assume "b + a = c + a"
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  then have "b + a + - a = c + a  + - a" by simp
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  then show "b = c" unfolding add_assoc by simp
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qed
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lemma minus_add_cancel: "- a + (a + b) = b"
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by (simp add: add_assoc [symmetric])
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lemma add_minus_cancel: "a + (- a + b) = b"
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by (simp add: add_assoc [symmetric])
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lemma minus_add: "- (a + b) = - b + - a"
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proof -
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  have "(a + b) + (- b + - a) = 0"
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    by (simp add: add_assoc add_minus_cancel)
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  thus "- (a + b) = - b + - a"
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    by (rule minus_unique)
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qed
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lemma right_minus_eq: "a - b = 0 \<longleftrightarrow> a = b"
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proof
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  assume "a - b = 0"
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  have "a = (a - b) + b" by (simp add:diff_minus add_assoc)
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  also have "\<dots> = b" using `a - b = 0` by simp
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  finally show "a = b" .
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next
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  assume "a = b" thus "a - b = 0" by (simp add: diff_minus)
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qed
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lemma diff_self [simp]: "a - a = 0"
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by (simp add: diff_minus)
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lemma diff_0 [simp]: "0 - a = - a"
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by (simp add: diff_minus)
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lemma diff_0_right [simp]: "a - 0 = a" 
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by (simp add: diff_minus)
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lemma diff_minus_eq_add [simp]: "a - - b = a + b"
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by (simp add: diff_minus)
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lemma neg_equal_iff_equal [simp]:
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  "- a = - b \<longleftrightarrow> a = b" 
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proof 
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  assume "- a = - b"
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  hence "- (- a) = - (- b)" by simp
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  thus "a = b" by simp
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next
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  assume "a = b"
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  thus "- a = - b" by simp
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qed
obua@14738
   367
haftmann@25062
   368
lemma neg_equal_0_iff_equal [simp]:
haftmann@25062
   369
  "- a = 0 \<longleftrightarrow> a = 0"
nipkow@29667
   370
by (subst neg_equal_iff_equal [symmetric], simp)
obua@14738
   371
haftmann@25062
   372
lemma neg_0_equal_iff_equal [simp]:
haftmann@25062
   373
  "0 = - a \<longleftrightarrow> 0 = a"
nipkow@29667
   374
by (subst neg_equal_iff_equal [symmetric], simp)
obua@14738
   375
obua@14738
   376
text{*The next two equations can make the simplifier loop!*}
obua@14738
   377
haftmann@25062
   378
lemma equation_minus_iff:
haftmann@25062
   379
  "a = - b \<longleftrightarrow> b = - a"
obua@14738
   380
proof -
haftmann@25062
   381
  have "- (- a) = - b \<longleftrightarrow> - a = b" by (rule neg_equal_iff_equal)
haftmann@25062
   382
  thus ?thesis by (simp add: eq_commute)
haftmann@25062
   383
qed
haftmann@25062
   384
haftmann@25062
   385
lemma minus_equation_iff:
haftmann@25062
   386
  "- a = b \<longleftrightarrow> - b = a"
haftmann@25062
   387
proof -
haftmann@25062
   388
  have "- a = - (- b) \<longleftrightarrow> a = -b" by (rule neg_equal_iff_equal)
obua@14738
   389
  thus ?thesis by (simp add: eq_commute)
obua@14738
   390
qed
obua@14738
   391
huffman@28130
   392
lemma diff_add_cancel: "a - b + b = a"
nipkow@29667
   393
by (simp add: diff_minus add_assoc)
huffman@28130
   394
huffman@28130
   395
lemma add_diff_cancel: "a + b - b = a"
nipkow@29667
   396
by (simp add: diff_minus add_assoc)
nipkow@29667
   397
haftmann@36348
   398
declare diff_minus[symmetric, algebra_simps, field_simps]
huffman@28130
   399
huffman@29914
   400
lemma eq_neg_iff_add_eq_0: "a = - b \<longleftrightarrow> a + b = 0"
huffman@29914
   401
proof
huffman@29914
   402
  assume "a = - b" then show "a + b = 0" by simp
huffman@29914
   403
next
huffman@29914
   404
  assume "a + b = 0"
huffman@29914
   405
  moreover have "a + (b + - b) = (a + b) + - b"
huffman@29914
   406
    by (simp only: add_assoc)
huffman@29914
   407
  ultimately show "a = - b" by simp
huffman@29914
   408
qed
huffman@29914
   409
huffman@44348
   410
lemma add_eq_0_iff: "x + y = 0 \<longleftrightarrow> y = - x"
huffman@44348
   411
  unfolding eq_neg_iff_add_eq_0 [symmetric]
huffman@44348
   412
  by (rule equation_minus_iff)
huffman@44348
   413
huffman@45548
   414
lemma minus_diff_eq [simp]: "- (a - b) = b - a"
huffman@45548
   415
  by (simp add: diff_minus minus_add)
huffman@45548
   416
huffman@45548
   417
lemma add_diff_eq[algebra_simps, field_simps]: "a + (b - c) = (a + b) - c"
huffman@45548
   418
  by (simp add: diff_minus add_assoc)
huffman@45548
   419
huffman@45548
   420
lemma diff_eq_eq[algebra_simps, field_simps]: "a - b = c \<longleftrightarrow> a = c + b"
huffman@45548
   421
  by (auto simp add: diff_minus add_assoc)
huffman@45548
   422
huffman@45548
   423
lemma eq_diff_eq[algebra_simps, field_simps]: "a = c - b \<longleftrightarrow> a + b = c"
huffman@45548
   424
  by (auto simp add: diff_minus add_assoc)
huffman@45548
   425
huffman@45548
   426
lemma diff_diff_eq2[algebra_simps, field_simps]: "a - (b - c) = (a + c) - b"
huffman@45548
   427
  by (simp add: diff_minus minus_add add_assoc)
huffman@45548
