src/HOL/Groups.thy
author wenzelm
Sun Sep 05 21:41:24 2010 +0200 (2010-09-05)
changeset 39134 917b4b6ba3d2
parent 37986 3b3187adf292
child 40368 47c186c8577d
permissions -rw-r--r--
turned show_sorts/show_types into proper configuration options;
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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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syntax
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  "_index1"  :: index    ("\<^sub>1")
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translations
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  (index) "\<^sub>1" => (index) "\<^bsub>\<struct>\<^esub>"
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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 show_sorts => 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 Syntax.const Syntax.constrainC $ Syntax.const c $ Syntax.term_of_typ show_sorts 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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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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lemma neg_equal_0_iff_equal [simp]:
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  "- a = 0 \<longleftrightarrow> a = 0"
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by (subst neg_equal_iff_equal [symmetric], simp)
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lemma neg_0_equal_iff_equal [simp]:
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  "0 = - a \<longleftrightarrow> 0 = a"
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by (subst neg_equal_iff_equal [symmetric], simp)
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text{*The next two equations can make the simplifier loop!*}
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haftmann@25062
   365
lemma equation_minus_iff:
haftmann@25062
   366
  "a = - b \<longleftrightarrow> b = - a"
obua@14738
   367
proof -
haftmann@25062
   368
  have "- (- a) = - b \<longleftrightarrow> - a = b" by (rule neg_equal_iff_equal)
haftmann@25062
   369
  thus ?thesis by (simp add: eq_commute)
haftmann@25062
   370
qed
haftmann@25062
   371
haftmann@25062
   372
lemma minus_equation_iff:
haftmann@25062
   373
  "- a = b \<longleftrightarrow> - b = a"
haftmann@25062
   374
proof -
haftmann@25062
   375
  have "- a = - (- b) \<longleftrightarrow> a = -b" by (rule neg_equal_iff_equal)
obua@14738
   376
  thus ?thesis by (simp add: eq_commute)
obua@14738
   377
qed
obua@14738
   378
huffman@28130
   379
lemma diff_add_cancel: "a - b + b = a"
nipkow@29667
   380
by (simp add: diff_minus add_assoc)
huffman@28130
   381
huffman@28130
   382
lemma add_diff_cancel: "a + b - b = a"
nipkow@29667
   383
by (simp add: diff_minus add_assoc)
nipkow@29667
   384
haftmann@36348
   385
declare diff_minus[symmetric, algebra_simps, field_simps]
huffman@28130
   386
huffman@29914
   387
lemma eq_neg_iff_add_eq_0: "a = - b \<longleftrightarrow> a + b = 0"
huffman@29914
   388
proof
huffman@29914
   389
  assume "a = - b" then show "a + b = 0" by simp
huffman@29914
   390
next
huffman@29914
   391
  assume "a + b = 0"
huffman@29914
   392
  moreover have "a + (b + - b) = (a + b) + - b"
huffman@29914
   393
    by (simp only: add_assoc)
huffman@29914
   394
  ultimately show "a = - b" by simp
huffman@29914
   395
qed
huffman@29914
   396
haftmann@25062
   397
end
haftmann@25062
   398
haftmann@25762
   399
class ab_group_add = minus + uminus + comm_monoid_add +
haftmann@25062
   400
  assumes ab_left_minus: "- a + a = 0"
haftmann@25062
   401
  assumes ab_diff_minus: "a - b = a + (- b)"
haftmann@25267
   402
