src/HOL/Groups.thy
author wenzelm
Mon Aug 22 23:39:05 2011 +0200 (2011-08-22)
changeset 44433 9fbee4aab115
parent 44348 40101794c52f
child 44848 f4d0b060c7ca
permissions -rw-r--r--
special treatment of structure index 1 in Pure, including legacy warning;
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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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  val name = "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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  val name = "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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  val name = "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
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   366
haftmann@25062
   367
lemma neg_equal_0_iff_equal [simp]:
haftmann@25062
   368
  "- a = 0 \<longleftrightarrow> a = 0"
nipkow@29667
   369
by (subst neg_equal_iff_equal [symmetric], simp)
obua@14738
   370
haftmann@25062
   371
lemma neg_0_equal_iff_equal [simp]:
haftmann@25062
   372
  "0 = - a \<longleftrightarrow> 0 = a"
nipkow@29667
   373
by (subst neg_equal_iff_equal [symmetric], simp)
obua@14738
   374
obua@14738
   375
text{*The next two equations can make the simplifier loop!*}
obua@14738
   376
haftmann@25062
   377
lemma equation_minus_iff:
haftmann@25062
   378
  "a = - b \<longleftrightarrow> b = - a"
obua@14738
   379
proof -
haftmann@25062
   380
  have "- (- a) = - b \<longleftrightarrow> - a = b" by (rule neg_equal_iff_equal)
haftmann@25062
   381
  thus ?thesis by (simp add: eq_commute)
haftmann@25062
   382
qed
haftmann@25062
   383
haftmann@25062
   384
lemma minus_equation_iff:
haftmann@25062
   385
  "- a = b \<longleftrightarrow> - b = a"
haftmann@25062
   386
proof -
haftmann@25062
   387
  have "- a = - (- b) \<longleftrightarrow> a = -b" by (rule neg_equal_iff_equal)
obua@14738
   388
  thus ?thesis by (simp add: eq_commute)
obua@14738
   389
qed
obua@14738
   390
huffman@28130
   391
lemma diff_add_cancel: "a - b + b = a"
nipkow@29667
   392
by (simp add: diff_minus add_assoc)
huffman@28130
   393
huffman@28130
   394
lemma add_diff_cancel: "a + b - b = a"
nipkow@29667
   395
by (simp add: diff_minus add_assoc)
nipkow@29667
   396
haftmann@36348
   397
declare diff_minus[symmetric, algebra_simps, field_simps]
huffman@28130
   398
huffman@29914
   399
lemma eq_neg_iff_add_eq_0: "a = - b \<longleftrightarrow> a + b = 0"
huffman@29914
   400
proof
huffman@29914
   401
  assume "a = - b" then show "a + b = 0" by simp
huffman@29914
   402
next
huffman@29914
   403
  assume "a + b = 0"
huffman@29914
   404
  moreover have "a + (b + - b) = (a + b) + - b"
huffman@29914
   405
    by (simp only: add_assoc)
huffman@29914
   406
  ultimately show "a = - b" by simp
huffman@29914
   407
qed
huffman@29914
   408
huffman@44348
   409
lemma add_eq_0_iff: "x + y = 0 \<longleftrightarrow> y = - x"
huffman@44348
   410
  unfolding eq_neg_iff_add_eq_0 [symmetric]
huffman@44348
   411
  by (rule equation_minus_iff)
huffman@44348
   412
haftmann@25062
   413
end
haftmann@25062
   414
haftmann@25762
   415
class ab_group_add = minus + uminus + comm_monoid_add +
haftmann@25062
   416
  assumes ab_left_minus: "- a + a = 0"
haftmann@25062
   417
  assumes ab_diff_minus: "a - b = a + (- b)"
haftmann@25267
   418
begin
haftmann@25062
   419
haftmann@25267
   420
subclass group_add
haftmann@28823
   421
  proof qed (simp_all add: ab_left_minus ab_diff_minus)
haftmann@25062
   422
huffman@29904
   423
subclass cancel_comm_monoid_add
haftmann@28823
   424
proof
haftmann@25062
   425
  fix a b c :: 'a
haftmann@25062
   426
  assume "a + b = a + c"
haftmann@25062
   427
  then have "- a + a + b = - a + a + c"
haftmann@25062
   428
    unfolding add_assoc by simp
haftmann@25062
   429
  then show "b = c" by simp
haftmann@25062
   430
qed
haftmann@25062
   431
haftmann@36348
   432
lemma uminus_add_conv_diff[algebra_simps, field_simps]:
haftmann@25062
   433
  "- a + b = b - a"
nipkow@29667
   434
by (simp add:diff_minus add_commute)
haftmann@25062
   435
haftmann@25062
   436
lemma minus_add_distrib [simp]:
haftmann@25062
   437
  "- (a + b) = - a + - b"
huffman@34146
   438
by (rule minus_unique) (simp add: add_ac)
haftmann@25062
   439
haftmann@25062
   440
lemma minus_diff_eq [simp]:
haftmann@25062
   441
  "- (a - b) = b - a"
nipkow@29667
   442
by (simp add: diff_minus add_commute)
haftmann@25077
   443
haftmann@36348
   444
lemma add_diff_eq[algebra_simps, field_simps]: "a + (b - c) = (a + b) - c"
nipkow@29667
   445
by (simp add: diff_minus add_ac)
haftmann@25077
   446
haftmann@36348
   447
lemma diff_add_eq[algebra_simps, field_simps]: "(a - b) + c = (a + c) - b"
nipkow@29667
   448
by (simp add: diff_minus add_ac)
haftmann@25077
   449
haftmann@36348
   450
lemma diff_eq_eq[algebra_simps, field_simps]: "a - b = c \<longleftrightarrow> a = c + b"
nipkow@29667
   451
by (auto simp add: diff_minus add_assoc)
haftmann@25077
   452
haftmann@36348
   453
lemma eq_diff_eq[algebra_simps, field_simps]: "a = c - b \<longleftrightarrow> a + b = c"
nipkow@29667
   454
by (auto simp add: diff_minus add_assoc)
haftmann@25077
   455
haftmann@36348
   456
lemma diff_diff_eq[algebra_simps, field_simps]: "(a - b) - c = a - (b + c)"
nipkow@29667
   457
by (simp add: diff_minus add_ac)
haftmann@25077
   458
haftmann@36348
   459
lemma diff_diff_eq2[algebra_simps, field_simps]: "a - (b - c) = (a + c) - b"
nipkow@29667
   460
by (simp add: diff_minus add_ac)
haftmann@25077
   461
haftmann@25077
   462
lemma eq_iff_diff_eq_0: "a = b \<longleftrightarrow> a - b = 0"
nipkow@29667
   463
by (simp add: algebra_simps)
haftmann@25077
   464
huffman@35216
   465
(* FIXME: duplicates right_minus_eq from class group_add *)
huffman@35216
   466
(* but only this one is declared as a simp rule. *)
blanchet@35828
   467
lemma diff_eq_0_iff_eq [simp, no_atp]: "a - b = 0 \<longleftrightarrow> a = b"
huffman@44348
   468
  by (rule right_minus_eq)
huffman@30629
   469
haftmann@37884
   470
lemma diff_eq_diff_eq:
haftmann@37884
   471
  "a - b = c - d \<Longrightarrow> a = b \<longleftrightarrow> c = d"
haftmann@37884
   472
  by (auto simp add: algebra_simps)
haftmann@37884
   473
  
haftmann@25062
   474
end
obua@14738
   475
haftmann@37884
   476
obua@14738
   477
subsection {* (Partially) Ordered Groups *} 
obua@14738
   478
haftmann@35301
   479
text {*
haftmann@35301
   480
  The theory of partially ordered groups is taken from the books:
haftmann@35301
   481
  \begin{itemize}
haftmann@35301
   482
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
haftmann@35301
   483
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
haftmann@35301
   484
  \end{itemize}
haftmann@35301
   485
  Most of the used notions can also be looked up in 
haftmann@35301
   486
  \begin{itemize}
haftmann@35301
   487
  \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
haftmann@35301
   488
  \item \emph{Algebra I} by van der Waerden, Springer.