   428
huffman@45548
   429
lemma eq_iff_diff_eq_0: "a = b \<longleftrightarrow> a - b = 0"
huffman@45548
   430
  by (fact right_minus_eq [symmetric])
huffman@45548
   431
huffman@45548
   432
lemma diff_eq_diff_eq:
huffman@45548
   433
  "a - b = c - d \<Longrightarrow> a = b \<longleftrightarrow> c = d"
huffman@45548
   434
  by (simp add: eq_iff_diff_eq_0 [of a b] eq_iff_diff_eq_0 [of c d])
huffman@45548
   435
haftmann@25062
   436
end
haftmann@25062
   437
haftmann@25762
   438
class ab_group_add = minus + uminus + comm_monoid_add +
haftmann@25062
   439
  assumes ab_left_minus: "- a + a = 0"
haftmann@25062
   440
  assumes ab_diff_minus: "a - b = a + (- b)"
haftmann@25267
   441
begin
haftmann@25062
   442
haftmann@25267
   443
subclass group_add
haftmann@28823
   444
  proof qed (simp_all add: ab_left_minus ab_diff_minus)
haftmann@25062
   445
huffman@29904
   446
subclass cancel_comm_monoid_add
haftmann@28823
   447
proof
haftmann@25062
   448
  fix a b c :: 'a
haftmann@25062
   449
  assume "a + b = a + c"
haftmann@25062
   450
  then have "- a + a + b = - a + a + c"
haftmann@25062
   451
    unfolding add_assoc by simp
haftmann@25062
   452
  then show "b = c" by simp
haftmann@25062
   453
qed
haftmann@25062
   454
haftmann@36348
   455
lemma uminus_add_conv_diff[algebra_simps, field_simps]:
haftmann@25062
   456
  "- a + b = b - a"
nipkow@29667
   457
by (simp add:diff_minus add_commute)
haftmann@25062
   458
haftmann@25062
   459
lemma minus_add_distrib [simp]:
haftmann@25062
   460
  "- (a + b) = - a + - b"
huffman@34146
   461
by (rule minus_unique) (simp add: add_ac)
haftmann@25062
   462
haftmann@36348
   463
lemma diff_add_eq[algebra_simps, field_simps]: "(a - b) + c = (a + c) - b"
nipkow@29667
   464
by (simp add: diff_minus add_ac)
haftmann@25077
   465
haftmann@36348
   466
lemma diff_diff_eq[algebra_simps, field_simps]: "(a - b) - c = a - (b + c)"
nipkow@29667
   467
by (simp add: diff_minus add_ac)
haftmann@25077
   468
huffman@35216
   469
(* FIXME: duplicates right_minus_eq from class group_add *)
huffman@35216
   470
(* but only this one is declared as a simp rule. *)
blanchet@35828
   471
lemma diff_eq_0_iff_eq [simp, no_atp]: "a - b = 0 \<longleftrightarrow> a = b"
huffman@44348
   472
  by (rule right_minus_eq)
huffman@30629
   473
huffman@48556
   474
lemma add_diff_cancel_left: "(c + a) - (c + b) = a - b"
huffman@48556
   475
  by (simp add: diff_minus add_ac)
huffman@48556
   476
haftmann@25062
   477
end
obua@14738
   478
haftmann@37884
   479
obua@14738
   480
subsection {* (Partially) Ordered Groups *} 
obua@14738
   481
haftmann@35301
   482
text {*
haftmann@35301
   483
  The theory of partially ordered groups is taken from the books:
haftmann@35301
   484
  \begin{itemize}
haftmann@35301
   485
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
haftmann@35301
   486
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
haftmann@35301
   487
  \end{itemize}
haftmann@35301
   488
  Most of the used notions can also be looked up in 
haftmann@35301
   489
  \begin{itemize}
haftmann@35301
   490
  \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
haftmann@35301
   491
  \item \emph{Algebra I} by van der Waerden, Springer.
haftmann@35301
   492
  \end{itemize}
haftmann@35301
   493
*}
haftmann@35301
   494
haftmann@35028
   495
class ordered_ab_semigroup_add = order + ab_semigroup_add +
haftmann@25062
   496
  assumes add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b"
haftmann@25062
   497
begin
haftmann@24380
   498
haftmann@25062
   499
lemma add_right_mono:
haftmann@25062
   500
  "a \<le> b \<Longrightarrow> a + c \<le> b + c"
nipkow@29667
   501
by (simp add: add_commute [of _ c] add_left_mono)
obua@14738
   502
obua@14738
   503
text {* non-strict, in both arguments *}
obua@14738
   504
lemma add_mono:
haftmann@25062
   505
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d"
obua@14738
   506
  apply (erule add_right_mono [THEN order_trans])
obua@14738
   507
  apply (simp add: add_commute add_left_mono)
obua@14738
   508
  done
obua@14738
   509
haftmann@25062
   510
end
haftmann@25062
   511
haftmann@35028
   512
class ordered_cancel_ab_semigroup_add =
haftmann@35028
   513
  ordered_ab_semigroup_add + cancel_ab_semigroup_add
haftmann@25062
   514
begin
haftmann@25062
   515
obua@14738
   516
lemma add_strict_left_mono:
haftmann@25062
   517
  "a < b \<Longrightarrow> c + a < c + b"
nipkow@29667
   518
by (auto simp add: less_le add_left_mono)
obua@14738
   519
obua@14738
   520
lemma add_strict_right_mono:
haftmann@25062
   521
  "a < b \<Longrightarrow> a + c < b + c"
nipkow@29667
   522
by (simp add: add_commute [of _ c] add_strict_left_mono)
obua@14738
   523
obua@14738
   524
text{*Strict monotonicity in both arguments*}
haftmann@25062
   525
lemma add_strict_mono:
haftmann@25062
   526
  "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
haftmann@25062
   527
apply (erule add_strict_right_mono [THEN less_trans])
obua@14738
   528
apply (erule add_strict_left_mono)
obua@14738
   529
done
obua@14738
   530
obua@14738
   531
lemma add_less_le_mono:
haftmann@25062
   532
  "a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c < b + d"
haftmann@25062
   533
apply (erule add_strict_right_mono [THEN less_le_trans])
haftmann@25062
   534
apply (erule add_left_mono)
obua@14738
   535
done
obua@14738
   536
obua@14738
   537
lemma add_le_less_mono:
haftmann@25062
   538
  "a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
haftmann@25062
   539
apply (erule add_right_mono [THEN le_less_trans])
obua@14738
   540
apply (erule add_strict_left_mono) 
obua@14738
   541
done
obua@14738
   542
haftmann@25062
   543
end
haftmann@25062
   544
haftmann@35028
   545
class ordered_ab_semigroup_add_imp_le =
haftmann@35028
   546
  ordered_cancel_ab_semigroup_add +
haftmann@25062
   547
  assumes add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b"
haftmann@25062
   548
begin
haftmann@25062
   549
obua@14738
   550
lemma add_less_imp_less_left:
nipkow@29667
   551
  assumes less: "c + a < c + b" shows "a < b"
obua@14738
   552
proof -
obua@14738
   553
  from less have le: "c + a <= c + b" by (simp add: order_le_less)
obua@14738
   554
  have "a <= b" 
obua@14738
   555
    apply (insert le)
obua@14738
   556
    apply (drule add_le_imp_le_left)
obua@14738
   557
    by (insert le, drule add_le_imp_le_left, assumption)
obua@14738
   558
  moreover have "a \<noteq> b"
obua@14738
   559
  proof (rule ccontr)
obua@14738
   560
    assume "~(a \<noteq> b)"
obua@14738
   561
    then have "a = b" by simp
obua@14738
   562
    then have "c + a = c + b" by simp