begin
haftmann@25062
   403
haftmann@25267
   404
subclass group_add
haftmann@28823
   405
  proof qed (simp_all add: ab_left_minus ab_diff_minus)
haftmann@25062
   406
huffman@29904
   407
subclass cancel_comm_monoid_add
haftmann@28823
   408
proof
haftmann@25062
   409
  fix a b c :: 'a
haftmann@25062
   410
  assume "a + b = a + c"
haftmann@25062
   411
  then have "- a + a + b = - a + a + c"
haftmann@25062
   412
    unfolding add_assoc by simp
haftmann@25062
   413
  then show "b = c" by simp
haftmann@25062
   414
qed
haftmann@25062
   415
haftmann@36348
   416
lemma uminus_add_conv_diff[algebra_simps, field_simps]:
haftmann@25062
   417
  "- a + b = b - a"
nipkow@29667
   418
by (simp add:diff_minus add_commute)
haftmann@25062
   419
haftmann@25062
   420
lemma minus_add_distrib [simp]:
haftmann@25062
   421
  "- (a + b) = - a + - b"
huffman@34146
   422
by (rule minus_unique) (simp add: add_ac)
haftmann@25062
   423
haftmann@25062
   424
lemma minus_diff_eq [simp]:
haftmann@25062
   425
  "- (a - b) = b - a"
nipkow@29667
   426
by (simp add: diff_minus add_commute)
haftmann@25077
   427
haftmann@36348
   428
lemma add_diff_eq[algebra_simps, field_simps]: "a + (b - c) = (a + b) - c"
nipkow@29667
   429
by (simp add: diff_minus add_ac)
haftmann@25077
   430
haftmann@36348
   431
lemma diff_add_eq[algebra_simps, field_simps]: "(a - b) + c = (a + c) - b"
nipkow@29667
   432
by (simp add: diff_minus add_ac)
haftmann@25077
   433
haftmann@36348
   434
lemma diff_eq_eq[algebra_simps, field_simps]: "a - b = c \<longleftrightarrow> a = c + b"
nipkow@29667
   435
by (auto simp add: diff_minus add_assoc)
haftmann@25077
   436
haftmann@36348
   437
lemma eq_diff_eq[algebra_simps, field_simps]: "a = c - b \<longleftrightarrow> a + b = c"
nipkow@29667
   438
by (auto simp add: diff_minus add_assoc)
haftmann@25077
   439
haftmann@36348
   440
lemma diff_diff_eq[algebra_simps, field_simps]: "(a - b) - c = a - (b + c)"
nipkow@29667
   441
by (simp add: diff_minus add_ac)
haftmann@25077
   442
haftmann@36348
   443
lemma diff_diff_eq2[algebra_simps, field_simps]: "a - (b - c) = (a + c) - b"
nipkow@29667
   444
by (simp add: diff_minus add_ac)
haftmann@25077
   445
haftmann@25077
   446
lemma eq_iff_diff_eq_0: "a = b \<longleftrightarrow> a - b = 0"
nipkow@29667
   447
by (simp add: algebra_simps)
haftmann@25077
   448
huffman@35216
   449
(* FIXME: duplicates right_minus_eq from class group_add *)
huffman@35216
   450
(* but only this one is declared as a simp rule. *)
blanchet@35828
   451
lemma diff_eq_0_iff_eq [simp, no_atp]: "a - b = 0 \<longleftrightarrow> a = b"
huffman@30629
   452
by (simp add: algebra_simps)
huffman@30629
   453
haftmann@37884
   454
lemma diff_eq_diff_eq:
haftmann@37884
   455
  "a - b = c - d \<Longrightarrow> a = b \<longleftrightarrow> c = d"
haftmann@37884
   456
  by (auto simp add: algebra_simps)
haftmann@37884
   457
  
haftmann@25062
   458
end
obua@14738
   459
haftmann@37884
   460
obua@14738
   461
subsection {* (Partially) Ordered Groups *} 
obua@14738
   462
haftmann@35301
   463
text {*
haftmann@35301
   464
  The theory of partially ordered groups is taken from the books:
haftmann@35301
   465
  \begin{itemize}
haftmann@35301
   466
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
haftmann@35301
   467
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
haftmann@35301
   468
  \end{itemize}
haftmann@35301
   469
  Most of the used notions can also be looked up in 
haftmann@35301
   470
  \begin{itemize}
haftmann@35301
   471
  \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
haftmann@35301
   472
  \item \emph{Algebra I} by van der Waerden, Springer.
