src/HOL/Groups.thy
author huffman
Thu Nov 17 18:31:00 2011 +0100 (2011-11-17)
changeset 45548 3e2722d66169
parent 45294 3c5d3d286055
child 48556 62a3fbf9d35b
permissions -rw-r--r--
Groups.thy: generalize several lemmas from class ab_group_add to class group_add
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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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uses ("Tools/abel_cancel.ML")
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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"
haftmann@25062
   366
  thus "- a = - b" by simp
obua@14738
   367
qed
obua@14738
   368
haftmann@25062
   369
lemma neg_equal_0_iff_equal [simp]:
haftmann@25062
   370
  "- a = 0 \<longleftrightarrow> a = 0"
nipkow@29667
   371
by (subst neg_equal_iff_equal [symmetric], simp)
obua@14738
   372
haftmann@25062
   373
lemma neg_0_equal_iff_equal [simp]:
haftmann@25062
   374
  "0 = - a \<longleftrightarrow> 0 = a"
nipkow@29667
   375
by (subst neg_equal_iff_equal [symmetric], simp)
obua@14738
   376
obua@14738
   377
text{*The next two equations can make the simplifier loop!*}
obua@14738
   378
haftmann@25062
   379
lemma equation_minus_iff:
haftmann@25062
   380
  "a = - b \<longleftrightarrow> b = - a"
obua@14738
   381
proof -
haftmann@25062
   382
  have "- (- a) = - b \<longleftrightarrow> - a = b" by (rule neg_equal_iff_equal)
haftmann@25062
   383
  thus ?thesis by (simp add: eq_commute)
haftmann@25062
   384
qed
haftmann@25062
   385
haftmann@25062
   386
lemma minus_equation_iff:
haftmann@25062
   387
  "- a = b \<longleftrightarrow> - b = a"
haftmann@25062
   388
proof -
haftmann@25062
   389
  have "- a = - (- b) \<longleftrightarrow> a = -b" by (rule neg_equal_iff_equal)
obua@14738
   390
  thus ?thesis by (simp add: eq_commute)
obua@14738
   391
qed
obua@14738
   392
huffman@28130
   393
lemma diff_add_cancel: "a - b + b = a"
nipkow@29667
   394
by (simp add: diff_minus add_assoc)
huffman@28130
   395
huffman@28130
   396
lemma add_diff_cancel: "a + b - b = a"
nipkow@29667
   397
by (simp add: diff_minus add_assoc)
nipkow@29667
   398
haftmann@36348
   399
declare diff_minus[symmetric, algebra_simps, field_simps]
huffman@28130
   400
huffman@29914
   401
lemma eq_neg_iff_add_eq_0: "a = - b \<longleftrightarrow> a + b = 0"
huffman@29914
   402
proof
huffman@29914
   403
  assume "a = - b" then show "a + b = 0" by simp
huffman@29914
   404
next
huffman@29914
   405
  assume "a + b = 0"
huffman@29914
   406
  moreover have "a + (b + - b) = (a + b) + - b"
huffman@29914
   407
    by (simp only: add_assoc)
huffman@29914
   408
  ultimately show "a = - b" by simp
huffman@29914
   409
qed
huffman@29914
   410
huffman@44348
   411
lemma add_eq_0_iff: "x + y = 0 \<longleftrightarrow> y = - x"
huffman@44348
   412
  unfolding eq_neg_iff_add_eq_0 [symmetric]
huffman@44348
   413
  by (rule equation_minus_iff)
huffman@44348
   414
huffman@45548
   415
lemma minus_diff_eq [simp]: "- (a - b) = b - a"
huffman@45548
   416
  by (simp add: diff_minus minus_add)
huffman@45548
   417
huffman@45548
   418
lemma add_diff_eq[algebra_simps, field_simps]: "a + (b - c) = (a + b) - c"
huffman@45548
   419
  by (simp add: diff_minus add_assoc)
huffman@45548
   420
huffman@45548
   421
lemma diff_eq_eq[algebra_simps, field_simps]: "a - b = c \<longleftrightarrow> a = c + b"
huffman@45548
   422
  by (auto simp add: diff_minus add_assoc)
huffman@45548
   423
huffman@45548
   424
lemma eq_diff_eq[algebra_simps, field_simps]: "a = c - b \<longleftrightarrow> a + b = c"
huffman@45548
   425
  by (auto simp add: diff_minus add_assoc)
huffman@45548
   426
huffman@45548
   427
lemma diff_diff_eq2[algebra_simps, field_simps]: "a - (b - c) = (a + c) - b"
huffman@45548
   428
  by (simp add: diff_minus minus_add add_assoc)
huffman@45548
   429
huffman@45548
   430
lemma eq_iff_diff_eq_0: "a = b \<longleftrightarrow> a - b = 0"
huffman@45548
   431
  by (fact right_minus_eq [symmetric])
huffman@45548
   432
huffman@45548
   433
lemma diff_eq_diff_eq:
huffman@45548
   434
  "a - b = c - d \<Longrightarrow> a = b \<longleftrightarrow> c = d"
huffman@45548
   435
  by (simp add: eq_iff_diff_eq_0 [of a b] eq_iff_diff_eq_0 [of c d])
huffman@45548
   436
haftmann@25062
   437
end
haftmann@25062
   438
haftmann@25762
   439
class ab_group_add = minus + uminus + comm_monoid_add +
haftmann@25062
   440
  assumes ab_left_minus: "- a + a = 0"
haftmann@25062
   441
  assumes ab_diff_minus: "a - b = a + (- b)"
haftmann@25267
   442
begin
haftmann@25062
   443
haftmann@25267
   444
subclass group_add
haftmann@28823
   445
  proof qed (simp_all add: ab_left_minus ab_diff_minus)
haftmann@25062
   446
huffman@29904
   447
subclass cancel_comm_monoid_add
haftmann@28823
   448
proof
haftmann@25062
   449
  fix a b c :: 'a
haftmann@25062
   450
  assume "a + b = a + c"
haftmann@25062
   451
  then have "- a + a + b = - a + a + c"
haftmann@25062
   452
    unfolding add_assoc by simp
haftmann@25062
   453
  then show "b = c" by simp
haftmann@25062
   454
qed
haftmann@25062
   455
haftmann@36348
   456
lemma uminus_add_conv_diff[algebra_simps, field_simps]:
haftmann@25062
   457
  "- a + b = b - a"
nipkow@29667
   458
by (simp add:diff_minus add_commute)
haftmann@25062
   459
haftmann@25062
   460
lemma minus_add_distrib [simp]:
haftmann@25062
   461
  "- (a + b) = - a + - b"
huffman@34146
   462
by (rule minus_unique) (simp add: add_ac)
haftmann@25062
   463
haftmann@36348
   464
lemma diff_add_eq[algebra_simps, field_simps]: "(a - b) + c = (a + c) - b"
nipkow@29667
   465
by (simp add: diff_minus add_ac)
haftmann@25077
   466
haftmann@36348
   467
lemma diff_diff_eq[algebra_simps, field_simps]: "(a - b) - c = a - (b + c)"
nipkow@29667
   468
by (simp add: diff_minus add_ac)
haftmann@25077
   469
huffman@35216
   470
(* FIXME: duplicates right_minus_eq from class group_add *)
huffman@35216
   471
(* but only this one is declared as a simp rule. *)
blanchet@35828
   472
lemma diff_eq_0_iff_eq [simp, no_atp]: "a - b = 0 \<longleftrightarrow> a = b"
huffman@44348
   473
  by (rule right_minus_eq)
huffman@30629
   474
haftmann@25062
   475
end
obua@14738
   476
haftmann@37884
   477
obua@14738
   478
subsection {* (Partially) Ordered Groups *} 
obua@14738
   479
haftmann@35301
   480
text {*
haftmann@35301
   481
  The theory of partially ordered groups is taken from the books:
haftmann@35301
   482
  \begin{itemize}
haftmann@35301
   483
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
haftmann@35301
   484
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
haftmann@35301
   485
  \end{itemize}
haftmann@35301
   486
  Most of the used notions can also be looked up in 
haftmann@35301
   487
  \begin{itemize}
haftmann@35301
   488
  \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
haftmann@35301
   489
  \item \emph{Algebra I} by van der Waerden, Springer.