haftmann@35301
   489
  \end{itemize}
haftmann@35301
   490
*}
haftmann@35301
   491
haftmann@35028
   492
class ordered_ab_semigroup_add = order + ab_semigroup_add +
haftmann@25062
   493
  assumes add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b"
haftmann@25062
   494
begin
haftmann@24380
   495
haftmann@25062
   496
lemma add_right_mono:
haftmann@25062
   497
  "a \<le> b \<Longrightarrow> a + c \<le> b + c"
nipkow@29667
   498
by (simp add: add_commute [of _ c] add_left_mono)
obua@14738
   499
obua@14738
   500
text {* non-strict, in both arguments *}
obua@14738
   501
lemma add_mono:
haftmann@25062
   502
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d"
obua@14738
   503
  apply (erule add_right_mono [THEN order_trans])
obua@14738
   504
  apply (simp add: add_commute add_left_mono)
obua@14738
   505
  done
obua@14738
   506
haftmann@25062
   507
end
haftmann@25062
   508
haftmann@35028
   509
class ordered_cancel_ab_semigroup_add =
haftmann@35028
   510
  ordered_ab_semigroup_add + cancel_ab_semigroup_add
haftmann@25062
   511
begin
haftmann@25062
   512
obua@14738
   513
lemma add_strict_left_mono:
haftmann@25062
   514
  "a < b \<Longrightarrow> c + a < c + b"
nipkow@29667
   515
by (auto simp add: less_le add_left_mono)
obua@14738
   516
obua@14738
   517
lemma add_strict_right_mono:
haftmann@25062
   518
  "a < b \<Longrightarrow> a + c < b + c"
nipkow@29667
   519
by (simp add: add_commute [of _ c] add_strict_left_mono)
obua@14738
   520
obua@14738
   521
text{*Strict monotonicity in both arguments*}
haftmann@25062
   522
lemma add_strict_mono:
haftmann@25062
   523
  "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
haftmann@25062
   524
apply (erule add_strict_right_mono [THEN less_trans])
obua@14738
   525
apply (erule add_strict_left_mono)
obua@14738
   526
done
obua@14738
   527
obua@14738
   528
lemma add_less_le_mono:
haftmann@25062
   529
  "a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c < b + d"
haftmann@25062
   530
apply (erule add_strict_right_mono [THEN less_le_trans])
haftmann@25062
   531
apply (erule add_left_mono)
obua@14738
   532
done
obua@14738
   533
obua@14738
   534
lemma add_le_less_mono:
haftmann@25062
   535
  "a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
haftmann@25062
   536
apply (erule add_right_mono [THEN le_less_trans])
obua@14738
   537
apply (erule add_strict_left_mono) 
obua@14738
   538
done
obua@14738
   539
haftmann@25062
   540
end
haftmann@25062
   541
haftmann@35028
   542
class ordered_ab_semigroup_add_imp_le =
haftmann@35028
   543
  ordered_cancel_ab_semigroup_add +
haftmann@25062
   544
  assumes add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b"
haftmann@25062
   545
begin
haftmann@25062
   546
obua@14738
   547
lemma add_less_imp_less_left:
nipkow@29667
   548
  assumes less: "c + a < c + b" shows "a < b"
obua@14738
   549
proof -
obua@14738
   550
  from less have le: "c + a <= c + b" by (simp add: order_le_less)
obua@14738
   551
  have "a <= b" 
obua@14738
   552
    apply (insert le)
obua@14738
   553
    apply (drule add_le_imp_le_left)
obua@14738
   554
    by (insert le, drule add_le_imp_le_left, assumption)
obua@14738
   555
  moreover have "a \<noteq> b"
obua@14738
   556
  proof (rule ccontr)
obua@14738
   557
    assume "~(a \<noteq> b)"
obua@14738
   558
    then have "a = b" by simp
obua@14738
   559
    then have "c + a = c + b" by simp
obua@14738
   560
    with less show "False"by simp
obua@14738
   561
  qed
obua@14738
   562
  ultimately show "a < b" by (simp add: order_le_less)
obua@14738
   563
qed
obua@14738
   564
obua@14738
   565
lemma add_less_imp_less_right:
haftmann@25062
   566
  "a + c < b + c \<Longrightarrow> a < b"
obua@14738
   567
apply (rule add_less_imp_less_left [of c])
obua@14738
   568
apply (simp add: add_commute)  
obua@14738
   569
done
obua@14738
   570
obua@14738
   571
lemma add_less_cancel_left [simp]:
haftmann@25062
   572
  "c + a < c + b \<longleftrightarrow> a < b"
nipkow@29667
   573
by (blast intro: add_less_imp_less_left add_strict_left_mono) 
obua@14738
   574
obua@14738
   575
lemma add_less_cancel_right [simp]:
haftmann@25062
   576
  "a + c < b + c \<longleftrightarrow> a < b"
nipkow@29667
   577
by (blast intro: add_less_imp_less_right add_strict_right_mono)
obua@14738
   578
obua@14738
   579
lemma add_le_cancel_left [simp]:
haftmann@25062
   580
  "c + a \<le> c + b \<longleftrightarrow> a \<le> b"
nipkow@29667
   581
by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono) 
obua@14738
   582
obua@14738
   583
lemma add_le_cancel_right [simp]:
haftmann@25062
   584
  "a + c \<le> b + c \<longleftrightarrow> a \<le> b"
nipkow@29667
   585
by (simp add: add_commute [of a c] add_commute [of b c])
obua@14738
   586
obua@14738
   587
lemma add_le_imp_le_right:
haftmann@25062
   588
  "a + c \<le> b + c \<Longrightarrow> a \<le> b"
nipkow@29667
   589
by simp
haftmann@25062
   590
haftmann@25077
   591
lemma max_add_distrib_left:
haftmann@25077
   592
  "max x y + z = max (x + z) (y + z)"
haftmann@25077
   593
  unfolding max_def by auto