obua@14738
   563
    with less show "False"by simp
obua@14738
   564
  qed
obua@14738
   565
  ultimately show "a < b" by (simp add: order_le_less)
obua@14738
   566
qed
obua@14738
   567
obua@14738
   568
lemma add_less_imp_less_right:
haftmann@25062
   569
  "a + c < b + c \<Longrightarrow> a < b"
obua@14738
   570
apply (rule add_less_imp_less_left [of c])
obua@14738
   571
apply (simp add: add_commute)  
obua@14738
   572
done
obua@14738
   573
obua@14738
   574
lemma add_less_cancel_left [simp]:
haftmann@25062
   575
  "c + a < c + b \<longleftrightarrow> a < b"
nipkow@29667
   576
by (blast intro: add_less_imp_less_left add_strict_left_mono) 
obua@14738
   577
obua@14738
   578
lemma add_less_cancel_right [simp]:
haftmann@25062
   579
  "a + c < b + c \<longleftrightarrow> a < b"
nipkow@29667
   580
by (blast intro: add_less_imp_less_right add_strict_right_mono)
obua@14738
   581
obua@14738
   582
lemma add_le_cancel_left [simp]:
haftmann@25062
   583
  "c + a \<le> c + b \<longleftrightarrow> a \<le> b"
nipkow@29667
   584
by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono) 
obua@14738
   585
obua@14738
   586
lemma add_le_cancel_right [simp]:
haftmann@25062
   587
  "a + c \<le> b + c \<longleftrightarrow> a \<le> b"
nipkow@29667
   588
by (simp add: add_commute [of a c] add_commute [of b c])
obua@14738
   589
obua@14738
   590
lemma add_le_imp_le_right:
haftmann@25062
   591
  "a + c \<le> b + c \<Longrightarrow> a \<le> b"
nipkow@29667
   592
by simp
haftmann@25062
   593
haftmann@25077
   594
lemma max_add_distrib_left:
haftmann@25077
   595
  "max x y + z = max (x + z) (y + z)"
haftmann@25077
   596
  unfolding max_def by auto
haftmann@25077
   597
haftmann@25077
   598
lemma min_add_distrib_left:
haftmann@25077
   599
  "min x y + z = min (x + z) (y + z)"
haftmann@25077
   600
  unfolding min_def by auto
haftmann@25077
   601
huffman@44848
   602
lemma max_add_distrib_right:
huffman@44848
   603
  "x + max y z = max (x + y) (x + z)"
huffman@44848
   604
  unfolding max_def by auto
huffman@44848
   605
huffman@44848
   606
lemma min_add_distrib_right:
huffman@44848
   607
  "x + min y z = min (x + y) (x + z)"
huffman@44848
   608
  unfolding min_def by auto
huffman@44848
   609
haftmann@25062
   610
end
haftmann@25062
   611
haftmann@25303
   612
subsection {* Support for reasoning about signs *}
haftmann@25303
   613
haftmann@35028
   614
class ordered_comm_monoid_add =
haftmann@35028
   615
  ordered_cancel_ab_semigroup_add + comm_monoid_add
haftmann@25303
   616
begin
haftmann@25303
   617
haftmann@25303
   618
lemma add_pos_nonneg:
nipkow@29667
   619
  assumes "0 < a" and "0 \<le> b" shows "0 < a + b"
haftmann@25303
   620
proof -
haftmann@25303
   621
  have "0 + 0 < a + b" 
haftmann@25303
   622
    using assms by (rule add_less_le_mono)
haftmann@25303
   623
  then show ?thesis by simp
haftmann@25303
   624
qed
haftmann@25303
   625
haftmann@25303
   626
lemma add_pos_pos:
nipkow@29667
   627
  assumes "0 < a" and "0 < b" shows "0 < a + b"
nipkow@29667
   628
by (rule add_pos_nonneg) (insert assms, auto)
haftmann@25303
   629
haftmann@25303
   630
lemma add_nonneg_pos:
nipkow@29667
   631
  assumes "0 \<le> a" and "0 < b" shows "0 < a + b"
haftmann@25303
   632
proof -
haftmann@25303
   633
  have "0 + 0 < a + b" 
haftmann@25303
   634
    using assms by (rule add_le_less_mono)
haftmann@25303
   635
  then show ?thesis by simp
haftmann@25303
   636
qed
haftmann@25303
   637
huffman@36977
   638
lemma add_nonneg_nonneg [simp]:
nipkow@29667
   639
  assumes "0 \<le> a" and "0 \<le> b" shows "0 \<le> a + b"
haftmann@25303
   640
proof -
haftmann@25303
   641
  have "0 + 0 \<le> a + b" 
haftmann@25303
   642
    using assms by (rule add_mono)
haftmann@25303
   643
  then show ?thesis by simp
haftmann@25303
   644
qed
haftmann@25303
   645
huffman@30691
   646
lemma add_neg_nonpos:
nipkow@29667
   647
  assumes "a < 0" and "b \<le> 0" shows "a + b < 0"
haftmann@25303
   648
proof -
haftmann@25303
   649
  have "a + b < 0 + 0"
haftmann@25303
   650
    using assms by (rule add_less_le_mono)
haftmann@25303
   651
  then show ?thesis by simp
haftmann@25303
   652
qed
haftmann@25303
   653
haftmann@25303
   654
lemma add_neg_neg: 
nipkow@29667
   655
  assumes "a < 0" and "b < 0" shows "a + b < 0"
nipkow@29667
   656
by (rule add_neg_nonpos) (insert assms, auto)
haftmann@25303
   657
haftmann@25303
   658
lemma add_nonpos_neg:
nipkow@29667
   659
  assumes "a \<le> 0" and "b < 0" shows "a + b < 0"
haftmann@25303
   660
proof -
haftmann@25303
   661
  have "a + b < 0 + 0"
haftmann@25303
   662
    using assms by (rule add_le_less_mono)
haftmann@25303
   663
  then show ?thesis by simp
haftmann@25303
   664
qed
haftmann@25303
   665
haftmann@25303
   666
lemma add_nonpos_nonpos:
nipkow@29667
   667
  assumes "a \<le> 0" and "b \<le> 0" shows "a + b \<le> 0"
haftmann@25303
   668
proof -
haftmann@25303
   669
  have "a + b \<le> 0 + 0"
haftmann@25303
   670
    using assms by (rule add_mono)
haftmann@25303
   671
  then show ?thesis by simp
haftmann@25303
   672
qed
haftmann@25303
   673
huffman@30691
   674
lemmas add_sign_intros =
huffman@30691
   675
  add_pos_nonneg add_pos_pos add_nonneg_pos add_nonneg_nonneg
huffman@30691
   676
  add_neg_nonpos add_neg_neg add_nonpos_neg add_nonpos_nonpos
huffman@30691
   677
huffman@29886
   678
lemma add_nonneg_eq_0_iff:
huffman@29886
   679
  assumes x: "0 \<le> x" and y: "0 \<le> y"
huffman@29886
   680
  shows "x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@29886
   681
proof (intro iffI conjI)
huffman@29886
   682
  have "x = x + 0" by simp
huffman@29886
   683
  also have "x + 0 \<le> x + y" using y by (rule add_left_mono)
huffman@29886
   684
  also assume "x + y = 0"
huffman@29886
   685
  also have "0 \<le> x" using x .
huffman@29886
   686
  finally show "x = 0" .
huffman@29886
   687
next
huffman@29886
   688
  have "y = 0 + y" by simp
huffman@29886
   689
  also have "0 + y \<le> x + y" using x by (rule add_right_mono)
huffman@29886
   690
  also assume "x + y = 0"
huffman@29886
   691
  also have "0 \<le> y" using y .
huffman@29886
   692
  finally show "y = 0" .
huffman@29886
   693
next
huffman@29886
   694
  assume "x = 0 \<and> y = 0"
huffman@29886
   695
  then show "x + y = 0" by simp
huffman@29886
   696
qed
huffman@29886
   697
haftmann@25303
   698
end
haftmann@25303
   699
haftmann@35028
   700
class ordered_ab_group_add =
haftmann@35028
   701
  ab_group_add + ordered_ab_semigroup_add
haftmann@25062
   702
begin
haftmann@25062
   703
haftmann@35028
   704
subclass ordered_cancel_ab_semigroup_add ..