haftmann@35301
   490
  \end{itemize}
haftmann@35301
   491
*}
haftmann@35301
   492
haftmann@35028
   493
class ordered_ab_semigroup_add = order + ab_semigroup_add +
haftmann@25062
   494
  assumes add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b"
haftmann@25062
   495
begin
haftmann@24380
   496
haftmann@25062
   497
lemma add_right_mono:
haftmann@25062
   498
  "a \<le> b \<Longrightarrow> a + c \<le> b + c"
nipkow@29667
   499
by (simp add: add_commute [of _ c] add_left_mono)
obua@14738
   500
obua@14738
   501
text {* non-strict, in both arguments *}
obua@14738
   502
lemma add_mono:
haftmann@25062
   503
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d"
obua@14738
   504
  apply (erule add_right_mono [THEN order_trans])
obua@14738
   505
  apply (simp add: add_commute add_left_mono)
obua@14738
   506
  done
obua@14738
   507
haftmann@25062
   508
end
haftmann@25062
   509
haftmann@35028
   510
class ordered_cancel_ab_semigroup_add =
haftmann@35028
   511
  ordered_ab_semigroup_add + cancel_ab_semigroup_add
haftmann@25062
   512
begin
haftmann@25062
   513
obua@14738
   514
lemma add_strict_left_mono:
haftmann@25062
   515
  "a < b \<Longrightarrow> c + a < c + b"
nipkow@29667
   516
by (auto simp add: less_le add_left_mono)
obua@14738
   517
obua@14738
   518
lemma add_strict_right_mono:
haftmann@25062
   519
  "a < b \<Longrightarrow> a + c < b + c"
nipkow@29667
   520
by (simp add: add_commute [of _ c] add_strict_left_mono)
obua@14738
   521
obua@14738
   522
text{*Strict monotonicity in both arguments*}
haftmann@25062
   523
lemma add_strict_mono:
haftmann@25062
   524
  "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
haftmann@25062
   525
apply (erule add_strict_right_mono [THEN less_trans])
obua@14738
   526
apply (erule add_strict_left_mono)
obua@14738
   527
done
obua@14738
   528
obua@14738
   529
lemma add_less_le_mono:
haftmann@25062
   530
  "a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c < b + d"
haftmann@25062
   531
apply (erule add_strict_right_mono [THEN less_le_trans])
haftmann@25062
   532
apply (erule add_left_mono)
obua@14738
   533
done
obua@14738
   534
obua@14738
   535
lemma add_le_less_mono:
haftmann@25062
   536
  "a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
haftmann@25062
   537
apply (erule add_right_mono [THEN le_less_trans])
obua@14738
   538
apply (erule add_strict_left_mono) 
obua@14738
   539
done
obua@14738
   540
haftmann@25062
   541
end
haftmann@25062
   542
haftmann@35028
   543
class ordered_ab_semigroup_add_imp_le =
haftmann@35028
   544
  ordered_cancel_ab_semigroup_add +
haftmann@25062
   545
  assumes add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b"
haftmann@25062
   546
begin
haftmann@25062
   547
obua@14738
   548
lemma add_less_imp_less_left:
nipkow@29667
   549
  assumes less: "c + a < c + b" shows "a < b"
obua@14738
   550
proof -
obua@14738
   551
  from less have le: "c + a <= c + b" by (simp add: order_le_less)
obua@14738
   552
  have "a <= b" 
obua@14738
   553
    apply (insert le)
obua@14738
   554
    apply (drule add_le_imp_le_left)
obua@14738
   555
    by (insert le, drule add_le_imp_le_left, assumption)
obua@14738
   556
  moreover have "a \<noteq> b"
obua@14738
   557
  proof (rule ccontr)
obua@14738
   558
    assume "~(a \<noteq> b)"
obua@14738
   559
    then have "a = b" by simp
obua@14738
   560
    then have "c + a = c + b" by simp
obua@14738
   561
    with less show "False"by simp
obua@14738
   562
  qed
obua@14738
   563
  ultimately show "a < b" by (simp add: order_le_less)
obua@14738
   564
qed
obua@14738
   565
obua@14738
   566
lemma add_less_imp_less_right:
haftmann@25062
   567
  "a + c < b + c \<Longrightarrow> a < b"
obua@14738
   568
apply (rule add_less_imp_less_left [of c])
obua@14738
   569
apply (simp add: add_commute)  
obua@14738
   570
done
obua@14738
   571
obua@14738
   572
lemma add_less_cancel_left [simp]:
haftmann@25062
   573
  "c + a < c + b \<longleftrightarrow> a < b"
nipkow@29667
   574
by (blast intro: add_less_imp_less_left add_strict_left_mono) 
obua@14738
   575
obua@14738
   576
lemma add_less_cancel_right [simp]:
haftmann@25062
   577
  "a + c < b + c \<longleftrightarrow> a < b"
nipkow@29667
   578
by (blast intro: add_less_imp_less_right add_strict_right_mono)
obua@14738
   579
obua@14738
   580
lemma add_le_cancel_left [simp]:
haftmann@25062
   581
  "c + a \<le> c + b \<longleftrightarrow> a \<le> b"
nipkow@29667
   582
by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono) 
obua@14738
   583
obua@14738
   584
lemma add_le_cancel_right [simp]:
haftmann@25062
   585
  "a + c \<le> b + c \<longleftrightarrow> a \<le> b"
nipkow@29667
   586
by (simp add: add_commute [of a c] add_commute [of b c])
obua@14738
   587
obua@14738
   588
lemma add_le_imp_le_right:
haftmann@25062
   589
  "a + c \<le> b + c \<Longrightarrow> a \<le> b"
nipkow@29667
   590
by simp
haftmann@25062
   591
haftmann@25077
   592
lemma max_add_distrib_left:
haftmann@25077
   593
  "max x y + z = max (x + z) (y + z)"
haftmann@25077
   594
  unfolding max_def by auto
haftmann@25077
   595
haftmann@25077
   596
lemma min_add_distrib_left:
haftmann@25077
   597
  "min x y + z = min (x + z) (y + z)"
haftmann@25077
   598
  unfolding min_def by auto