haftmann@25077
   594
haftmann@25077
   595
lemma min_add_distrib_left:
haftmann@25077
   596
  "min x y + z = min (x + z) (y + z)"
haftmann@25077
   597
  unfolding min_def by auto
haftmann@25077
   598
haftmann@25062
   599
end
haftmann@25062
   600
haftmann@25303
   601
subsection {* Support for reasoning about signs *}
haftmann@25303
   602
haftmann@35028
   603
class ordered_comm_monoid_add =
haftmann@35028
   604
  ordered_cancel_ab_semigroup_add + comm_monoid_add
haftmann@25303
   605
begin
haftmann@25303
   606
haftmann@25303
   607
lemma add_pos_nonneg:
nipkow@29667
   608
  assumes "0 < a" and "0 \<le> b" shows "0 < a + b"
haftmann@25303
   609
proof -
haftmann@25303
   610
  have "0 + 0 < a + b" 
haftmann@25303
   611
    using assms by (rule add_less_le_mono)
haftmann@25303
   612
  then show ?thesis by simp
haftmann@25303
   613
qed
haftmann@25303
   614
haftmann@25303
   615
lemma add_pos_pos:
nipkow@29667
   616
  assumes "0 < a" and "0 < b" shows "0 < a + b"
nipkow@29667
   617
by (rule add_pos_nonneg) (insert assms, auto)
haftmann@25303
   618
haftmann@25303
   619
lemma add_nonneg_pos:
nipkow@29667
   620
  assumes "0 \<le> a" and "0 < b" shows "0 < a + b"
haftmann@25303
   621
proof -
haftmann@25303
   622
  have "0 + 0 < a + b" 
haftmann@25303
   623
    using assms by (rule add_le_less_mono)
haftmann@25303
   624
  then show ?thesis by simp
haftmann@25303
   625
qed
haftmann@25303
   626
huffman@36977
   627
lemma add_nonneg_nonneg [simp]:
nipkow@29667
   628
  assumes "0 \<le> a" and "0 \<le> b" shows "0 \<le> a + b"
haftmann@25303
   629
proof -
haftmann@25303
   630
  have "0 + 0 \<le> a + b" 
haftmann@25303
   631
    using assms by (rule add_mono)
haftmann@25303
   632
  then show ?thesis by simp
haftmann@25303
   633
qed
haftmann@25303
   634
huffman@30691
   635
lemma add_neg_nonpos:
nipkow@29667
   636
  assumes "a < 0" and "b \<le> 0" shows "a + b < 0"
haftmann@25303
   637
proof -
haftmann@25303
   638
  have "a + b < 0 + 0"
haftmann@25303
   639
    using assms by (rule add_less_le_mono)
haftmann@25303
   640
  then show ?thesis by simp
haftmann@25303
   641
qed
haftmann@25303
   642
haftmann@25303
   643
lemma add_neg_neg: 
nipkow@29667
   644
  assumes "a < 0" and "b < 0" shows "a + b < 0"
nipkow@29667
   645
by (rule add_neg_nonpos) (insert assms, auto)
haftmann@25303
   646
haftmann@25303
   647
lemma add_nonpos_neg:
nipkow@29667
   648
  assumes "a \<le> 0" and "b < 0" shows "a + b < 0"
haftmann@25303
   649
proof -
haftmann@25303
   650
  have "a + b < 0 + 0"
haftmann@25303
   651
    using assms by (rule add_le_less_mono)
haftmann@25303
   652
  then show ?thesis by simp
haftmann@25303
   653
qed
haftmann@25303
   654
haftmann@25303
   655
lemma add_nonpos_nonpos:
nipkow@29667
   656
  assumes "a \<le> 0" and "b \<le> 0" shows "a + b \<le> 0"
haftmann@25303
   657
proof -
haftmann@25303
   658
  have "a + b \<le> 0 + 0"
haftmann@25303
   659
    using assms by (rule add_mono)
haftmann@25303
   660
  then show ?thesis by simp
haftmann@25303
   661
qed
haftmann@25303
   662
huffman@30691
   663
lemmas add_sign_intros =
huffman@30691
   664
  add_pos_nonneg add_pos_pos add_nonneg_pos add_nonneg_nonneg
huffman@30691
   665
  add_neg_nonpos add_neg_neg add_nonpos_neg add_nonpos_nonpos
huffman@30691
   666
huffman@29886
   667
lemma add_nonneg_eq_0_iff:
huffman@29886
   668
  assumes x: "0 \<le> x" and y: "0 \<le> y"
huffman@29886
   669
  shows "x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@29886
   670
proof (intro iffI conjI)
huffman@29886
   671
  have "x = x + 0" by simp
huffman@29886
   672
  also have "x + 0 \<le> x + y" using y by (rule add_left_mono)
huffman@29886
   673
  also assume "x + y = 0"
huffman@29886
   674
  also have "0 \<le> x" using x .
huffman@29886
   675
  finally show "x = 0" .
huffman@29886
   676
next
huffman@29886
   677
  have "y = 0 + y" by simp
huffman@29886
   678
  also have "0 + y \<le> x + y" using x by (rule add_right_mono)
huffman@29886
   679
  also assume "x + y = 0"
huffman@29886
   680
  also have "0 \<le> y" using y .
huffman@29886
   681
  finally show "y = 0" .
huffman@29886
   682
next
huffman@29886
   683
  assume "x = 0 \<and> y = 0"
huffman@29886
   684
  then show "x + y = 0" by simp
huffman@29886
   685
qed
huffman@29886
   686
haftmann@25303
   687
end
haftmann@25303
   688
haftmann@35028
   689
class ordered_ab_group_add =
haftmann@35028
   690
  ab_group_add + ordered_ab_semigroup_add
haftmann@25062
   691
begin
haftmann@25062
   692
haftmann@35028
   693
subclass ordered_cancel_ab_semigroup_add ..
haftmann@25062
   694
haftmann@35028
   695
subclass ordered_ab_semigroup_add_imp_le
haftmann@28823
   696
proof
haftmann@25062
   697
  fix a b c :: 'a
haftmann@25062
   698
  assume "c + a \<le> c + b"
haftmann@25062
   699
  hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono)
haftmann@25062
   700
  hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc)
haftmann@25062
   701
  thus "a \<le> b" by simp
haftmann@25062
   702
qed
haftmann@25062
   703
haftmann@35028
   704
subclass ordered_comm_monoid_add ..