haftmann@35301
   473
  \end{itemize}
haftmann@35301
   474
*}
haftmann@35301
   475
haftmann@35028
   476
class ordered_ab_semigroup_add = order + ab_semigroup_add +
haftmann@25062
   477
  assumes add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b"
haftmann@25062
   478
begin
haftmann@24380
   479
haftmann@25062
   480
lemma add_right_mono:
haftmann@25062
   481
  "a \<le> b \<Longrightarrow> a + c \<le> b + c"
nipkow@29667
   482
by (simp add: add_commute [of _ c] add_left_mono)
obua@14738
   483
obua@14738
   484
text {* non-strict, in both arguments *}
obua@14738
   485
lemma add_mono:
haftmann@25062
   486
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d"
obua@14738
   487
  apply (erule add_right_mono [THEN order_trans])
obua@14738
   488
  apply (simp add: add_commute add_left_mono)
obua@14738
   489
  done
obua@14738
   490
haftmann@25062
   491
end
haftmann@25062
   492
haftmann@35028
   493
class ordered_cancel_ab_semigroup_add =
haftmann@35028
   494
  ordered_ab_semigroup_add + cancel_ab_semigroup_add
haftmann@25062
   495
begin
haftmann@25062
   496
obua@14738
   497
lemma add_strict_left_mono:
haftmann@25062
   498
  "a < b \<Longrightarrow> c + a < c + b"
nipkow@29667
   499
by (auto simp add: less_le add_left_mono)
obua@14738
   500
obua@14738
   501
lemma add_strict_right_mono:
haftmann@25062
   502
  "a < b \<Longrightarrow> a + c < b + c"
nipkow@29667
   503
by (simp add: add_commute [of _ c] add_strict_left_mono)
obua@14738
   504
obua@14738
   505
text{*Strict monotonicity in both arguments*}
haftmann@25062
   506
lemma add_strict_mono:
haftmann@25062
   507
  "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
haftmann@25062
   508
apply (erule add_strict_right_mono [THEN less_trans])
obua@14738
   509
apply (erule add_strict_left_mono)
obua@14738
   510
done
obua@14738
   511
obua@14738
   512
lemma add_less_le_mono:
haftmann@25062
   513
  "a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c < b + d"
haftmann@25062
   514
apply (erule add_strict_right_mono [THEN less_le_trans])
haftmann@25062
   515
apply (erule add_left_mono)
obua@14738
   516
done
obua@14738
   517
obua@14738
   518
lemma add_le_less_mono:
haftmann@25062
   519
  "a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
haftmann@25062
   520
apply (erule add_right_mono [THEN le_less_trans])
obua@14738
   521
apply (erule add_strict_left_mono) 
obua@14738
   522
done
obua@14738
   523
haftmann@25062
   524
end
haftmann@25062
   525
haftmann@35028
   526
class ordered_ab_semigroup_add_imp_le =
haftmann@35028
   527
  ordered_cancel_ab_semigroup_add +
haftmann@25062
   528
  assumes add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b"
haftmann@25062
   529
begin
haftmann@25062
   530
obua@14738
   531
lemma add_less_imp_less_left:
nipkow@29667
   532
  assumes less: "c + a < c + b" shows "a < b"
obua@14738
   533
proof -
obua@14738
   534
  from less have le: "c + a <= c + b" by (simp add: order_le_less)
obua@14738
   535
  have "a <= b" 
obua@14738
   536
    apply (insert le)
obua@14738
   537
    apply (drule add_le_imp_le_left)
obua@14738
   538
    by (insert le, drule add_le_imp_le_left, assumption)
obua@14738
   539
  moreover have "a \<noteq> b"
obua@14738
   540
  proof (rule ccontr)
obua@14738
   541
    assume "~(a \<noteq> b)"
obua@14738
   542
    then have "a = b" by simp
obua@14738
   543
    then have "c + a = c + b" by simp
obua@14738
   544
    with less show "False"by simp
obua@14738
   545
  qed
obua@14738
   546
  ultimately show "a < b" by (simp add: order_le_less)
obua@14738
   547
qed
obua@14738
   548
obua@14738
   549
lemma add_less_imp_less_right:
haftmann@25062
   550
  "a + c < b + c \<Longrightarrow> a < b"
obua@14738
   551
apply (rule add_less_imp_less_left [of c])
obua@14738
   552
apply (simp add: add_commute)  
obua@14738
   553
done
obua@14738
   554
obua@14738
   555
lemma add_less_cancel_left [simp]:
haftmann@25062
   556
  "c + a < c + b \<longleftrightarrow> a < b"
nipkow@29667
   557
by (blast intro: add_less_imp_less_left add_strict_left_mono) 
obua@14738
   558
obua@14738
   559
lemma add_less_cancel_right [simp]:
haftmann@25062
   560
  "a + c < b + c \<longleftrightarrow> a < b"
nipkow@29667
   561
by (blast intro: add_less_imp_less_right add_strict_right_mono)
obua@14738
   562
obua@14738
   563
lemma add_le_cancel_left [simp]:
haftmann@25062
   564
  "c + a \<le> c + b \<longleftrightarrow> a \<le> b"