haftmann@25077
   599
huffman@44848
   600
lemma max_add_distrib_right:
huffman@44848
   601
  "x + max y z = max (x + y) (x + z)"
huffman@44848
   602
  unfolding max_def by auto
huffman@44848
   603
huffman@44848
   604
lemma min_add_distrib_right:
huffman@44848
   605
  "x + min y z = min (x + y) (x + z)"
huffman@44848
   606
  unfolding min_def by auto
huffman@44848
   607
haftmann@25062
   608
end
haftmann@25062
   609
haftmann@25303
   610
subsection {* Support for reasoning about signs *}
haftmann@25303
   611
haftmann@35028
   612
class ordered_comm_monoid_add =
haftmann@35028
   613
  ordered_cancel_ab_semigroup_add + comm_monoid_add
haftmann@25303
   614
begin
haftmann@25303
   615
haftmann@25303
   616
lemma add_pos_nonneg:
nipkow@29667
   617
  assumes "0 < a" and "0 \<le> b" shows "0 < a + b"
haftmann@25303
   618
proof -
haftmann@25303
   619
  have "0 + 0 < a + b" 
haftmann@25303
   620
    using assms by (rule add_less_le_mono)
haftmann@25303
   621
  then show ?thesis by simp
haftmann@25303
   622
qed
haftmann@25303
   623
haftmann@25303
   624
lemma add_pos_pos:
nipkow@29667
   625
  assumes "0 < a" and "0 < b" shows "0 < a + b"
nipkow@29667
   626
by (rule add_pos_nonneg) (insert assms, auto)
haftmann@25303
   627
haftmann@25303
   628
lemma add_nonneg_pos:
nipkow@29667
   629
  assumes "0 \<le> a" and "0 < b" shows "0 < a + b"
haftmann@25303
   630
proof -
haftmann@25303
   631
  have "0 + 0 < a + b" 
haftmann@25303
   632
    using assms by (rule add_le_less_mono)
haftmann@25303
   633
  then show ?thesis by simp
haftmann@25303
   634
qed
haftmann@25303
   635
huffman@36977
   636
lemma add_nonneg_nonneg [simp]:
nipkow@29667
   637
  assumes "0 \<le> a" and "0 \<le> b" shows "0 \<le> a + b"
haftmann@25303
   638
proof -
haftmann@25303
   639
  have "0 + 0 \<le> a + b" 
haftmann@25303
   640
    using assms by (rule add_mono)
haftmann@25303
   641
  then show ?thesis by simp
haftmann@25303
   642
qed
haftmann@25303
   643
huffman@30691
   644
lemma add_neg_nonpos:
nipkow@29667
   645
  assumes "a < 0" and "b \<le> 0" shows "a + b < 0"
haftmann@25303
   646
proof -
haftmann@25303
   647
  have "a + b < 0 + 0"
haftmann@25303
   648
    using assms by (rule add_less_le_mono)
haftmann@25303
   649
  then show ?thesis by simp
haftmann@25303
   650
qed
haftmann@25303
   651
haftmann@25303
   652
lemma add_neg_neg: 
nipkow@29667
   653
  assumes "a < 0" and "b < 0" shows "a + b < 0"
nipkow@29667
   654
by (rule add_neg_nonpos) (insert assms, auto)
haftmann@25303
   655
haftmann@25303
   656
lemma add_nonpos_neg:
nipkow@29667
   657
  assumes "a \<le> 0" and "b < 0" shows "a + b < 0"
haftmann@25303
   658
proof -
haftmann@25303
   659
  have "a + b < 0 + 0"
haftmann@25303
   660
    using assms by (rule add_le_less_mono)
haftmann@25303
   661
  then show ?thesis by simp
haftmann@25303
   662
qed
haftmann@25303
   663
haftmann@25303
   664
lemma add_nonpos_nonpos:
nipkow@29667
   665
  assumes "a \<le> 0" and "b \<le> 0" shows "a + b \<le> 0"
haftmann@25303
   666
proof -
haftmann@25303
   667
  have "a + b \<le> 0 + 0"
haftmann@25303
   668
    using assms by (rule add_mono)
haftmann@25303
   669
  then show ?thesis by simp
haftmann@25303
   670
qed
haftmann@25303
   671
huffman@30691
   672
lemmas add_sign_intros =
huffman@30691
   673
  add_pos_nonneg add_pos_pos add_nonneg_pos add_nonneg_nonneg
huffman@30691
   674
  add_neg_nonpos add_neg_neg add_nonpos_neg add_nonpos_nonpos
huffman@30691
   675
huffman@29886
   676
lemma add_nonneg_eq_0_iff:
huffman@29886
   677
  assumes x: "0 \<le> x" and y: "0 \<le> y"
huffman@29886
   678
  shows "x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@29886
   679
proof (intro iffI conjI)
huffman@29886
   680
  have "x = x + 0" by simp
huffman@29886
   681
  also have "x + 0 \<le> x + y" using y by (rule add_left_mono)
huffman@29886
   682
  also assume "x + y = 0"
huffman@29886
   683
  also have "0 \<le> x" using x .
huffman@29886
   684
  finally show "x = 0" .
huffman@29886
   685
next
huffman@29886
   686
  have "y = 0 + y" by simp
huffman@29886
   687
  also have "0 + y \<le> x + y" using x by (rule add_right_mono)
huffman@29886
   688
  also assume "x + y = 0"
huffman@29886
   689
  also have "0 \<le> y" using y .
huffman@29886
   690
  finally show "y = 0" .
huffman@29886
   691
next
huffman@29886
   692
  assume "x = 0 \<and> y = 0"
huffman@29886
   693
  then show "x + y = 0" by simp
huffman@29886
   694
qed
huffman@29886
   695
haftmann@25303
   696
end
haftmann@25303
   697
haftmann@35028
   698
class ordered_ab_group_add =
haftmann@35028
   699
  ab_group_add + ordered_ab_semigroup_add
haftmann@25062
   700
begin
haftmann@25062
   701
haftmann@35028
   702
subclass ordered_cancel_ab_semigroup_add ..
haftmann@25062
   703
haftmann@35028
   704
subclass ordered_ab_semigroup_add_imp_le
haftmann@28823
   705
proof
haftmann@25062
   706
  fix a b c :: 'a
haftmann@25062
   707
  assume "c + a \<le> c + b"
haftmann@25062
   708
  hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono)
haftmann@25062
   709
  hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc)
haftmann@25062
   710
  thus "a \<le> b" by simp
haftmann@25062
   711
qed
haftmann@25062
   712
haftmann@35028
   713
subclass ordered_comm_monoid_add ..