haftmann@25303
   705
haftmann@25077
   706
lemma max_diff_distrib_left:
haftmann@25077
   707
  shows "max x y - z = max (x - z) (y - z)"
nipkow@29667
   708
by (simp add: diff_minus, rule max_add_distrib_left) 
haftmann@25077
   709
haftmann@25077
   710
lemma min_diff_distrib_left:
haftmann@25077
   711
  shows "min x y - z = min (x - z) (y - z)"
nipkow@29667
   712
by (simp add: diff_minus, rule min_add_distrib_left) 
haftmann@25077
   713
haftmann@25077
   714
lemma le_imp_neg_le:
nipkow@29667
   715
  assumes "a \<le> b" shows "-b \<le> -a"
haftmann@25077
   716
proof -
nipkow@29667
   717
  have "-a+a \<le> -a+b" using `a \<le> b` by (rule add_left_mono) 
nipkow@29667
   718
  hence "0 \<le> -a+b" by simp
nipkow@29667
   719
  hence "0 + (-b) \<le> (-a + b) + (-b)" by (rule add_right_mono) 
nipkow@29667
   720
  thus ?thesis by (simp add: add_assoc)
haftmann@25077
   721
qed
haftmann@25077
   722
haftmann@25077
   723
lemma neg_le_iff_le [simp]: "- b \<le> - a \<longleftrightarrow> a \<le> b"
haftmann@25077
   724
proof 
haftmann@25077
   725
  assume "- b \<le> - a"
nipkow@29667
   726
  hence "- (- a) \<le> - (- b)" by (rule le_imp_neg_le)
haftmann@25077
   727
  thus "a\<le>b" by simp
haftmann@25077
   728
next
haftmann@25077
   729
  assume "a\<le>b"
haftmann@25077
   730
  thus "-b \<le> -a" by (rule le_imp_neg_le)
haftmann@25077
   731
qed
haftmann@25077
   732
haftmann@25077
   733
lemma neg_le_0_iff_le [simp]: "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"
nipkow@29667
   734
by (subst neg_le_iff_le [symmetric], simp)
haftmann@25077
   735
haftmann@25077
   736
lemma neg_0_le_iff_le [simp]: "0 \<le> - a \<longleftrightarrow> a \<le> 0"
nipkow@29667
   737
by (subst neg_le_iff_le [symmetric], simp)
haftmann@25077
   738
haftmann@25077
   739
lemma neg_less_iff_less [simp]: "- b < - a \<longleftrightarrow> a < b"
nipkow@29667
   740
by (force simp add: less_le) 
haftmann@25077
   741
haftmann@25077
   742
lemma neg_less_0_iff_less [simp]: "- a < 0 \<longleftrightarrow> 0 < a"
nipkow@29667
   743
by (subst neg_less_iff_less [symmetric], simp)
haftmann@25077
   744
haftmann@25077
   745
lemma neg_0_less_iff_less [simp]: "0 < - a \<longleftrightarrow> a < 0"
nipkow@29667
   746
by (subst neg_less_iff_less [symmetric], simp)
haftmann@25077
   747
haftmann@25077
   748
text{*The next several equations can make the simplifier loop!*}
haftmann@25077
   749
haftmann@25077
   750
lemma less_minus_iff: "a < - b \<longleftrightarrow> b < - a"
haftmann@25077
   751
proof -
haftmann@25077
   752
  have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
haftmann@25077
   753
  thus ?thesis by simp
haftmann@25077
   754
qed
haftmann@25077
   755
haftmann@25077
   756
lemma minus_less_iff: "- a < b \<longleftrightarrow> - b < a"
haftmann@25077
   757
proof -
haftmann@25077
   758
  have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
haftmann@25077
   759
  thus ?thesis by simp
haftmann@25077
   760
qed
haftmann@25077
   761
haftmann@25077
   762
lemma le_minus_iff: "a \<le> - b \<longleftrightarrow> b \<le> - a"
haftmann@25077
   763
proof -
haftmann@25077
   764
  have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff)
haftmann@25077
   765
  have "(- (- a) <= -b) = (b <= - a)" 
haftmann@25077
   766
    apply (auto simp only: le_less)
haftmann@25077
   767
    apply (drule mm)
haftmann@25077
   768
    apply (simp_all)
haftmann@25077
   769
    apply (drule mm[simplified], assumption)
haftmann@25077
   770
    done
haftmann@25077
   771
  then show ?thesis by simp
haftmann@25077
   772
qed
haftmann@25077
   773
haftmann@25077
   774
lemma minus_le_iff: "- a \<le> b \<longleftrightarrow> - b \<le> a"
nipkow@29667
   775
by (auto simp add: le_less minus_less_iff)
haftmann@25077
   776
haftmann@37884
   777
lemma diff_less_0_iff_less [simp, no_atp]:
haftmann@37884
   778
  "a - b < 0 \<longleftrightarrow> a < b"
haftmann@25077
   779
proof -
haftmann@37884
   780
  have "a - b < 0 \<longleftrightarrow> a + (- b) < b + (- b)" by (simp add: diff_minus)
haftmann@37884
   781
  also have "... \<longleftrightarrow> a < b" by (simp only: add_less_cancel_right)
haftmann@25077
   782
  finally show ?thesis .
haftmann@25077
   783
qed
haftmann@25077
   784
haftmann@37884
   785
lemmas less_iff_diff_less_0 = diff_less_0_iff_less [symmetric]
haftmann@37884
   786
haftmann@36348
   787
lemma diff_less_eq[algebra_simps, field_simps]: "a - b < c \<longleftrightarrow> a < c + b"
haftmann@25077
   788
apply (subst less_iff_diff_less_0 [of a])
haftmann@25077
   789
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
haftmann@25077
   790
apply (simp add: diff_minus add_ac)
haftmann@25077
   791
done
haftmann@25077
   792
haftmann@36348
   793
lemma less_diff_eq[algebra_simps, field_simps]: "a < c - b \<longleftrightarrow> a + b < c"
haftmann@36302
   794
apply (subst less_iff_diff_less_0 [of "a + b"])
haftmann@25077
   795
apply (subst less_iff_diff_less_0 [of a])
haftmann@25077
   796
apply (simp add: diff_minus add_ac)
haftmann@25077
   797
done
haftmann@25077
   798
haftmann@36348
   799
lemma diff_le_eq[algebra_simps, field_simps]: "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
nipkow@29667
   800
by (auto simp add: le_less diff_less_eq diff_add_cancel add_diff_cancel)
haftmann@25077
   801
haftmann@36348
   802
lemma le_diff_eq[algebra_simps, field_simps]: "a \<le> c - b \<longleftrightarrow> a + b \<le> c"
nipkow@29667
   803
by (auto simp add: le_less less_diff_eq diff_add_cancel add_diff_cancel)
haftmann@25077
   804
haftmann@37884
   805
lemma diff_le_0_iff_le [simp, no_atp]:
haftmann@37884
   806
  "a - b \<le> 0 \<longleftrightarrow> a \<le> b"
haftmann@37884
   807
  by (simp add: algebra_simps)
haftmann@37884
   808
haftmann@37884
   809
lemmas le_iff_diff_le_0 = diff_le_0_iff_le [symmetric]
haftmann@37884
   810
haftmann@37884
   811
lemma diff_eq_diff_less:
haftmann@37884
   812
  "a - b = c - d \<Longrightarrow> a < b \<longleftrightarrow> c < d"
haftmann@37884
   813
  by (auto simp only: less_iff_diff_less_0 [of a b] less_iff_diff_less_0 [of c d])
haftmann@37884
   814
haftmann@37889
   815
lemma diff_eq_diff_less_eq:
haftmann@37889
   816
  "a - b = c - d \<Longrightarrow> a \<le> b \<longleftrightarrow> c \<le> d"
haftmann@37889
   817
  by (auto simp only: le_iff_diff_le_0 [of a b] le_iff_diff_le_0 [of c d])
haftmann@25077
   818
haftmann@25077
   819
end
haftmann@25077
   820
wenzelm@37986
   821
use "Tools/abel_cancel.ML"
haftmann@37884
   822
haftmann@37889
   823
simproc_setup abel_cancel_sum
haftmann@37889
   824
  ("a + b::'a::ab_group_add" | "a - b::'a::ab_group_add") =
haftmann@37889
   825
  {* fn phi => Abel_Cancel.sum_proc *}
haftmann@37889
   826
haftmann@37889
   827
simproc_setup abel_cancel_relation
haftmann@37889
   828
  ("a < (b::'a::ordered_ab_group_add)" | "a \<le> (b::'a::ordered_ab_group_add)" | "c = (d::'b::ab_group_add)") =
haftmann@37889
   829
  {* fn phi => Abel_Cancel.rel_proc *}
haftmann@37884
   830
haftmann@35028
   831
class linordered_ab_semigroup_add =
haftmann@35028
   832
  linorder + ordered_ab_semigroup_add
haftmann@25062
   833
haftmann@35028
   834
class linordered_cancel_ab_semigroup_add =
haftmann@35028
   835
  linorder + ordered_cancel_ab_semigroup_add
haftmann@25267
   836
begin
haftmann@25062
   837
haftmann@35028
   838
subclass linordered_ab_semigroup_add ..