nipkow@29667
   565
by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono) 
obua@14738
   566
obua@14738
   567
lemma add_le_cancel_right [simp]:
haftmann@25062
   568
  "a + c \<le> b + c \<longleftrightarrow> a \<le> b"
nipkow@29667
   569
by (simp add: add_commute [of a c] add_commute [of b c])
obua@14738
   570
obua@14738
   571
lemma add_le_imp_le_right:
haftmann@25062
   572
  "a + c \<le> b + c \<Longrightarrow> a \<le> b"
nipkow@29667
   573
by simp
haftmann@25062
   574
haftmann@25077
   575
lemma max_add_distrib_left:
haftmann@25077
   576
  "max x y + z = max (x + z) (y + z)"
haftmann@25077
   577
  unfolding max_def by auto
haftmann@25077
   578
haftmann@25077
   579
lemma min_add_distrib_left:
haftmann@25077
   580
  "min x y + z = min (x + z) (y + z)"
haftmann@25077
   581
  unfolding min_def by auto
haftmann@25077
   582
haftmann@25062
   583
end
haftmann@25062
   584
haftmann@25303
   585
subsection {* Support for reasoning about signs *}
haftmann@25303
   586
haftmann@35028
   587
class ordered_comm_monoid_add =
haftmann@35028
   588
  ordered_cancel_ab_semigroup_add + comm_monoid_add
haftmann@25303
   589
begin
haftmann@25303
   590
haftmann@25303
   591
lemma add_pos_nonneg:
nipkow@29667
   592
  assumes "0 < a" and "0 \<le> b" shows "0 < a + b"
haftmann@25303
   593
proof -
haftmann@25303
   594
  have "0 + 0 < a + b" 
haftmann@25303
   595
    using assms by (rule add_less_le_mono)
haftmann@25303
   596
  then show ?thesis by simp
haftmann@25303
   597
qed
haftmann@25303
   598
haftmann@25303
   599
lemma add_pos_pos:
nipkow@29667
   600
  assumes "0 < a" and "0 < b" shows "0 < a + b"
nipkow@29667
   601
by (rule add_pos_nonneg) (insert assms, auto)
haftmann@25303
   602
haftmann@25303
   603
lemma add_nonneg_pos:
nipkow@29667
   604
  assumes "0 \<le> a" and "0 < b" shows "0 < a + b"
haftmann@25303
   605
proof -
haftmann@25303
   606
  have "0 + 0 < a + b" 
haftmann@25303
   607
    using assms by (rule add_le_less_mono)
haftmann@25303
   608
  then show ?thesis by simp
haftmann@25303
   609
qed
haftmann@25303
   610
huffman@36977
   611
lemma add_nonneg_nonneg [simp]:
nipkow@29667
   612
  assumes "0 \<le> a" and "0 \<le> b" shows "0 \<le> a + b"
haftmann@25303
   613
proof -
haftmann@25303
   614
  have "0 + 0 \<le> a + b" 
haftmann@25303
   615
    using assms by (rule add_mono)
haftmann@25303
   616
  then show ?thesis by simp
haftmann@25303
   617
qed
haftmann@25303
   618
huffman@30691
   619
lemma add_neg_nonpos:
nipkow@29667
   620
  assumes "a < 0" and "b \<le> 0" shows "a + b < 0"
haftmann@25303
   621
proof -
haftmann@25303
   622
  have "a + b < 0 + 0"
haftmann@25303
   623
    using assms by (rule add_less_le_mono)
haftmann@25303
   624
  then show ?thesis by simp
haftmann@25303
   625
qed
haftmann@25303
   626
haftmann@25303
   627
lemma add_neg_neg: 
nipkow@29667
   628
  assumes "a < 0" and "b < 0" shows "a + b < 0"
nipkow@29667
   629
by (rule add_neg_nonpos) (insert assms, auto)
haftmann@25303
   630
haftmann@25303
   631
lemma add_nonpos_neg:
nipkow@29667
   632
  assumes "a \<le> 0" and "b < 0" shows "a + b < 0"
haftmann@25303
   633
proof -
haftmann@25303
   634
  have "a + b < 0 + 0"
haftmann@25303
   635
    using assms by (rule add_le_less_mono)
haftmann@25303
   636
  then show ?thesis by simp
haftmann@25303
   637
qed
haftmann@25303
   638
haftmann@25303
   639
lemma add_nonpos_nonpos:
nipkow@29667
   640
  assumes "a \<le> 0" and "b \<le> 0" shows "a + b \<le> 0"
haftmann@25303
   641
proof -
haftmann@25303
   642
  have "a + b \<le> 0 + 0"
haftmann@25303
   643
    using assms by (rule add_mono)
haftmann@25303
   644
  then show ?thesis by simp
haftmann@25303
   645
qed
haftmann@25303
   646
huffman@30691
   647
lemmas add_sign_intros =
huffman@30691
   648
  add_pos_nonneg add_pos_pos add_nonneg_pos add_nonneg_nonneg
huffman@30691
   649
  add_neg_nonpos add_neg_neg add_nonpos_neg add_nonpos_nonpos
huffman@30691
   650
huffman@29886
   651
lemma add_nonneg_eq_0_iff:
huffman@29886
   652
  assumes x: "0 \<le> x" and y: "0 \<le> y"
huffman@29886
   653
  shows "x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@29886
   654
proof (intro iffI conjI)
huffman@29886
   655
  have "x = x + 0" by simp
huffman@29886
   656
  also have "x + 0 \<le> x + y" using y by (rule add_left_mono)
huffman@29886
   657
  also assume "x + y = 0"
huffman@29886
   658
  also have "0 \<le> x" using x .