haftmann@25303
   714
haftmann@25077
   715
lemma max_diff_distrib_left:
haftmann@25077
   716
  shows "max x y - z = max (x - z) (y - z)"
nipkow@29667
   717
by (simp add: diff_minus, rule max_add_distrib_left) 
haftmann@25077
   718
haftmann@25077
   719
lemma min_diff_distrib_left:
haftmann@25077
   720
  shows "min x y - z = min (x - z) (y - z)"
nipkow@29667
   721
by (simp add: diff_minus, rule min_add_distrib_left) 
haftmann@25077
   722
haftmann@25077
   723
lemma le_imp_neg_le:
nipkow@29667
   724
  assumes "a \<le> b" shows "-b \<le> -a"
haftmann@25077
   725
proof -
nipkow@29667
   726
  have "-a+a \<le> -a+b" using `a \<le> b` by (rule add_left_mono) 
nipkow@29667
   727
  hence "0 \<le> -a+b" by simp
nipkow@29667
   728
  hence "0 + (-b) \<le> (-a + b) + (-b)" by (rule add_right_mono) 
nipkow@29667
   729
  thus ?thesis by (simp add: add_assoc)
haftmann@25077
   730
qed
haftmann@25077
   731
haftmann@25077
   732
lemma neg_le_iff_le [simp]: "- b \<le> - a \<longleftrightarrow> a \<le> b"
haftmann@25077
   733
proof 
haftmann@25077
   734
  assume "- b \<le> - a"
nipkow@29667
   735
  hence "- (- a) \<le> - (- b)" by (rule le_imp_neg_le)
haftmann@25077
   736
  thus "a\<le>b" by simp
haftmann@25077
   737
next
haftmann@25077
   738
  assume "a\<le>b"
haftmann@25077
   739
  thus "-b \<le> -a" by (rule le_imp_neg_le)
haftmann@25077
   740
qed
haftmann@25077
   741
haftmann@25077
   742
lemma neg_le_0_iff_le [simp]: "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"
nipkow@29667
   743
by (subst neg_le_iff_le [symmetric], simp)
haftmann@25077
   744
haftmann@25077
   745
lemma neg_0_le_iff_le [simp]: "0 \<le> - a \<longleftrightarrow> a \<le> 0"
nipkow@29667
   746
by (subst neg_le_iff_le [symmetric], simp)
haftmann@25077
   747
haftmann@25077
   748
lemma neg_less_iff_less [simp]: "- b < - a \<longleftrightarrow> a < b"
nipkow@29667
   749
by (force simp add: less_le) 
haftmann@25077
   750
haftmann@25077
   751
lemma neg_less_0_iff_less [simp]: "- a < 0 \<longleftrightarrow> 0 < a"
nipkow@29667
   752
by (subst neg_less_iff_less [symmetric], simp)
haftmann@25077
   753
haftmann@25077
   754
lemma neg_0_less_iff_less [simp]: "0 < - a \<longleftrightarrow> a < 0"
nipkow@29667
   755
by (subst neg_less_iff_less [symmetric], simp)
haftmann@25077
   756
haftmann@25077
   757
text{*The next several equations can make the simplifier loop!*}
haftmann@25077
   758
haftmann@25077
   759
lemma less_minus_iff: "a < - b \<longleftrightarrow> b < - a"
haftmann@25077
   760
proof -
haftmann@25077
   761
  have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
haftmann@25077
   762
  thus ?thesis by simp
haftmann@25077
   763
qed
haftmann@25077
   764
haftmann@25077
   765
lemma minus_less_iff: "- a < b \<longleftrightarrow> - b < a"
haftmann@25077
   766
proof -
haftmann@25077
   767
  have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
haftmann@25077
   768
  thus ?thesis by simp
haftmann@25077
   769
qed
haftmann@25077
   770
haftmann@25077
   771
lemma le_minus_iff: "a \<le> - b \<longleftrightarrow> b \<le> - a"
haftmann@25077
   772
proof -
haftmann@25077
   773
  have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff)
haftmann@25077
   774
  have "(- (- a) <= -b) = (b <= - a)" 
haftmann@25077
   775
    apply (auto simp only: le_less)
haftmann@25077
   776
    apply (drule mm)
haftmann@25077
   777
    apply (simp_all)
haftmann@25077
   778
    apply (drule mm[simplified], assumption)
haftmann@25077
   779
    done
haftmann@25077
   780
  then show ?thesis by simp
haftmann@25077
   781
qed
haftmann@25077
   782
haftmann@25077
   783
lemma minus_le_iff: "- a \<le> b \<longleftrightarrow> - b \<le> a"
nipkow@29667
   784
by (auto simp add: le_less minus_less_iff)
haftmann@25077
   785
haftmann@37884
   786
lemma diff_less_0_iff_less [simp, no_atp]:
haftmann@37884
   787
  "a - b < 0 \<longleftrightarrow> a < b"
haftmann@25077
   788
proof -
haftmann@37884
   789
  have "a - b < 0 \<longleftrightarrow> a + (- b) < b + (- b)" by (simp add: diff_minus)
haftmann@37884
   790
  also have "... \<longleftrightarrow> a < b" by (simp only: add_less_cancel_right)
haftmann@25077
   791
  finally show ?thesis .