haftmann@25062
   839
haftmann@35028
   840
subclass ordered_ab_semigroup_add_imp_le
haftmann@28823
   841
proof
haftmann@25062
   842
  fix a b c :: 'a
haftmann@25062
   843
  assume le: "c + a <= c + b"  
haftmann@25062
   844
  show "a <= b"
haftmann@25062
   845
  proof (rule ccontr)
haftmann@25062
   846
    assume w: "~ a \<le> b"
haftmann@25062
   847
    hence "b <= a" by (simp add: linorder_not_le)
haftmann@25062
   848
    hence le2: "c + b <= c + a" by (rule add_left_mono)
haftmann@25062
   849
    have "a = b" 
haftmann@25062
   850
      apply (insert le)
haftmann@25062
   851
      apply (insert le2)
haftmann@25062
   852
      apply (drule antisym, simp_all)
haftmann@25062
   853
      done
haftmann@25062
   854
    with w show False 
haftmann@25062
   855
      by (simp add: linorder_not_le [symmetric])
haftmann@25062
   856
  qed
haftmann@25062
   857
qed
haftmann@25062
   858
haftmann@25267
   859
end
haftmann@25267
   860
haftmann@35028
   861
class linordered_ab_group_add = linorder + ordered_ab_group_add
haftmann@25267
   862
begin
haftmann@25230
   863
haftmann@35028
   864
subclass linordered_cancel_ab_semigroup_add ..
haftmann@25230
   865
haftmann@35036
   866
lemma neg_less_eq_nonneg [simp]:
haftmann@25303
   867
  "- a \<le> a \<longleftrightarrow> 0 \<le> a"
haftmann@25303
   868
proof
haftmann@25303
   869
  assume A: "- a \<le> a" show "0 \<le> a"
haftmann@25303
   870
  proof (rule classical)
haftmann@25303
   871
    assume "\<not> 0 \<le> a"
haftmann@25303
   872
    then have "a < 0" by auto
haftmann@25303
   873
    with A have "- a < 0" by (rule le_less_trans)
haftmann@25303
   874
    then show ?thesis by auto
haftmann@25303
   875
  qed
haftmann@25303
   876
next
haftmann@25303
   877
  assume A: "0 \<le> a" show "- a \<le> a"
haftmann@25303
   878
  proof (rule order_trans)
haftmann@25303
   879
    show "- a \<le> 0" using A by (simp add: minus_le_iff)
haftmann@25303
   880
  next
haftmann@25303
   881
    show "0 \<le> a" using A .
haftmann@25303
   882
  qed
haftmann@25303
   883
qed
haftmann@35036
   884
haftmann@35036
   885
lemma neg_less_nonneg [simp]:
haftmann@35036
   886
  "- a < a \<longleftrightarrow> 0 < a"
haftmann@35036
   887
proof
haftmann@35036
   888
  assume A: "- a < a" show "0 < a"
haftmann@35036
   889
  proof (rule classical)
haftmann@35036
   890
    assume "\<not> 0 < a"
haftmann@35036
   891
    then have "a \<le> 0" by auto
haftmann@35036
   892
    with A have "- a < 0" by (rule less_le_trans)
haftmann@35036
   893
    then show ?thesis by auto
haftmann@35036
   894
  qed
haftmann@35036
   895
next
haftmann@35036
   896
  assume A: "0 < a" show "- a < a"
haftmann@35036
   897
  proof (rule less_trans)
haftmann@35036
   898
    show "- a < 0" using A by (simp add: minus_le_iff)
haftmann@35036
   899
  next
haftmann@35036
   900
    show "0 < a" using A .
haftmann@35036
   901
  qed
haftmann@35036
   902
qed
haftmann@35036
   903
haftmann@35036
   904
lemma less_eq_neg_nonpos [simp]:
haftmann@25303
   905
  "a \<le> - a \<longleftrightarrow> a \<le> 0"
haftmann@25303
   906
proof
haftmann@25303
   907
  assume A: "a \<le> - a" show "a \<le> 0"
haftmann@25303
   908
  proof (rule classical)
haftmann@25303
   909
    assume "\<not> a \<le> 0"
haftmann@25303
   910
    then have "0 < a" by auto
haftmann@25303
   911
    then have "0 < - a" using A by (rule less_le_trans)
haftmann@25303
   912
    then show ?thesis by auto
haftmann@25303
   913
  qed
haftmann@25303
   914
next
haftmann@25303
   915
  assume A: "a \<le> 0" show "a \<le> - a"
haftmann@25303
   916
  proof (rule order_trans)
haftmann@25303
   917
    show "0 \<le> - a" using A by (simp add: minus_le_iff)
haftmann@25303
   918
  next
haftmann@25303
   919
    show "a \<le> 0" using A .