haftmann@25062
   705
haftmann@35028
   706
subclass ordered_ab_semigroup_add_imp_le
haftmann@28823
   707
proof
haftmann@25062
   708
  fix a b c :: 'a
haftmann@25062
   709
  assume "c + a \<le> c + b"
haftmann@25062
   710
  hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono)
haftmann@25062
   711
  hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc)
haftmann@25062
   712
  thus "a \<le> b" by simp
haftmann@25062
   713
qed
haftmann@25062
   714
haftmann@35028
   715
subclass ordered_comm_monoid_add ..
haftmann@25303
   716
haftmann@25077
   717
lemma max_diff_distrib_left:
haftmann@25077
   718
  shows "max x y - z = max (x - z) (y - z)"
nipkow@29667
   719
by (simp add: diff_minus, rule max_add_distrib_left) 
haftmann@25077
   720
haftmann@25077
   721
lemma min_diff_distrib_left:
haftmann@25077
   722
  shows "min x y - z = min (x - z) (y - z)"
nipkow@29667
   723
by (simp add: diff_minus, rule min_add_distrib_left) 
haftmann@25077
   724
haftmann@25077
   725
lemma le_imp_neg_le:
nipkow@29667
   726
  assumes "a \<le> b" shows "-b \<le> -a"
haftmann@25077
   727
proof -
nipkow@29667
   728
  have "-a+a \<le> -a+b" using `a \<le> b` by (rule add_left_mono) 
nipkow@29667
   729
  hence "0 \<le> -a+b" by simp
nipkow@29667
   730
  hence "0 + (-b) \<le> (-a + b) + (-b)" by (rule add_right_mono) 
nipkow@29667
   731
  thus ?thesis by (simp add: add_assoc)
haftmann@25077
   732
qed
haftmann@25077
   733
haftmann@25077
   734
lemma neg_le_iff_le [simp]: "- b \<le> - a \<longleftrightarrow> a \<le> b"
haftmann@25077
   735
proof 
haftmann@25077
   736
  assume "- b \<le> - a"
nipkow@29667
   737
  hence "- (- a) \<le> - (- b)" by (rule le_imp_neg_le)
haftmann@25077
   738
  thus "a\<le>b" by simp
haftmann@25077
   739
next
haftmann@25077
   740
  assume "a\<le>b"
haftmann@25077
   741
  thus "-b \<le> -a" by (rule le_imp_neg_le)
haftmann@25077
   742
qed
haftmann@25077
   743
haftmann@25077
   744
lemma neg_le_0_iff_le [simp]: "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"
nipkow@29667
   745
by (subst neg_le_iff_le [symmetric], simp)
haftmann@25077
   746
haftmann@25077
   747
lemma neg_0_le_iff_le [simp]: "0 \<le> - a \<longleftrightarrow> a \<le> 0"
nipkow@29667
   748
by (subst neg_le_iff_le [symmetric], simp)
haftmann@25077
   749
haftmann@25077
   750
lemma neg_less_iff_less [simp]: "- b < - a \<longleftrightarrow> a < b"
nipkow@29667
   751
by (force simp add: less_le) 
haftmann@25077
   752
haftmann@25077
   753
lemma neg_less_0_iff_less [simp]: "- a < 0 \<longleftrightarrow> 0 < a"
nipkow@29667
   754
by (subst neg_less_iff_less [symmetric], simp)
haftmann@25077
   755
haftmann@25077
   756
lemma neg_0_less_iff_less [simp]: "0 < - a \<longleftrightarrow> a < 0"
nipkow@29667
   757
by (subst neg_less_iff_less [symmetric], simp)
haftmann@25077
   758
haftmann@25077
   759
text{*The next several equations can make the simplifier loop!*}
haftmann@25077
   760
haftmann@25077
   761
lemma less_minus_iff: "a < - b \<longleftrightarrow> b < - a"
haftmann@25077
   762
proof -
haftmann@25077
   763
  have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
haftmann@25077
   764
  thus ?thesis by simp
haftmann@25077
   765
qed
haftmann@25077
   766
haftmann@25077
   767
lemma minus_less_iff: "- a < b \<longleftrightarrow> - b < a"
haftmann@25077
   768
proof -
haftmann@25077
   769
  have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
haftmann@25077
   770
  thus ?thesis by simp
haftmann@25077
   771
qed
haftmann@25077
   772
haftmann@25077
   773
lemma le_minus_iff: "a \<le> - b \<longleftrightarrow> b \<le> - a"
haftmann@25077
   774
proof -
haftmann@25077
   775
  have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff)
haftmann@25077
   776
  have "(- (- a) <= -b) = (b <= - a)" 
haftmann@25077
   777
    apply (auto simp only: le_less)
haftmann@25077
   778
    apply (drule mm)
haftmann@25077
   779
    apply (simp_all)
haftmann@25077
   780
    apply (drule mm[simplified], assumption)
haftmann@25077
   781
    done
haftmann@25077
   782
  then show ?thesis by simp
haftmann@25077
   783
qed
haftmann@25077
   784
haftmann@25077
   785
lemma minus_le_iff: "- a \<le> b \<longleftrightarrow> - b \<le> a"
nipkow@29667
   786
by (auto simp add: le_less minus_less_iff)
haftmann@25077
   787
haftmann@37884
   788
lemma diff_less_0_iff_less [simp, no_atp]:
haftmann@37884
   789
  "a - b < 0 \<longleftrightarrow> a < b"
haftmann@25077
   790
proof -
haftmann@37884
   791
  have "a - b < 0 \<longleftrightarrow> a + (- b) < b + (- b)" by (simp add: diff_minus)
haftmann@37884
   792
  also have "... \<longleftrightarrow> a < b" by (simp only: add_less_cancel_right)
haftmann@25077
   793
  finally show ?thesis .