haftmann@25077
   792
qed
haftmann@25077
   793
haftmann@37884
   794
lemmas less_iff_diff_less_0 = diff_less_0_iff_less [symmetric]
haftmann@37884
   795
haftmann@36348
   796
lemma diff_less_eq[algebra_simps, field_simps]: "a - b < c \<longleftrightarrow> a < c + b"
haftmann@25077
   797
apply (subst less_iff_diff_less_0 [of a])
haftmann@25077
   798
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
haftmann@25077
   799
apply (simp add: diff_minus add_ac)
haftmann@25077
   800
done
haftmann@25077
   801
haftmann@36348
   802
lemma less_diff_eq[algebra_simps, field_simps]: "a < c - b \<longleftrightarrow> a + b < c"
haftmann@36302
   803
apply (subst less_iff_diff_less_0 [of "a + b"])
haftmann@25077
   804
apply (subst less_iff_diff_less_0 [of a])
haftmann@25077
   805
apply (simp add: diff_minus add_ac)
haftmann@25077
   806
done
haftmann@25077
   807
haftmann@36348
   808
lemma diff_le_eq[algebra_simps, field_simps]: "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
nipkow@29667
   809
by (auto simp add: le_less diff_less_eq diff_add_cancel add_diff_cancel)
haftmann@25077
   810
haftmann@36348
   811
lemma le_diff_eq[algebra_simps, field_simps]: "a \<le> c - b \<longleftrightarrow> a + b \<le> c"
nipkow@29667
   812
by (auto simp add: le_less less_diff_eq diff_add_cancel add_diff_cancel)
haftmann@25077
   813
haftmann@37884
   814
lemma diff_le_0_iff_le [simp, no_atp]:
haftmann@37884
   815
  "a - b \<le> 0 \<longleftrightarrow> a \<le> b"
haftmann@37884
   816
  by (simp add: algebra_simps)
haftmann@37884
   817
haftmann@37884
   818
lemmas le_iff_diff_le_0 = diff_le_0_iff_le [symmetric]
haftmann@37884
   819
haftmann@37884
   820
lemma diff_eq_diff_less:
haftmann@37884
   821
  "a - b = c - d \<Longrightarrow> a < b \<longleftrightarrow> c < d"
haftmann@37884
   822
  by (auto simp only: less_iff_diff_less_0 [of a b] less_iff_diff_less_0 [of c d])
haftmann@37884
   823
haftmann@37889
   824
lemma diff_eq_diff_less_eq:
haftmann@37889
   825
  "a - b = c - d \<Longrightarrow> a \<le> b \<longleftrightarrow> c \<le> d"
haftmann@37889
   826
  by (auto simp only: le_iff_diff_le_0 [of a b] le_iff_diff_le_0 [of c d])
haftmann@25077
   827
haftmann@25077
   828
end
haftmann@25077
   829
wenzelm@37986
   830
use "Tools/abel_cancel.ML"
haftmann@37884
   831
haftmann@37889
   832
simproc_setup abel_cancel_sum
haftmann@37889
   833
  ("a + b::'a::ab_group_add" | "a - b::'a::ab_group_add") =
haftmann@37889
   834
  {* fn phi => Abel_Cancel.sum_proc *}
haftmann@37889
   835
haftmann@37889
   836
simproc_setup abel_cancel_relation
haftmann@37889
   837
  ("a < (b::'a::ordered_ab_group_add)" | "a \<le> (b::'a::ordered_ab_group_add)" | "c = (d::'b::ab_group_add)") =
haftmann@37889
   838
  {* fn phi => Abel_Cancel.rel_proc *}
haftmann@37884
   839
haftmann@35028
   840
class linordered_ab_semigroup_add =
haftmann@35028
   841
  linorder + ordered_ab_semigroup_add
haftmann@25062
   842
haftmann@35028
   843
class linordered_cancel_ab_semigroup_add =
haftmann@35028
   844
  linorder + ordered_cancel_ab_semigroup_add
haftmann@25267
   845
begin
haftmann@25062
   846
haftmann@35028
   847
subclass linordered_ab_semigroup_add ..
haftmann@25062
   848
haftmann@35028
   849
subclass ordered_ab_semigroup_add_imp_le
haftmann@28823
   850
proof
haftmann@25062
   851
  fix a b c :: 'a
haftmann@25062
   852
  assume le: "c + a <= c + b"  
haftmann@25062
   853
  show "a <= b"
haftmann@25062
   854
  proof (rule ccontr)
haftmann@25062
   855
    assume w: "~ a \<le> b"
haftmann@25062
   856
    hence "b <= a" by (simp add: linorder_not_le)
haftmann@25062
   857
    hence le2: "c + b <= c + a" by (rule add_left_mono)
haftmann@25062
   858
    have "a = b" 
haftmann@25062
   859
      apply (insert le)
haftmann@25062
   860
      apply (insert le2)
haftmann@25062
   861
      apply (drule antisym, simp_all)
haftmann@25062
   862
      done
haftmann@25062
   863
    with w show False 
haftmann@25062
   864
      by (simp add: linorder_not_le [symmetric])
haftmann@25062
   865
  qed
haftmann@25062
   866
qed
haftmann@25062
   867
haftmann@25267
   868
end
haftmann@25267
   869
haftmann@35028
   870
class linordered_ab_group_add = linorder + ordered_ab_group_add
haftmann@25267
   871
begin
haftmann@25230
   872
haftmann@35028
   873
subclass linordered_cancel_ab_semigroup_add ..
haftmann@25230
   874
haftmann@35036
   875
lemma neg_less_eq_nonneg [simp]:
haftmann@25303
   876
  "- a \<le> a \<longleftrightarrow> 0 \<le> a"
haftmann@25303
   877
proof
haftmann@25303
   878
  assume A: "- a \<le> a" show "0 \<le> a"
haftmann@25303
   879
  proof (rule classical)
haftmann@25303
   880
    assume "\<not> 0 \<le> a"
haftmann@25303
   881
    then have "a < 0" by auto
haftmann@25303
   882
    with A have "- a < 0" by (rule le_less_trans)
haftmann@25303
   883
    then show ?thesis by auto
haftmann@25303
   884
  qed
haftmann@25303
   885
next
haftmann@25303
   886
  assume A: "0 \<le> a" show "- a \<le> a"
haftmann@25303
   887
  proof (rule order_trans)
haftmann@25303
   888
    show "- a \<le> 0" using A by (simp add: minus_le_iff)
haftmann@25303
   889
  next
haftmann@25303
   890
    show "0 \<le> a" using A .
haftmann@25303
   891
  qed
haftmann@25303
   892
qed
haftmann@35036
   893
haftmann@35036
   894
lemma neg_less_nonneg [simp]:
haftmann@35036
   895
  "- a < a \<longleftrightarrow> 0 < a"
haftmann@35036
   896
proof
haftmann@35036
   897
  assume A: "- a < a" show "0 < a"
haftmann@35036
   898
  proof (rule classical)
haftmann@35036
   899
    assume "\<not> 0 < a"
haftmann@35036
   900
    then have "a \<le> 0" by auto
haftmann@35036
   901
    with A have "- a < 0" by (rule less_le_trans)
haftmann@35036
   902
    then show ?thesis by auto
haftmann@35036
   903
  qed
haftmann@35036
   904
next
haftmann@35036
   905
  assume A: "0 < a" show "- a < a"
haftmann@35036
   906
  proof (rule less_trans)
haftmann@35036
   907
    show "- a < 0" using A by (simp add: minus_le_iff)
haftmann@35036
   908
  next
haftmann@35036
   909
    show "0 < a" using A .
haftmann@35036
   910
  qed
haftmann@35036
   911
qed
haftmann@35036
   912
haftmann@35036
   913
lemma less_eq_neg_nonpos [simp]:
haftmann@25303
   914
  "a \<le> - a \<longleftrightarrow> a \<le> 0"
haftmann@25303
   915
proof
haftmann@25303
   916
  assume A: "a \<le> - a" show "a \<le> 0"
haftmann@25303
   917
  proof (rule classical)
haftmann@25303
   918
    assume "\<not> a \<le> 0"
haftmann@25303
   919
    then have "0 < a" by auto
haftmann@25303
   920
    then have "0 < - a" using A by (rule less_le_trans)
haftmann@25303
   921
    then show ?thesis by auto
haftmann@25303
   922
  qed
haftmann@25303
   923
next
haftmann@25303
   924
  assume A: "a \<le> 0" show "a \<le> - a"
haftmann@25303
   925
  proof (rule order_trans)
haftmann@25303
   926
    show "0 \<le> - a" using A by (simp add: minus_le_iff)
haftmann@25303
   927
  next
haftmann@25303
   928
    show "a \<le> 0" using A .