haftmann@25303
   920
  qed
haftmann@25303
   921
qed
haftmann@25303
   922
haftmann@35036
   923
lemma equal_neg_zero [simp]:
haftmann@25303
   924
  "a = - a \<longleftrightarrow> a = 0"
haftmann@25303
   925
proof
haftmann@25303
   926
  assume "a = 0" then show "a = - a" by simp
haftmann@25303
   927
next
haftmann@25303
   928
  assume A: "a = - a" show "a = 0"
haftmann@25303
   929
  proof (cases "0 \<le> a")
haftmann@25303
   930
    case True with A have "0 \<le> - a" by auto
haftmann@25303
   931
    with le_minus_iff have "a \<le> 0" by simp
haftmann@25303
   932
    with True show ?thesis by (auto intro: order_trans)
haftmann@25303
   933
  next
haftmann@25303
   934
    case False then have B: "a \<le> 0" by auto
haftmann@25303
   935
    with A have "- a \<le> 0" by auto
haftmann@25303
   936
    with B show ?thesis by (auto intro: order_trans)
haftmann@25303
   937
  qed
haftmann@25303
   938
qed
haftmann@25303
   939
haftmann@35036
   940
lemma neg_equal_zero [simp]:
haftmann@25303
   941
  "- a = a \<longleftrightarrow> a = 0"
haftmann@35036
   942
  by (auto dest: sym)
haftmann@35036
   943
haftmann@35036
   944
lemma double_zero [simp]:
haftmann@35036
   945
  "a + a = 0 \<longleftrightarrow> a = 0"
haftmann@35036
   946
proof
haftmann@35036
   947
  assume assm: "a + a = 0"
haftmann@35036
   948
  then have a: "- a = a" by (rule minus_unique)
huffman@35216
   949
  then show "a = 0" by (simp only: neg_equal_zero)
haftmann@35036
   950
qed simp
haftmann@35036
   951
haftmann@35036
   952
lemma double_zero_sym [simp]:
haftmann@35036
   953
  "0 = a + a \<longleftrightarrow> a = 0"
haftmann@35036
   954
  by (rule, drule sym) simp_all
haftmann@35036
   955
haftmann@35036
   956
lemma zero_less_double_add_iff_zero_less_single_add [simp]:
haftmann@35036
   957
  "0 < a + a \<longleftrightarrow> 0 < a"
haftmann@35036
   958
proof
haftmann@35036
   959
  assume "0 < a + a"
haftmann@35036
   960
  then have "0 - a < a" by (simp only: diff_less_eq)
haftmann@35036
   961
  then have "- a < a" by simp
huffman@35216
   962
  then show "0 < a" by (simp only: neg_less_nonneg)
haftmann@35036
   963
next
haftmann@35036
   964
  assume "0 < a"
haftmann@35036
   965
  with this have "0 + 0 < a + a"
haftmann@35036
   966
    by (rule add_strict_mono)
haftmann@35036
   967
  then show "0 < a + a" by simp
haftmann@35036
   968
qed
haftmann@35036
   969
haftmann@35036
   970
lemma zero_le_double_add_iff_zero_le_single_add [simp]:
haftmann@35036
   971
  "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
haftmann@35036
   972
  by (auto simp add: le_less)
haftmann@35036
   973
haftmann@35036
   974
lemma double_add_less_zero_iff_single_add_less_zero [simp]:
haftmann@35036
   975
  "a + a < 0 \<longleftrightarrow> a < 0"
haftmann@35036
   976
proof -
haftmann@35036
   977
  have "\<not> a + a < 0 \<longleftrightarrow> \<not> a < 0"
haftmann@35036
   978
    by (simp add: not_less)
haftmann@35036
   979
  then show ?thesis by simp
haftmann@35036
   980
qed
haftmann@35036
   981
haftmann@35036
   982
lemma double_add_le_zero_iff_single_add_le_zero [simp]:
haftmann@35036
   983
  "a + a \<le> 0 \<longleftrightarrow> a \<le> 0" 
haftmann@35036
   984
proof -
haftmann@35036
   985
  have "\<not> a + a \<le> 0 \<longleftrightarrow> \<not> a \<le> 0"
haftmann@35036
   986
    by (simp add: not_le)
haftmann@35036
   987
  then show ?thesis by simp
haftmann@35036
   988
qed
haftmann@35036
   989
haftmann@35036
   990
lemma le_minus_self_iff:
haftmann@35036
   991
  "a \<le> - a \<longleftrightarrow> a \<le> 0"
haftmann@35036
   992
proof -
haftmann@35036
   993
  from add_le_cancel_left [of "- a" "a + a" 0]
haftmann@35036
   994
  have "a \<le> - a \<longleftrightarrow> a + a \<le> 0" 
haftmann@35036
   995
    by (simp add: add_assoc [symmetric])
haftmann@35036
   996
  thus ?thesis by simp
haftmann@35036
   997
qed
haftmann@35036
   998
haftmann@35036
   999
lemma minus_le_self_iff:
haftmann@35036
  1000
  "- a \<le> a \<longleftrightarrow> 0 \<le> a"
haftmann@35036
  1001
proof -
haftmann@35036
  1002
  from add_le_cancel_left [of "- a" 0 "a + a"]
haftmann@35036
  1003
  have "- a \<le> a \<longleftrightarrow> 0 \<le> a + a" 
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_max_eq_min:
haftmann@35036
  1009
  "- max x y = min (-x) (-y)"
haftmann@35036
  1010
  by (auto simp add: max_def min_def)
haftmann@35036
  1011
haftmann@35036
  1012
lemma minus_min_eq_max:
haftmann@35036
  1013
  "- min x y = max (-x) (-y)"
haftmann@35036
  1014
  by (auto simp add: max_def min_def)
haftmann@25303
  1015
haftmann@25267
  1016
end
haftmann@25267
  1017
haftmann@36302
  1018
context ordered_comm_monoid_add
haftmann@36302
  1019
begin
obua@14738
  1020
paulson@15234
  1021
lemma add_increasing:
haftmann@36302
  1022
  "0 \<le> a \<Longrightarrow> b \<le> c \<Longrightarrow> b \<le> a + c"
haftmann@36302
  1023
  by (insert add_mono [of 0 a b c], simp)
obua@14738
  1024
nipkow@15539
  1025
lemma add_increasing2:
haftmann@36302
  1026
  "0 \<le> c \<Longrightarrow> b \<le> a \<Longrightarrow> b \<le> a + c"
haftmann@36302
  1027
  by (simp add: add_increasing add_commute [of a])
nipkow@15539
  1028
paulson@15234
  1029
lemma add_strict_increasing:
haftmann@36302
  1030
  "0 < a \<Longrightarrow> b \<le> c \<Longrightarrow> b < a + c"
haftmann@36302
  1031
  by (insert add_less_le_mono [of 0 a b c], simp)
paulson@15234
  1032
paulson@15234
  1033
lemma add_strict_increasing2:
haftmann@36302
  1034