haftmann@25077
   794
qed
haftmann@25077
   795
haftmann@37884
   796
lemmas less_iff_diff_less_0 = diff_less_0_iff_less [symmetric]
haftmann@37884
   797
haftmann@36348
   798
lemma diff_less_eq[algebra_simps, field_simps]: "a - b < c \<longleftrightarrow> a < c + b"
haftmann@25077
   799
apply (subst less_iff_diff_less_0 [of a])
haftmann@25077
   800
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
haftmann@25077
   801
apply (simp add: diff_minus add_ac)
haftmann@25077
   802
done
haftmann@25077
   803
haftmann@36348
   804
lemma less_diff_eq[algebra_simps, field_simps]: "a < c - b \<longleftrightarrow> a + b < c"
haftmann@36302
   805
apply (subst less_iff_diff_less_0 [of "a + b"])
haftmann@25077
   806
apply (subst less_iff_diff_less_0 [of a])
haftmann@25077
   807
apply (simp add: diff_minus add_ac)
haftmann@25077
   808
done
haftmann@25077
   809
haftmann@36348
   810
lemma diff_le_eq[algebra_simps, field_simps]: "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
nipkow@29667
   811
by (auto simp add: le_less diff_less_eq diff_add_cancel add_diff_cancel)
haftmann@25077
   812
haftmann@36348
   813
lemma le_diff_eq[algebra_simps, field_simps]: "a \<le> c - b \<longleftrightarrow> a + b \<le> c"
nipkow@29667
   814
by (auto simp add: le_less less_diff_eq diff_add_cancel add_diff_cancel)
haftmann@25077
   815
haftmann@37884
   816
lemma diff_le_0_iff_le [simp, no_atp]:
haftmann@37884
   817
  "a - b \<le> 0 \<longleftrightarrow> a \<le> b"
haftmann@37884
   818
  by (simp add: algebra_simps)
haftmann@37884
   819
haftmann@37884
   820
lemmas le_iff_diff_le_0 = diff_le_0_iff_le [symmetric]
haftmann@37884
   821
haftmann@37884
   822
lemma diff_eq_diff_less:
haftmann@37884
   823
  "a - b = c - d \<Longrightarrow> a < b \<longleftrightarrow> c < d"
haftmann@37884
   824
  by (auto simp only: less_iff_diff_less_0 [of a b] less_iff_diff_less_0 [of c d])
haftmann@37884
   825
haftmann@37889
   826
lemma diff_eq_diff_less_eq:
haftmann@37889
   827
  "a - b = c - d \<Longrightarrow> a \<le> b \<longleftrightarrow> c \<le> d"
haftmann@37889
   828
  by (auto simp only: le_iff_diff_le_0 [of a b] le_iff_diff_le_0 [of c d])
haftmann@25077
   829
haftmann@25077
   830
end
haftmann@25077
   831
wenzelm@48891
   832
ML_file "Tools/group_cancel.ML"
huffman@48556
   833
huffman@48556
   834
simproc_setup group_cancel_add ("a + b::'a::ab_group_add") =
huffman@48556
   835
  {* fn phi => fn ss => try Group_Cancel.cancel_add_conv *}
huffman@48556
   836
huffman@48556
   837
simproc_setup group_cancel_diff ("a - b::'a::ab_group_add") =
huffman@48556
   838
  {* fn phi => fn ss => try Group_Cancel.cancel_diff_conv *}
haftmann@37884
   839
huffman@48556
   840
simproc_setup group_cancel_eq ("a = (b::'a::ab_group_add)") =
huffman@48556
   841
  {* fn phi => fn ss => try Group_Cancel.cancel_eq_conv *}
haftmann@37889
   842
huffman@48556
   843
simproc_setup group_cancel_le ("a \<le> (b::'a::ordered_ab_group_add)") =
huffman@48556
   844
  {* fn phi => fn ss => try Group_Cancel.cancel_le_conv *}
huffman@48556
   845
huffman@48556
   846
simproc_setup group_cancel_less ("a < (b::'a::ordered_ab_group_add)") =
huffman@48556
   847
  {* fn phi => fn ss => try Group_Cancel.cancel_less_conv *}
haftmann@37884
   848
haftmann@35028
   849
class linordered_ab_semigroup_add =
haftmann@35028
   850
  linorder + ordered_ab_semigroup_add
haftmann@25062
   851
haftmann@35028
   852
class linordered_cancel_ab_semigroup_add =
haftmann@35028
   853
  linorder + ordered_cancel_ab_semigroup_add
haftmann@25267
   854
begin
haftmann@25062
   855
haftmann@35028
   856
subclass linordered_ab_semigroup_add ..
haftmann@25062
   857
haftmann@35028
   858
subclass ordered_ab_semigroup_add_imp_le
haftmann@28823
   859
proof
haftmann@25062
   860
  fix a b c :: 'a
haftmann@25062
   861
  assume le: "c + a <= c + b"  
haftmann@25062
   862
  show "a <= b"
haftmann@25062
   863
  proof (rule ccontr)
haftmann@25062
   864
    assume w: "~ a \<le> b"
haftmann@25062
   865
    hence "b <= a" by (simp add: linorder_not_le)
haftmann@25062
   866
    hence le2: "c + b <= c + a" by (rule add_left_mono)
haftmann@25062
   867
    have "a = b" 
haftmann@25062
   868
      apply (insert le)
haftmann@25062
   869
      apply (insert le2)
haftmann@25062
   870
      apply (drule antisym, simp_all)
haftmann@25062
   871
      done
haftmann@25062
   872
    with w show False 
haftmann@25062
   873
      by (simp add: linorder_not_le [symmetric])
haftmann@25062
   874
  qed
haftmann@25062
   875
qed
haftmann@25062
   876
haftmann@25267
   877
end
haftmann@25267
   878
haftmann@35028
   879
class linordered_ab_group_add = linorder + ordered_ab_group_add
haftmann@25267
   880
begin
haftmann@25230
   881
haftmann@35028
   882
subclass linordered_cancel_ab_semigroup_add ..
haftmann@25230
   883
haftmann@35036
   884
lemma neg_less_eq_nonneg [simp]:
haftmann@25303
   885
  "- a \<le> a \<longleftrightarrow> 0 \<le> a"
haftmann@25303
   886
proof
haftmann@25303
   887
  assume A: "- a \<le> a" show "0 \<le> a"
haftmann@25303
   888
  proof (rule classical)
haftmann@25303
   889
    assume "\<not> 0 \<le> a"
haftmann@25303
   890
    then have "a < 0" by auto
haftmann@25303
   891
    with A have "- a < 0" by (rule le_less_trans)
haftmann@25303
   892
    then show ?thesis by auto
haftmann@25303
   893
  qed
haftmann@25303
   894
next
haftmann@25303
   895
  assume A: "0 \<le> a" show "- a \<le> a"
haftmann@25303
   896
  proof (rule order_trans)
haftmann@25303
   897
    show "- a \<le> 0" using A by (simp add: minus_le_iff)
haftmann@25303
   898
  next
haftmann@25303
   899
    show "0 \<le> a" using A .
haftmann@25303
   900
  qed
haftmann@25303
   901
qed
haftmann@35036
   902
haftmann@35036
   903
lemma neg_less_nonneg [simp]:
haftmann@35036
   904
  "- a < a \<longleftrightarrow> 0 < a"
haftmann@35036
   905
proof
haftmann@35036
   906
  assume A: "- a < a" show "0 < a"
haftmann@35036
   907
  proof (rule classical)
haftmann@35036
   908
    assume "\<not> 0 < a"
haftmann@35036
   909
    then have "a \<le> 0" by auto
haftmann@35036
   910
    with A have "- a < 0" by (rule less_le_trans)
haftmann@35036
   911
    then show ?thesis by auto
haftmann@35036
   912
  qed
haftmann@35036
   913
next
haftmann@35036
   914
  assume A: "0 < a" show "- a < a"
haftmann@35036
   915
  proof (rule less_trans)
haftmann@35036
   916
    show "- a < 0" using A by (simp add: minus_le_iff)
haftmann@35036
   917
  next
haftmann@35036
   918
    show "0 < a" using A .