haftmann@25303
   929
  qed
haftmann@25303
   930
qed
haftmann@25303
   931
haftmann@35036
   932
lemma equal_neg_zero [simp]:
haftmann@25303
   933
  "a = - a \<longleftrightarrow> a = 0"
haftmann@25303
   934
proof
haftmann@25303
   935
  assume "a = 0" then show "a = - a" by simp
haftmann@25303
   936
next
haftmann@25303
   937
  assume A: "a = - a" show "a = 0"
haftmann@25303
   938
  proof (cases "0 \<le> a")
haftmann@25303
   939
    case True with A have "0 \<le> - a" by auto
haftmann@25303
   940
    with le_minus_iff have "a \<le> 0" by simp
haftmann@25303
   941
    with True show ?thesis by (auto intro: order_trans)
haftmann@25303
   942
  next
haftmann@25303
   943
    case False then have B: "a \<le> 0" by auto
haftmann@25303
   944
    with A have "- a \<le> 0" by auto
haftmann@25303
   945
    with B show ?thesis by (auto intro: order_trans)
haftmann@25303
   946
  qed
haftmann@25303
   947
qed
haftmann@25303
   948
haftmann@35036
   949
lemma neg_equal_zero [simp]:
haftmann@25303
   950
  "- a = a \<longleftrightarrow> a = 0"
haftmann@35036
   951
  by (auto dest: sym)
haftmann@35036
   952
haftmann@35036
   953
lemma double_zero [simp]:
haftmann@35036
   954
  "a + a = 0 \<longleftrightarrow> a = 0"
haftmann@35036
   955
proof
haftmann@35036
   956
  assume assm: "a + a = 0"
haftmann@35036
   957
  then have a: "- a = a" by (rule minus_unique)
huffman@35216
   958
  then show "a = 0" by (simp only: neg_equal_zero)
haftmann@35036
   959
qed simp
haftmann@35036
   960
haftmann@35036
   961
lemma double_zero_sym [simp]:
haftmann@35036
   962
  "0 = a + a \<longleftrightarrow> a = 0"
haftmann@35036
   963
  by (rule, drule sym) simp_all
haftmann@35036
   964
haftmann@35036
   965
lemma zero_less_double_add_iff_zero_less_single_add [simp]:
haftmann@35036
   966
  "0 < a + a \<longleftrightarrow> 0 < a"
haftmann@35036
   967
proof
haftmann@35036
   968
  assume "0 < a + a"
haftmann@35036
   969
  then have "0 - a < a" by (simp only: diff_less_eq)
haftmann@35036
   970
  then have "- a < a" by simp
huffman@35216
   971
  then show "0 < a" by (simp only: neg_less_nonneg)
haftmann@35036
   972
next
haftmann@35036
   973
  assume "0 < a"
haftmann@35036
   974
  with this have "0 + 0 < a + a"
haftmann@35036
   975
    by (rule add_strict_mono)
haftmann@35036
   976
  then show "0 < a + a" by simp
haftmann@35036
   977
qed
haftmann@35036
   978
haftmann@35036
   979
lemma zero_le_double_add_iff_zero_le_single_add [simp]:
haftmann@35036
   980
  "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
haftmann@35036
   981
  by (auto simp add: le_less)
haftmann@35036
   982
haftmann@35036
   983
lemma double_add_less_zero_iff_single_add_less_zero [simp]:
haftmann@35036
   984
  "a + a < 0 \<longleftrightarrow> a < 0"
haftmann@35036
   985
proof -
haftmann@35036
   986
  have "\<not> a + a < 0 \<longleftrightarrow> \<not> a < 0"
haftmann@35036
   987
    by (simp add: not_less)
haftmann@35036
   988
  then show ?thesis by simp
haftmann@35036
   989
qed
haftmann@35036
   990
haftmann@35036
   991
lemma double_add_le_zero_iff_single_add_le_zero [simp]:
haftmann@35036
   992
  "a + a \<le> 0 \<longleftrightarrow> a \<le> 0" 
haftmann@35036
   993
proof -
haftmann@35036
   994
  have "\<not> a + a \<le> 0 \<longleftrightarrow> \<not> a \<le> 0"
haftmann@35036
   995
    by (simp add: not_le)
haftmann@35036
   996
  then show ?thesis by simp
haftmann@35036
   997
qed
haftmann@35036
   998
haftmann@35036
   999
lemma le_minus_self_iff:
haftmann@35036
  1000
  "a \<le> - a \<longleftrightarrow> a \<le> 0"
haftmann@35036
  1001
proof -
haftmann@35036
  1002
  from add_le_cancel_left [of "- a" "a + a" 0]
haftmann@35036
  1003
  have "a \<le> - a \<longleftrightarrow> a + a \<le> 0" 
haftmann@35036
  1004
    by (simp add: add_assoc [symmetric])
haftmann@35036
  1005
  thus ?thesis by simp
haftmann@35036
  1006
qed
haftmann@35036
  1007
haftmann@35036
  1008
lemma minus_le_self_iff:
haftmann@35036
  1009
  "- a \<le> a \<longleftrightarrow> 0 \<le> a"
haftmann@35036
  1010
proof -
haftmann@35036
  1011
  from add_le_cancel_left [of "- a" 0 "a + a"]
haftmann@35036
  1012
  have "- a \<le> a \<longleftrightarrow> 0 \<le> a + a" 
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_max_eq_min:
haftmann@35036
  1018
  "- max x y = min (-x) (-y)"
haftmann@35036
  1019
  by (auto simp add: max_def min_def)
haftmann@35036
  1020
haftmann@35036
  1021
lemma minus_min_eq_max:
haftmann@35036
  1022
  "- min x y = max (-x) (-y)"
haftmann@35036
  1023
  by (auto simp add: max_def min_def)
haftmann@25303
  1024
haftmann@25267
  1025
end
haftmann@25267
  1026
haftmann@36302
  1027
context ordered_comm_monoid_add
haftmann@36302
  1028
begin
obua@14738
  1029
paulson@15234
  1030
lemma add_increasing:
haftmann@36302
  1031
  "0 \<le> a \<Longrightarrow> b \<le> c \<Longrightarrow> b \<le> a + c"
haftmann@36302
  1032
  by (insert add_mono [of 0 a b c], simp)
obua@14738
  1033
nipkow@15539
  1034
lemma add_increasing2:
haftmann@36302
  1035
  "0 \<le> c \<Longrightarrow> b \<le> a \<Longrightarrow> b \<le> a + c"
haftmann@36302
  1036
  by (simp add: add_increasing add_commute [of a])
nipkow@15539
  1037
paulson@15234
  1038
lemma add_strict_increasing:
haftmann@36302
  1039
  "0 < a \<Longrightarrow> b \<le> c \<Longrightarrow> b < a + c"
haftmann@36302
  1040
  by (insert add_less_le_mono [of 0 a b c], simp)
paulson@15234
  1041
paulson@15234
  1042
lemma add_strict_increasing2:
haftmann@36302
  1043