  "0 \<le> a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@36302
  1035
  by (insert add_le_less_mono [of 0 a b c], simp)
haftmann@36302
  1036
haftmann@36302
  1037
end
paulson@15234
  1038
haftmann@35092
  1039
class abs =
haftmann@35092
  1040
  fixes abs :: "'a \<Rightarrow> 'a"
haftmann@35092
  1041
begin
haftmann@35092
  1042
haftmann@35092
  1043
notation (xsymbols)
haftmann@35092
  1044
  abs  ("\<bar>_\<bar>")
haftmann@35092
  1045
haftmann@35092
  1046
notation (HTML output)
haftmann@35092
  1047
  abs  ("\<bar>_\<bar>")
haftmann@35092
  1048
haftmann@35092
  1049
end
haftmann@35092
  1050
haftmann@35092
  1051
class sgn =
haftmann@35092
  1052
  fixes sgn :: "'a \<Rightarrow> 'a"
haftmann@35092
  1053
haftmann@35092
  1054
class abs_if = minus + uminus + ord + zero + abs +
haftmann@35092
  1055
  assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"
haftmann@35092
  1056
haftmann@35092
  1057
class sgn_if = minus + uminus + zero + one + ord + sgn +
haftmann@35092
  1058
  assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
haftmann@35092
  1059
begin
haftmann@35092
  1060
haftmann@35092
  1061
lemma sgn0 [simp]: "sgn 0 = 0"
haftmann@35092
  1062
  by (simp add:sgn_if)
haftmann@35092
  1063
haftmann@35092
  1064
end
obua@14738
  1065
haftmann@35028
  1066
class ordered_ab_group_add_abs = ordered_ab_group_add + abs +
haftmann@25303
  1067
  assumes abs_ge_zero [simp]: "\<bar>a\<bar> \<ge> 0"
haftmann@25303
  1068
    and abs_ge_self: "a \<le> \<bar>a\<bar>"
haftmann@25303
  1069
    and abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
haftmann@25303
  1070
    and abs_minus_cancel [simp]: "\<bar>-a\<bar> = \<bar>a\<bar>"
haftmann@25303
  1071
    and abs_triangle_ineq: "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25303
  1072
begin
haftmann@25303
  1073
haftmann@25307
  1074
lemma abs_minus_le_zero: "- \<bar>a\<bar> \<le> 0"
haftmann@25307
  1075
  unfolding neg_le_0_iff_le by simp
haftmann@25307
  1076
haftmann@25307
  1077
lemma abs_of_nonneg [simp]:
nipkow@29667
  1078
  assumes nonneg: "0 \<le> a" shows "\<bar>a\<bar> = a"
haftmann@25307
  1079
proof (rule antisym)
haftmann@25307
  1080
  from nonneg le_imp_neg_le have "- a \<le> 0" by simp
haftmann@25307
  1081
  from this nonneg have "- a \<le> a" by (rule order_trans)
haftmann@25307
  1082
  then show "\<bar>a\<bar> \<le> a" by (auto intro: abs_leI)
haftmann@25307
  1083
qed (rule abs_ge_self)
haftmann@25307
  1084
haftmann@25307
  1085
lemma abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
nipkow@29667
  1086
by (rule antisym)
haftmann@36302
  1087
   (auto intro!: abs_ge_self abs_leI order_trans [of "- \<bar>a\<bar>" 0 "\<bar>a\<bar>"])
haftmann@25307
  1088
haftmann@25307
  1089
lemma abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
haftmann@25307
  1090
proof -
haftmann@25307
  1091
  have "\<bar>a\<bar> = 0 \<Longrightarrow> a = 0"
haftmann@25307
  1092
  proof (rule antisym)
haftmann@25307
  1093
    assume zero: "\<bar>a\<bar> = 0"
haftmann@25307
  1094
    with abs_ge_self show "a \<le> 0" by auto
haftmann@25307
  1095
    from zero have "\<bar>-a\<bar> = 0" by simp
haftmann@36302
  1096
    with abs_ge_self [of "- a"] have "- a \<le> 0" by auto
haftmann@25307
  1097
    with neg_le_0_iff_le show "0 \<le> a" by auto
haftmann@25307
  1098
  qed
haftmann@25307
  1099
  then show ?thesis by auto
haftmann@25307
  1100
qed
haftmann@25307
  1101
haftmann@25303
  1102
lemma abs_zero [simp]: "\<bar>0\<bar> = 0"
nipkow@29667
  1103
by simp
avigad@16775
  1104
blanchet@35828
  1105
lemma abs_0_eq [simp, no_atp]: "0 = \<bar>a\<bar> \<longleftrightarrow> a = 0"
haftmann@25303
  1106
proof -
haftmann@25303
  1107
  have "0 = \<bar>a\<bar> \<longleftrightarrow> \<bar>a\<bar> = 0" by (simp only: eq_ac)
haftmann@25303
  1108
  thus ?thesis by simp
haftmann@25303
  1109
qed
haftmann@25303
  1110
haftmann@25303
  1111
lemma abs_le_zero_iff [simp]: "\<bar>a\<bar> \<le> 0 \<longleftrightarrow> a = 0" 
haftmann@25303
  1112
proof
haftmann@25303
  1113
  assume "\<bar>a\<bar> \<le> 0"
haftmann@25303
  1114
  then have "\<bar>a\<bar> = 0" by (rule antisym) simp
haftmann@25303
  1115
  thus "a = 0" by simp
haftmann@25303
  1116
next
haftmann@25303
  1117
  assume "a = 0"
haftmann@25303
  1118
  thus "\<bar>a\<bar> \<le> 0" by simp
haftmann@25303
  1119
qed
haftmann@25303
  1120
haftmann@25303
  1121
lemma zero_less_abs_iff [simp]: "0 < \<bar>a\<bar> \<longleftrightarrow> a \<noteq> 0"
nipkow@29667
  1122
by (simp add: less_le)
haftmann@25303
  1123
haftmann@25303
  1124
lemma abs_not_less_zero [simp]: "\<not> \<bar>a\<bar> < 0"
haftmann@25303
  1125
proof -
haftmann@25303
  1126
  have a: "\<And>x y. x \<le> y \<Longrightarrow> \<not> y < x" by auto
haftmann@25303
  1127
  show ?thesis by (simp add: a)
haftmann@25303
  1128
qed
avigad@16775
  1129
haftmann@25303
  1130
lemma abs_ge_minus_self: "- a \<le> \<bar>a\<bar>"
haftmann@25303
  1131
proof -
haftmann@25303
  1132
  have "- a \<le> \<bar>-a\<bar>" by (rule abs_ge_self)
haftmann@25303
  1133
  then show ?thesis by simp
haftmann@25303
  1134
qed
haftmann@25303
  1135
haftmann@25303
  1136
lemma abs_minus_commute: 
haftmann@25303
  1137
  "\<bar>a - b\<bar> = \<bar>b - a\<bar>"
haftmann@25303
  1138
proof -
haftmann@25303
  1139
  have "\<bar>a - b\<bar> = \<bar>- (a - b)\<bar>" by (simp only: abs_minus_cancel)
haftmann@25303
  1140
  also have "... = \<bar>b - a\<bar>" by simp
haftmann@25303
  1141
  finally show ?thesis .