haftmann@35036
   919
  qed
haftmann@35036
   920
qed
haftmann@35036
   921
haftmann@35036
   922
lemma less_eq_neg_nonpos [simp]:
haftmann@25303
   923
  "a \<le> - a \<longleftrightarrow> a \<le> 0"
haftmann@25303
   924
proof
haftmann@25303
   925
  assume A: "a \<le> - a" show "a \<le> 0"
haftmann@25303
   926
  proof (rule classical)
haftmann@25303
   927
    assume "\<not> a \<le> 0"
haftmann@25303
   928
    then have "0 < a" by auto
haftmann@25303
   929
    then have "0 < - a" using A by (rule less_le_trans)
haftmann@25303
   930
    then show ?thesis by auto
haftmann@25303
   931
  qed
haftmann@25303
   932
next
haftmann@25303
   933
  assume A: "a \<le> 0" show "a \<le> - a"
haftmann@25303
   934
  proof (rule order_trans)
haftmann@25303
   935
    show "0 \<le> - a" using A by (simp add: minus_le_iff)
haftmann@25303
   936
  next
haftmann@25303
   937
    show "a \<le> 0" using A .
haftmann@25303
   938
  qed
haftmann@25303
   939
qed
haftmann@25303
   940
haftmann@35036
   941
lemma equal_neg_zero [simp]:
haftmann@25303
   942
  "a = - a \<longleftrightarrow> a = 0"
haftmann@25303
   943
proof
haftmann@25303
   944
  assume "a = 0" then show "a = - a" by simp
haftmann@25303
   945
next
haftmann@25303
   946
  assume A: "a = - a" show "a = 0"
haftmann@25303
   947
  proof (cases "0 \<le> a")
haftmann@25303
   948
    case True with A have "0 \<le> - a" by auto
haftmann@25303
   949
    with le_minus_iff have "a \<le> 0" by simp
haftmann@25303
   950
    with True show ?thesis by (auto intro: order_trans)
haftmann@25303
   951
  next
haftmann@25303
   952
    case False then have B: "a \<le> 0" by auto
haftmann@25303
   953
    with A have "- a \<le> 0" by auto
haftmann@25303
   954
    with B show ?thesis by (auto intro: order_trans)
haftmann@25303
   955
  qed
haftmann@25303
   956
qed
haftmann@25303
   957
haftmann@35036
   958
lemma neg_equal_zero [simp]:
haftmann@25303
   959
  "- a = a \<longleftrightarrow> a = 0"
haftmann@35036
   960
  by (auto dest: sym)
haftmann@35036
   961
haftmann@35036
   962
lemma double_zero [simp]:
haftmann@35036
   963
  "a + a = 0 \<longleftrightarrow> a = 0"
haftmann@35036
   964
proof
haftmann@35036
   965
  assume assm: "a + a = 0"
haftmann@35036
   966
  then have a: "- a = a" by (rule minus_unique)
huffman@35216
   967
  then show "a = 0" by (simp only: neg_equal_zero)
haftmann@35036
   968
qed simp
haftmann@35036
   969
haftmann@35036
   970
lemma double_zero_sym [simp]:
haftmann@35036
   971
  "0 = a + a \<longleftrightarrow> a = 0"
haftmann@35036
   972
  by (rule, drule sym) simp_all
haftmann@35036
   973
haftmann@35036
   974
lemma zero_less_double_add_iff_zero_less_single_add [simp]:
haftmann@35036
   975
  "0 < a + a \<longleftrightarrow> 0 < a"
haftmann@35036
   976
proof
haftmann@35036
   977
  assume "0 < a + a"
haftmann@35036
   978
  then have "0 - a < a" by (simp only: diff_less_eq)
haftmann@35036
   979
  then have "- a < a" by simp
huffman@35216
   980
  then show "0 < a" by (simp only: neg_less_nonneg)
haftmann@35036
   981
next
haftmann@35036
   982
  assume "0 < a"
haftmann@35036
   983
  with this have "0 + 0 < a + a"
haftmann@35036
   984
    by (rule add_strict_mono)
haftmann@35036
   985
  then show "0 < a + a" by simp
haftmann@35036
   986
qed
haftmann@35036
   987
haftmann@35036
   988
lemma zero_le_double_add_iff_zero_le_single_add [simp]:
haftmann@35036
   989
  "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
haftmann@35036
   990
  by (auto simp add: le_less)
haftmann@35036
   991
haftmann@35036
   992
lemma double_add_less_zero_iff_single_add_less_zero [simp]:
haftmann@35036
   993
  "a + a < 0 \<longleftrightarrow> a < 0"
haftmann@35036
   994
proof -
haftmann@35036
   995
  have "\<not> a + a < 0 \<longleftrightarrow> \<not> a < 0"
haftmann@35036
   996
    by (simp add: not_less)
haftmann@35036
   997
  then show ?thesis by simp
haftmann@35036
   998
qed
haftmann@35036
   999
haftmann@35036
  1000
lemma double_add_le_zero_iff_single_add_le_zero [simp]:
haftmann@35036
  1001
  "a + a \<le> 0 \<longleftrightarrow> a \<le> 0" 
haftmann@35036
  1002
proof -
haftmann@35036
  1003
  have "\<not> a + a \<le> 0 \<longleftrightarrow> \<not> a \<le> 0"
haftmann@35036
  1004
    by (simp add: not_le)
haftmann@35036
  1005
  then show ?thesis by simp
haftmann@35036
  1006
qed
haftmann@35036
  1007
haftmann@35036
  1008
lemma le_minus_self_iff:
haftmann@35036
  1009
  "a \<le> - a \<longleftrightarrow> a \<le> 0"
haftmann@35036
  1010
proof -
haftmann@35036
  1011
  from add_le_cancel_left [of "- a" "a + a" 0]
haftmann@35036
  1012
  have "a \<le> - a \<longleftrightarrow> a + a \<le> 0" 
haftmann@35036
  1013
    by (simp add: add_assoc [symmetric])
haftmann@35036
  1014
  thus ?thesis by simp
haftmann@35036
  1015
qed
haftmann@35036
  1016
haftmann@35036
  1017
lemma minus_le_self_iff:
haftmann@35036
  1018
  "- a \<le> a \<longleftrightarrow> 0 \<le> a"
haftmann@35036
  1019
proof -
haftmann@35036
  1020
  from add_le_cancel_left [of "- a" 0 "a + a"]
haftmann@35036
  1021
  have "- a \<le> a \<longleftrightarrow> 0 \<le> a + a" 
haftmann@35036
  1022
    by (simp add: add_assoc [symmetric])
haftmann@35036
  1023
  thus ?thesis by simp
haftmann@35036
  1024
qed
haftmann@35036
  1025
haftmann@35036
  1026
lemma minus_max_eq_min:
haftmann@35036
  1027
  "- max x y = min (-x) (-y)"
haftmann@35036
  1028
  by (auto simp add: max_def min_def)
haftmann@35036
  1029
haftmann@35036
  1030
lemma minus_min_eq_max:
haftmann@35036
  1031
  "- min x y = max (-x) (-y)"
haftmann@35036
  1032
  by (auto simp add: max_def min_def)
haftmann@25303
  1033
haftmann@25267
  1034
end
haftmann@25267
  1035
haftmann@36302
  1036
context ordered_comm_monoid_add
haftmann@36302
  1037
begin
obua@14738
  1038
paulson@15234
  1039
lemma add_increasing:
haftmann@36302
  1040
  "0 \<le> a \<Longrightarrow> b \<le> c \<Longrightarrow> b \<le> a + c"
haftmann@36302
  1041
  by (insert add_mono [of 0 a b c], simp)
obua@14738
  1042
nipkow@15539