  "0 \<le> a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@36302
  1044
  by (insert add_le_less_mono [of 0 a b c], simp)
haftmann@36302
  1045
haftmann@36302
  1046
end
paulson@15234
  1047
haftmann@35092
  1048
class abs =
haftmann@35092
  1049
  fixes abs :: "'a \<Rightarrow> 'a"
haftmann@35092
  1050
begin
haftmann@35092
  1051
haftmann@35092
  1052
notation (xsymbols)
haftmann@35092
  1053
  abs  ("\<bar>_\<bar>")
haftmann@35092
  1054
haftmann@35092
  1055
notation (HTML output)
haftmann@35092
  1056
  abs  ("\<bar>_\<bar>")
haftmann@35092
  1057
haftmann@35092
  1058
end
haftmann@35092
  1059
haftmann@35092
  1060
class sgn =
haftmann@35092
  1061
  fixes sgn :: "'a \<Rightarrow> 'a"
haftmann@35092
  1062
haftmann@35092
  1063
class abs_if = minus + uminus + ord + zero + abs +
haftmann@35092
  1064
  assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"
haftmann@35092
  1065
haftmann@35092
  1066
class sgn_if = minus + uminus + zero + one + ord + sgn +
haftmann@35092
  1067
  assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
haftmann@35092
  1068
begin
haftmann@35092
  1069
haftmann@35092
  1070
lemma sgn0 [simp]: "sgn 0 = 0"
haftmann@35092
  1071
  by (simp add:sgn_if)
haftmann@35092
  1072
haftmann@35092
  1073
end
obua@14738
  1074
haftmann@35028
  1075
class ordered_ab_group_add_abs = ordered_ab_group_add + abs +
haftmann@25303
  1076
  assumes abs_ge_zero [simp]: "\<bar>a\<bar> \<ge> 0"
haftmann@25303
  1077
    and abs_ge_self: "a \<le> \<bar>a\<bar>"
haftmann@25303
  1078
    and abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
haftmann@25303
  1079
    and abs_minus_cancel [simp]: "\<bar>-a\<bar> = \<bar>a\<bar>"
haftmann@25303
  1080
    and abs_triangle_ineq: "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25303
  1081
begin
haftmann@25303
  1082
haftmann@25307
  1083
lemma abs_minus_le_zero: "- \<bar>a\<bar> \<le> 0"
haftmann@25307
  1084
  unfolding neg_le_0_iff_le by simp
haftmann@25307
  1085
haftmann@25307
  1086
lemma abs_of_nonneg [simp]:
nipkow@29667
  1087
  assumes nonneg: "0 \<le> a" shows "\<bar>a\<bar> = a"
haftmann@25307
  1088
proof (rule antisym)
haftmann@25307
  1089
  from nonneg le_imp_neg_le have "- a \<le> 0" by simp
haftmann@25307
  1090
  from this nonneg have "- a \<le> a" by (rule order_trans)
haftmann@25307
  1091
  then show "\<bar>a\<bar> \<le> a" by (auto intro: abs_leI)
haftmann@25307
  1092
qed (rule abs_ge_self)
haftmann@25307
  1093
haftmann@25307
  1094
lemma abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
nipkow@29667
  1095
by (rule antisym)
haftmann@36302
  1096
   (auto intro!: abs_ge_self abs_leI order_trans [of "- \<bar>a\<bar>" 0 "\<bar>a\<bar>"])
haftmann@25307
  1097
haftmann@25307
  1098
lemma abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
haftmann@25307
  1099
proof -
haftmann@25307
  1100
  have "\<bar>a\<bar> = 0 \<Longrightarrow> a = 0"
haftmann@25307
  1101
  proof (rule antisym)
haftmann@25307
  1102
    assume zero: "\<bar>a\<bar> = 0"
haftmann@25307
  1103
    with abs_ge_self show "a \<le> 0" by auto
haftmann@25307
  1104
    from zero have "\<bar>-a\<bar> = 0" by simp
haftmann@36302
  1105
    with abs_ge_self [of "- a"] have "- a \<le> 0" by auto
haftmann@25307
  1106
    with neg_le_0_iff_le show "0 \<le> a" by auto
haftmann@25307
  1107
  qed
haftmann@25307
  1108
  then show ?thesis by auto
haftmann@25307
  1109
qed
haftmann@25307
  1110
haftmann@25303
  1111
lemma abs_zero [simp]: "\<bar>0\<bar> = 0"
nipkow@29667
  1112
by simp
avigad@16775
  1113
blanchet@35828
  1114
lemma abs_0_eq [simp, no_atp]: "0 = \<bar>a\<bar> \<longleftrightarrow> a = 0"
haftmann@25303
  1115
proof -
haftmann@25303
  1116
  have "0 = \<bar>a\<bar> \<longleftrightarrow> \<bar>a\<bar> = 0" by (simp only: eq_ac)
haftmann@25303
  1117
  thus ?thesis by simp
haftmann@25303
  1118
qed
haftmann@25303
  1119
haftmann@25303
  1120
lemma abs_le_zero_iff [simp]: "\<bar>a\<bar> \<le> 0 \<longleftrightarrow> a = 0" 
haftmann@25303
  1121
proof
haftmann@25303
  1122
  assume "\<bar>a\<bar> \<le> 0"
haftmann@25303
  1123
  then have "\<bar>a\<bar> = 0" by (rule antisym) simp
haftmann@25303
  1124
  thus "a = 0" by simp
haftmann@25303
  1125
next
haftmann@25303
  1126
  assume "a = 0"
haftmann@25303
  1127
  thus "\<bar>a\<bar> \<le> 0" by simp
haftmann@25303
  1128
qed
haftmann@25303
  1129
haftmann@25303
  1130
lemma zero_less_abs_iff [simp]: "0 < \<bar>a\<bar> \<longleftrightarrow> a \<noteq> 0"
nipkow@29667
  1131
by (simp add: less_le)
haftmann@25303
  1132
haftmann@25303
  1133
lemma abs_not_less_zero [simp]: "\<not> \<bar>a\<bar> < 0"
haftmann@25303
  1134
proof -
haftmann@25303
  1135
  have a: "\<And>x y. x \<le> y \<Longrightarrow> \<not> y < x" by auto
haftmann@25303
  1136
  show ?thesis by (simp add: a)
haftmann@25303
  1137
qed
avigad@16775
  1138
haftmann@25303
  1139
lemma abs_ge_minus_self: "- a \<le> \<bar>a\<bar>"
haftmann@25303
  1140
proof -
haftmann@25303
  1141
  have "- a \<le> \<bar>-a\<bar>" by (rule abs_ge_self)
haftmann@25303
  1142
  then show ?thesis by simp
haftmann@25303
  1143
qed
haftmann@25303
  1144
haftmann@25303
  1145
lemma abs_minus_commute: 
haftmann@25303
  1146
  "\<bar>a - b\<bar> = \<bar>b - a\<bar>"
haftmann@25303
  1147
proof -
haftmann@25303
  1148
  have "\<bar>a - b\<bar> = \<bar>- (a - b)\<bar>" by (simp only: abs_minus_cancel)
haftmann@25303
  1149
  also have "... = \<bar>b - a\<bar>" by simp
haftmann@25303
  1150
  finally show ?thesis .