haftmann@25303
  1142
qed
haftmann@25303
  1143
haftmann@25303
  1144
lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a"
nipkow@29667
  1145
by (rule abs_of_nonneg, rule less_imp_le)
avigad@16775
  1146
haftmann@25303
  1147
lemma abs_of_nonpos [simp]:
nipkow@29667
  1148
  assumes "a \<le> 0" shows "\<bar>a\<bar> = - a"
haftmann@25303
  1149
proof -
haftmann@25303
  1150
  let ?b = "- a"
haftmann@25303
  1151
  have "- ?b \<le> 0 \<Longrightarrow> \<bar>- ?b\<bar> = - (- ?b)"
haftmann@25303
  1152
  unfolding abs_minus_cancel [of "?b"]
haftmann@25303
  1153
  unfolding neg_le_0_iff_le [of "?b"]
haftmann@25303
  1154
  unfolding minus_minus by (erule abs_of_nonneg)
haftmann@25303
  1155
  then show ?thesis using assms by auto
haftmann@25303
  1156
qed
haftmann@25303
  1157
  
haftmann@25303
  1158
lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a"
nipkow@29667
  1159
by (rule abs_of_nonpos, rule less_imp_le)
haftmann@25303
  1160
haftmann@25303
  1161
lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b"
nipkow@29667
  1162
by (insert abs_ge_self, blast intro: order_trans)
haftmann@25303
  1163
haftmann@25303
  1164
lemma abs_le_D2: "\<bar>a\<bar> \<le> b \<Longrightarrow> - a \<le> b"
haftmann@36302
  1165
by (insert abs_le_D1 [of "- a"], simp)
haftmann@25303
  1166
haftmann@25303
  1167
lemma abs_le_iff: "\<bar>a\<bar> \<le> b \<longleftrightarrow> a \<le> b \<and> - a \<le> b"
nipkow@29667
  1168
by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
haftmann@25303
  1169
haftmann@25303
  1170
lemma abs_triangle_ineq2: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>a - b\<bar>"
haftmann@36302
  1171
proof -
haftmann@36302
  1172
  have "\<bar>a\<bar> = \<bar>b + (a - b)\<bar>"
haftmann@36302
  1173
    by (simp add: algebra_simps add_diff_cancel)
haftmann@36302
  1174
  then have "\<bar>a\<bar> \<le> \<bar>b\<bar> + \<bar>a - b\<bar>"
haftmann@36302
  1175
    by (simp add: abs_triangle_ineq)
haftmann@36302
  1176
  then show ?thesis
haftmann@36302
  1177
    by (simp add: algebra_simps)
haftmann@36302
  1178
qed
haftmann@36302
  1179
haftmann@36302
  1180
lemma abs_triangle_ineq2_sym: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>b - a\<bar>"
haftmann@36302
  1181
  by (simp only: abs_minus_commute [of b] abs_triangle_ineq2)
avigad@16775
  1182
haftmann@25303
  1183
lemma abs_triangle_ineq3: "\<bar>\<bar>a\<bar> - \<bar>b\<bar>\<bar> \<le> \<bar>a - b\<bar>"
haftmann@36302
  1184
  by (simp add: abs_le_iff abs_triangle_ineq2 abs_triangle_ineq2_sym)
avigad@16775
  1185
haftmann@25303
  1186
lemma abs_triangle_ineq4: "\<bar>a - b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25303
  1187
proof -
haftmann@36302
  1188
  have "\<bar>a - b\<bar> = \<bar>a + - b\<bar>" by (subst diff_minus, rule refl)
haftmann@36302
  1189
  also have "... \<le> \<bar>a\<bar> + \<bar>- b\<bar>" by (rule abs_triangle_ineq)
nipkow@29667
  1190
  finally show ?thesis by simp
haftmann@25303
  1191
qed
avigad@16775
  1192
haftmann@25303
  1193
lemma abs_diff_triangle_ineq: "\<bar>a + b - (c + d)\<bar> \<le> \<bar>a - c\<bar> + \<bar>b - d\<bar>"
haftmann@25303
  1194
proof -
haftmann@25303
  1195
  have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: diff_minus add_ac)
haftmann@25303
  1196
  also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq)
haftmann@25303
  1197
  finally show ?thesis .
haftmann@25303
  1198
qed
avigad@16775
  1199
haftmann@25303
  1200
lemma abs_add_abs [simp]:
haftmann@25303
  1201
  "\<bar>\<bar>a\<bar> + \<bar>b\<bar>\<bar> = \<bar>a\<bar> + \<bar>b\<bar>" (is "?L = ?R")
haftmann@25303
  1202
proof (rule antisym)
haftmann@25303
  1203
  show "?L \<ge> ?R" by(rule abs_ge_self)
haftmann@25303
  1204
next
haftmann@25303
  1205
  have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq)
haftmann@25303
  1206
  also have "\<dots> = ?R" by simp
haftmann@25303
  1207
  finally show "?L \<le> ?R" .
haftmann@25303
  1208
qed
haftmann@25303
  1209
haftmann@25303
  1210
end
obua@14738
  1211
obua@15178
  1212
haftmann@25090
  1213
subsection {* Tools setup *}
haftmann@25090
  1214
blanchet@35828
  1215
lemma add_mono_thms_linordered_semiring [no_atp]:
haftmann@35028
  1216
  fixes i j k :: "'a\<Colon>ordered_ab_semigroup_add"
haftmann@25077
  1217
  shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1218
    and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1219
    and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1220
    and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"
haftmann@25077
  1221
by (rule add_mono, clarify+)+
haftmann@25077
  1222
blanchet@35828
  1223
lemma add_mono_thms_linordered_field [no_atp]:
haftmann@35028
  1224
  fixes i j k :: "'a\<Colon>ordered_cancel_ab_semigroup_add"
haftmann@25077
  1225
  shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1226
    and "i = j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1227
    and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1228
    and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1229
    and "i < j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1230
by (auto intro: add_strict_right_mono add_strict_left_mono
haftmann@25077
  1231
  add_less_le_mono add_le_less_mono add_strict_mono)
haftmann@25077
  1232
haftmann@33364
  1233
code_modulename SML
haftmann@35050
  1234
  Groups Arith
haftmann@33364
  1235
haftmann@33364
  1236
code_modulename OCaml
haftmann@35050
  1237
  Groups Arith
haftmann@33364
  1238
haftmann@33364
  1239
code_modulename Haskell
haftmann@35050
  1240
  Groups Arith
haftmann@33364
  1241
haftmann@37889
  1242
haftmann@37889
  1243
text {* Legacy *}
haftmann@37889
  1244
haftmann@37889
  1245
lemmas diff_def = diff_minus
haftmann@37889
  1246
obua@14738
  1247
end