  1043
lemma add_increasing2:
haftmann@36302
  1044
  "0 \<le> c \<Longrightarrow> b \<le> a \<Longrightarrow> b \<le> a + c"
haftmann@36302
  1045
  by (simp add: add_increasing add_commute [of a])
nipkow@15539
  1046
paulson@15234
  1047
lemma add_strict_increasing:
haftmann@36302
  1048
  "0 < a \<Longrightarrow> b \<le> c \<Longrightarrow> b < a + c"
haftmann@36302
  1049
  by (insert add_less_le_mono [of 0 a b c], simp)
paulson@15234
  1050
paulson@15234
  1051
lemma add_strict_increasing2:
haftmann@36302
  1052
  "0 \<le> a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@36302
  1053
  by (insert add_le_less_mono [of 0 a b c], simp)
haftmann@36302
  1054
haftmann@36302
  1055
end
paulson@15234
  1056
haftmann@35092
  1057
class abs =
haftmann@35092
  1058
  fixes abs :: "'a \<Rightarrow> 'a"
haftmann@35092
  1059
begin
haftmann@35092
  1060
haftmann@35092
  1061
notation (xsymbols)
haftmann@35092
  1062
  abs  ("\<bar>_\<bar>")
haftmann@35092
  1063
haftmann@35092
  1064
notation (HTML output)
haftmann@35092
  1065
  abs  ("\<bar>_\<bar>")
haftmann@35092
  1066
haftmann@35092
  1067
end
haftmann@35092
  1068
haftmann@35092
  1069
class sgn =
haftmann@35092
  1070
  fixes sgn :: "'a \<Rightarrow> 'a"
haftmann@35092
  1071
haftmann@35092
  1072
class abs_if = minus + uminus + ord + zero + abs +
haftmann@35092
  1073
  assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"
haftmann@35092
  1074
haftmann@35092
  1075
class sgn_if = minus + uminus + zero + one + ord + sgn +
haftmann@35092
  1076
  assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
haftmann@35092
  1077
begin
haftmann@35092
  1078
haftmann@35092
  1079
lemma sgn0 [simp]: "sgn 0 = 0"
haftmann@35092
  1080
  by (simp add:sgn_if)
haftmann@35092
  1081
haftmann@35092
  1082
end
obua@14738
  1083
haftmann@35028
  1084
class ordered_ab_group_add_abs = ordered_ab_group_add + abs +
haftmann@25303
  1085
  assumes abs_ge_zero [simp]: "\<bar>a\<bar> \<ge> 0"
haftmann@25303
  1086
    and abs_ge_self: "a \<le> \<bar>a\<bar>"
haftmann@25303
  1087
    and abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
haftmann@25303
  1088
    and abs_minus_cancel [simp]: "\<bar>-a\<bar> = \<bar>a\<bar>"
haftmann@25303
  1089
    and abs_triangle_ineq: "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25303
  1090
begin
haftmann@25303
  1091
haftmann@25307
  1092
lemma abs_minus_le_zero: "- \<bar>a\<bar> \<le> 0"
haftmann@25307
  1093
  unfolding neg_le_0_iff_le by simp
haftmann@25307
  1094
haftmann@25307
  1095
lemma abs_of_nonneg [simp]:
nipkow@29667
  1096
  assumes nonneg: "0 \<le> a" shows "\<bar>a\<bar> = a"
haftmann@25307
  1097
proof (rule antisym)
haftmann@25307
  1098
  from nonneg le_imp_neg_le have "- a \<le> 0" by simp
haftmann@25307
  1099
  from this nonneg have "- a \<le> a" by (rule order_trans)
haftmann@25307
  1100
  then show "\<bar>a\<bar> \<le> a" by (auto intro: abs_leI)
haftmann@25307
  1101
qed (rule abs_ge_self)
haftmann@25307
  1102
haftmann@25307
  1103
lemma abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
nipkow@29667
  1104
by (rule antisym)
haftmann@36302
  1105
   (auto intro!: abs_ge_self abs_leI order_trans [of "- \<bar>a\<bar>" 0 "\<bar>a\<bar>"])
haftmann@25307
  1106
haftmann@25307
  1107
lemma abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
haftmann@25307
  1108
proof -
haftmann@25307
  1109
  have "\<bar>a\<bar> = 0 \<Longrightarrow> a = 0"
haftmann@25307
  1110
  proof (rule antisym)
haftmann@25307
  1111
    assume zero: "\<bar>a\<bar> = 0"
haftmann@25307
  1112
    with abs_ge_self show "a \<le> 0" by auto
haftmann@25307
  1113
    from zero have "\<bar>-a\<bar> = 0" by simp
haftmann@36302
  1114
    with abs_ge_self [of "- a"] have "- a \<le> 0" by auto
haftmann@25307
  1115
    with neg_le_0_iff_le show "0 \<le> a" by auto
haftmann@25307
  1116
  qed
haftmann@25307
  1117
  then show ?thesis by auto
haftmann@25307
  1118
qed
haftmann@25307
  1119
haftmann@25303
  1120
lemma abs_zero [simp]: "\<bar>0\<bar> = 0"
nipkow@29667
  1121
by simp
avigad@16775
  1122
blanchet@35828
  1123
lemma abs_0_eq [simp, no_atp]: "0 = \<bar>a\<bar> \<longleftrightarrow> a = 0"
haftmann@25303
  1124
proof -
haftmann@25303
  1125
  have "0 = \<bar>a\<bar> \<longleftrightarrow> \<bar>a\<bar> = 0" by (simp only: eq_ac)
haftmann@25303
  1126
  thus ?thesis by simp
haftmann@25303
  1127
qed
haftmann@25303
  1128
haftmann@25303
  1129
lemma abs_le_zero_iff [simp]: "\<bar>a\<bar> \<le> 0 \<longleftrightarrow> a = 0" 
haftmann@25303
  1130
proof
haftmann@25303
  1131
  assume "\<bar>a\<bar> \<le> 0"
haftmann@25303
  1132
  then have "\<bar>a\<bar> = 0" by (rule antisym) simp
haftmann@25303
  1133
  thus "a = 0" by simp
haftmann@25303
  1134
next
haftmann@25303
  1135
  assume "a = 0"
haftmann@25303
  1136
  thus "\<bar>a\<bar> \<le> 0" by simp
haftmann@25303
  1137
qed
haftmann@25303
  1138
haftmann@25303
  1139
lemma zero_less_abs_iff [simp]: "0 < \<bar>a\<bar> \<longleftrightarrow> a \<noteq> 0"
nipkow@29667
  1140
by (simp add: less_le)
haftmann@25303
  1141
haftmann@25303
  1142
lemma abs_not_less_zero [simp]: "\<not> \<bar>a\<bar> < 0"
haftmann@25303
  1143
proof -
haftmann@25303
  1144
  have a: "\<And>x y. x \<le> y \<Longrightarrow> \<not> y < x" by auto
haftmann@25303
  1145
  show ?thesis by (simp add: a)
haftmann@25303
  1146
qed
avigad@16775
  1147
haftmann@25303
  1148
lemma abs_ge_minus_self: "- a \<le> \<bar>a\<bar>"
haftmann@25303
  1149
proof -
haftmann@25303
  1150
  have "- a \<le> \<bar>-a\<bar>" by (rule abs_ge_self)
haftmann@25303
  1151
  then show ?thesis by simp
haftmann@25303
  1152
qed
haftmann@25303
  1153
haftmann@25303
  1154
lemma abs_minus_commute: 
haftmann@25303
  1155
  "\<bar>a - b\<bar> = \<bar>b - a\<bar>"
haftmann@25303
  1156
proof -
haftmann@25303
  1157
  have "\<bar>a - b\<bar> = \<bar>- (a - b)\<bar>" by (simp only: abs_minus_cancel)
haftmann@25303
  1158
  also have "... = \<bar>b - a\<bar>" by simp
haftmann@25303
  1159
  finally show ?thesis .