haftmann@25303
  1151
qed
haftmann@25303
  1152
haftmann@25303
  1153
lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a"
nipkow@29667
  1154
by (rule abs_of_nonneg, rule less_imp_le)
avigad@16775
  1155
haftmann@25303
  1156
lemma abs_of_nonpos [simp]:
nipkow@29667
  1157
  assumes "a \<le> 0" shows "\<bar>a\<bar> = - a"
haftmann@25303
  1158
proof -
haftmann@25303
  1159
  let ?b = "- a"
haftmann@25303
  1160
  have "- ?b \<le> 0 \<Longrightarrow> \<bar>- ?b\<bar> = - (- ?b)"
haftmann@25303
  1161
  unfolding abs_minus_cancel [of "?b"]
haftmann@25303
  1162
  unfolding neg_le_0_iff_le [of "?b"]
haftmann@25303
  1163
  unfolding minus_minus by (erule abs_of_nonneg)
haftmann@25303
  1164
  then show ?thesis using assms by auto
haftmann@25303
  1165
qed
haftmann@25303
  1166
  
haftmann@25303
  1167
lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a"
nipkow@29667
  1168
by (rule abs_of_nonpos, rule less_imp_le)
haftmann@25303
  1169
haftmann@25303
  1170
lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b"
nipkow@29667
  1171
by (insert abs_ge_self, blast intro: order_trans)
haftmann@25303
  1172
haftmann@25303
  1173
lemma abs_le_D2: "\<bar>a\<bar> \<le> b \<Longrightarrow> - a \<le> b"
haftmann@36302
  1174
by (insert abs_le_D1 [of "- a"], simp)
haftmann@25303
  1175
haftmann@25303
  1176
lemma abs_le_iff: "\<bar>a\<bar> \<le> b \<longleftrightarrow> a \<le> b \<and> - a \<le> b"
nipkow@29667
  1177
by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
haftmann@25303
  1178
haftmann@25303
  1179
lemma abs_triangle_ineq2: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>a - b\<bar>"
haftmann@36302
  1180
proof -
haftmann@36302
  1181
  have "\<bar>a\<bar> = \<bar>b + (a - b)\<bar>"
haftmann@36302
  1182
    by (simp add: algebra_simps add_diff_cancel)
haftmann@36302
  1183
  then have "\<bar>a\<bar> \<le> \<bar>b\<bar> + \<bar>a - b\<bar>"
haftmann@36302
  1184
    by (simp add: abs_triangle_ineq)
haftmann@36302
  1185
  then show ?thesis
haftmann@36302
  1186
    by (simp add: algebra_simps)
haftmann@36302
  1187
qed
haftmann@36302
  1188
haftmann@36302
  1189
lemma abs_triangle_ineq2_sym: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>b - a\<bar>"
haftmann@36302
  1190
  by (simp only: abs_minus_commute [of b] abs_triangle_ineq2)
avigad@16775
  1191
haftmann@25303
  1192
lemma abs_triangle_ineq3: "\<bar>\<bar>a\<bar> - \<bar>b\<bar>\<bar> \<le> \<bar>a - b\<bar>"
haftmann@36302
  1193
  by (simp add: abs_le_iff abs_triangle_ineq2 abs_triangle_ineq2_sym)
avigad@16775
  1194
haftmann@25303
  1195
lemma abs_triangle_ineq4: "\<bar>a - b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25303
  1196
proof -
haftmann@36302
  1197
  have "\<bar>a - b\<bar> = \<bar>a + - b\<bar>" by (subst diff_minus, rule refl)
haftmann@36302
  1198
  also have "... \<le> \<bar>a\<bar> + \<bar>- b\<bar>" by (rule abs_triangle_ineq)
nipkow@29667
  1199
  finally show ?thesis by simp
haftmann@25303
  1200
qed
avigad@16775
  1201
haftmann@25303
  1202
lemma abs_diff_triangle_ineq: "\<bar>a + b - (c + d)\<bar> \<le> \<bar>a - c\<bar> + \<bar>b - d\<bar>"
haftmann@25303
  1203
proof -
haftmann@25303
  1204
  have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: diff_minus add_ac)
haftmann@25303
  1205
  also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq)
haftmann@25303
  1206
  finally show ?thesis .
haftmann@25303
  1207
qed
avigad@16775
  1208
haftmann@25303
  1209
lemma abs_add_abs [simp]:
haftmann@25303
  1210
  "\<bar>\<bar>a\<bar> + \<bar>b\<bar>\<bar> = \<bar>a\<bar> + \<bar>b\<bar>" (is "?L = ?R")
haftmann@25303
  1211
proof (rule antisym)
haftmann@25303
  1212
  show "?L \<ge> ?R" by(rule abs_ge_self)
haftmann@25303
  1213
next
haftmann@25303
  1214
  have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq)
haftmann@25303
  1215
  also have "\<dots> = ?R" by simp
haftmann@25303
  1216
  finally show "?L \<le> ?R" .
haftmann@25303
  1217
qed
haftmann@25303
  1218
haftmann@25303
  1219
end
obua@14738
  1220
obua@15178
  1221
haftmann@25090
  1222
subsection {* Tools setup *}
haftmann@25090
  1223
blanchet@35828
  1224
lemma add_mono_thms_linordered_semiring [no_atp]:
haftmann@35028
  1225
  fixes i j k :: "'a\<Colon>ordered_ab_semigroup_add"
haftmann@25077
  1226
  shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1227
    and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1228
    and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1229
    and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"
haftmann@25077
  1230
by (rule add_mono, clarify+)+
haftmann@25077
  1231
blanchet@35828
  1232
lemma add_mono_thms_linordered_field [no_atp]:
haftmann@35028
  1233
  fixes i j k :: "'a\<Colon>ordered_cancel_ab_semigroup_add"
haftmann@25077
  1234
  shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1235
    and "i = j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1236
    and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1237
    and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1238
    and "i < j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1239
by (auto intro: add_strict_right_mono add_strict_left_mono
haftmann@25077
  1240
  add_less_le_mono add_le_less_mono add_strict_mono)
haftmann@25077
  1241
haftmann@33364
  1242
code_modulename SML
haftmann@35050
  1243
  Groups Arith
haftmann@33364
  1244
haftmann@33364
  1245
code_modulename OCaml
haftmann@35050
  1246
  Groups Arith
haftmann@33364
  1247
haftmann@33364
  1248
code_modulename Haskell
haftmann@35050
  1249
  Groups Arith
haftmann@33364
  1250
haftmann@37889
  1251
haftmann@37889
  1252
text {* Legacy *}
haftmann@37889
  1253
haftmann@37889
  1254
lemmas diff_def = diff_minus
haftmann@37889
  1255
obua@14738
  1256
end