haftmann@25303
  1160
qed
haftmann@25303
  1161
haftmann@25303
  1162
lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a"
nipkow@29667
  1163
by (rule abs_of_nonneg, rule less_imp_le)
avigad@16775
  1164
haftmann@25303
  1165
lemma abs_of_nonpos [simp]:
nipkow@29667
  1166
  assumes "a \<le> 0" shows "\<bar>a\<bar> = - a"
haftmann@25303
  1167
proof -
haftmann@25303
  1168
  let ?b = "- a"
haftmann@25303
  1169
  have "- ?b \<le> 0 \<Longrightarrow> \<bar>- ?b\<bar> = - (- ?b)"
haftmann@25303
  1170
  unfolding abs_minus_cancel [of "?b"]
haftmann@25303
  1171
  unfolding neg_le_0_iff_le [of "?b"]
haftmann@25303
  1172
  unfolding minus_minus by (erule abs_of_nonneg)
haftmann@25303
  1173
  then show ?thesis using assms by auto
haftmann@25303
  1174
qed
haftmann@25303
  1175
  
haftmann@25303
  1176
lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a"
nipkow@29667
  1177
by (rule abs_of_nonpos, rule less_imp_le)
haftmann@25303
  1178
haftmann@25303
  1179
lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b"
nipkow@29667
  1180
by (insert abs_ge_self, blast intro: order_trans)
haftmann@25303
  1181
haftmann@25303
  1182
lemma abs_le_D2: "\<bar>a\<bar> \<le> b \<Longrightarrow> - a \<le> b"
haftmann@36302
  1183
by (insert abs_le_D1 [of "- a"], simp)
haftmann@25303
  1184
haftmann@25303
  1185
lemma abs_le_iff: "\<bar>a\<bar> \<le> b \<longleftrightarrow> a \<le> b \<and> - a \<le> b"
nipkow@29667
  1186
by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
haftmann@25303
  1187
haftmann@25303
  1188
lemma abs_triangle_ineq2: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>a - b\<bar>"
haftmann@36302
  1189
proof -
haftmann@36302
  1190
  have "\<bar>a\<bar> = \<bar>b + (a - b)\<bar>"
haftmann@36302
  1191
    by (simp add: algebra_simps add_diff_cancel)
haftmann@36302
  1192
  then have "\<bar>a\<bar> \<le> \<bar>b\<bar> + \<bar>a - b\<bar>"
haftmann@36302
  1193
    by (simp add: abs_triangle_ineq)
haftmann@36302
  1194
  then show ?thesis
haftmann@36302
  1195
    by (simp add: algebra_simps)
haftmann@36302
  1196
qed
haftmann@36302
  1197
haftmann@36302
  1198
lemma abs_triangle_ineq2_sym: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>b - a\<bar>"
haftmann@36302
  1199
  by (simp only: abs_minus_commute [of b] abs_triangle_ineq2)
avigad@16775
  1200
haftmann@25303
  1201
lemma abs_triangle_ineq3: "\<bar>\<bar>a\<bar> - \<bar>b\<bar>\<bar> \<le> \<bar>a - b\<bar>"
haftmann@36302
  1202
  by (simp add: abs_le_iff abs_triangle_ineq2 abs_triangle_ineq2_sym)
avigad@16775
  1203
haftmann@25303
  1204
lemma abs_triangle_ineq4: "\<bar>a - b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25303
  1205
proof -
haftmann@36302
  1206
  have "\<bar>a - b\<bar> = \<bar>a + - b\<bar>" by (subst diff_minus, rule refl)
haftmann@36302
  1207
  also have "... \<le> \<bar>a\<bar> + \<bar>- b\<bar>" by (rule abs_triangle_ineq)
nipkow@29667
  1208
  finally show ?thesis by simp
haftmann@25303
  1209
qed
avigad@16775
  1210
haftmann@25303
  1211
lemma abs_diff_triangle_ineq: "\<bar>a + b - (c + d)\<bar> \<le> \<bar>a - c\<bar> + \<bar>b - d\<bar>"
haftmann@25303
  1212
proof -
haftmann@25303
  1213
  have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: diff_minus add_ac)
haftmann@25303
  1214
  also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq)
haftmann@25303
  1215
  finally show ?thesis .
haftmann@25303
  1216
qed
avigad@16775
  1217
haftmann@25303
  1218
lemma abs_add_abs [simp]:
haftmann@25303
  1219
  "\<bar>\<bar>a\<bar> + \<bar>b\<bar>\<bar> = \<bar>a\<bar> + \<bar>b\<bar>" (is "?L = ?R")
haftmann@25303
  1220
proof (rule antisym)
haftmann@25303
  1221
  show "?L \<ge> ?R" by(rule abs_ge_self)
haftmann@25303
  1222
next
haftmann@25303
  1223
  have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq)
haftmann@25303
  1224
  also have "\<dots> = ?R" by simp
haftmann@25303
  1225
  finally show "?L \<le> ?R" .
haftmann@25303
  1226
qed
haftmann@25303
  1227
haftmann@25303
  1228
end
obua@14738
  1229
obua@15178
  1230
haftmann@25090
  1231
subsection {* Tools setup *}
haftmann@25090
  1232
blanchet@35828
  1233
lemma add_mono_thms_linordered_semiring [no_atp]:
haftmann@35028
  1234
  fixes i j k :: "'a\<Colon>ordered_ab_semigroup_add"
haftmann@25077
  1235
  shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1236
    and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1237
    and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1238
    and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"
haftmann@25077
  1239
by (rule add_mono, clarify+)+
haftmann@25077
  1240
blanchet@35828
  1241
lemma add_mono_thms_linordered_field [no_atp]:
haftmann@35028
  1242
  fixes i j k :: "'a\<Colon>ordered_cancel_ab_semigroup_add"
haftmann@25077
  1243
  shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1244
    and "i = j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1245
    and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1246
    and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1247
    and "i < j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1248
by (auto intro: add_strict_right_mono add_strict_left_mono
haftmann@25077
  1249
  add_less_le_mono add_le_less_mono add_strict_mono)
haftmann@25077
  1250
haftmann@33364
  1251
code_modulename SML
haftmann@35050
  1252
  Groups Arith
haftmann@33364
  1253
haftmann@33364
  1254
code_modulename OCaml
haftmann@35050
  1255
  Groups Arith
haftmann@33364
  1256
haftmann@33364
  1257
code_modulename Haskell
haftmann@35050
  1258
  Groups Arith
haftmann@33364
  1259
haftmann@37889
  1260
haftmann@37889
  1261
text {* Legacy *}
haftmann@37889
  1262
haftmann@37889
  1263
lemmas diff_def = diff_minus
haftmann@37889
  1264
obua@14738
  1265
end