huffman@29886
   659
  finally show "x = 0" .
huffman@29886
   660
next
huffman@29886
   661
  have "y = 0 + y" by simp
huffman@29886
   662
  also have "0 + y \<le> x + y" using x by (rule add_right_mono)
huffman@29886
   663
  also assume "x + y = 0"
huffman@29886
   664
  also have "0 \<le> y" using y .
huffman@29886
   665
  finally show "y = 0" .
huffman@29886
   666
next
huffman@29886
   667
  assume "x = 0 \<and> y = 0"
huffman@29886
   668
  then show "x + y = 0" by simp
huffman@29886
   669
qed
huffman@29886
   670
haftmann@25303
   671
end
haftmann@25303
   672
haftmann@35028
   673
class ordered_ab_group_add =
haftmann@35028
   674
  ab_group_add + ordered_ab_semigroup_add
haftmann@25062
   675
begin
haftmann@25062
   676
haftmann@35028
   677
subclass ordered_cancel_ab_semigroup_add ..
haftmann@25062
   678
haftmann@35028
   679
subclass ordered_ab_semigroup_add_imp_le
haftmann@28823
   680
proof
haftmann@25062
   681
  fix a b c :: 'a
haftmann@25062
   682
  assume "c + a \<le> c + b"
haftmann@25062
   683
  hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono)
haftmann@25062
   684
  hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc)
haftmann@25062
   685
  thus "a \<le> b" by simp
haftmann@25062
   686
qed
haftmann@25062
   687
haftmann@35028
   688
subclass ordered_comm_monoid_add ..
haftmann@25303
   689
haftmann@25077
   690
lemma max_diff_distrib_left:
haftmann@25077
   691
  shows "max x y - z = max (x - z) (y - z)"
nipkow@29667
   692
by (simp add: diff_minus, rule max_add_distrib_left) 
haftmann@25077
   693
haftmann@25077
   694
lemma min_diff_distrib_left:
haftmann@25077
   695
  shows "min x y - z = min (x - z) (y - z)"
nipkow@29667
   696
by (simp add: diff_minus, rule min_add_distrib_left) 
haftmann@25077
   697
haftmann@25077
   698
lemma le_imp_neg_le:
nipkow@29667
   699
  assumes "a \<le> b" shows "-b \<le> -a"
haftmann@25077
   700
proof -
nipkow@29667
   701
  have "-a+a \<le> -a+b" using `a \<le> b` by (rule add_left_mono) 
nipkow@29667
   702
  hence "0 \<le> -a+b" by simp
nipkow@29667
   703
  hence "0 + (-b) \<le> (-a + b) + (-b)" by (rule add_right_mono) 
nipkow@29667
   704
  thus ?thesis by (simp add: add_assoc)
haftmann@25077
   705
qed
haftmann@25077
   706
haftmann@25077
   707
lemma neg_le_iff_le [simp]: "- b \<le> - a \<longleftrightarrow> a \<le> b"
haftmann@25077
   708
proof 
haftmann@25077
   709
  assume "- b \<le> - a"
nipkow@29667
   710
  hence "- (- a) \<le> - (- b)" by (rule le_imp_neg_le)
haftmann@25077
   711
  thus "a\<le>b" by simp
haftmann@25077
   712
next
haftmann@25077
   713
  assume "a\<le>b"
haftmann@25077
   714
  thus "-b \<le> -a" by (rule le_imp_neg_le)
haftmann@25077
   715
qed
haftmann@25077
   716
haftmann@25077
   717
lemma neg_le_0_iff_le [simp]: "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"
nipkow@29667
   718
by (subst neg_le_iff_le [symmetric], simp)
haftmann@25077
   719
haftmann@25077
   720
lemma neg_0_le_iff_le [simp]: "0 \<le> - a \<longleftrightarrow> a \<le> 0"
nipkow@29667
   721
by (subst neg_le_iff_le [symmetric], simp)
haftmann@25077
   722
haftmann@25077
   723
lemma neg_less_iff_less [simp]: "- b < - a \<longleftrightarrow> a < b"
nipkow@29667
   724
by (force simp add: less_le) 
haftmann@25077
   725
haftmann@25077
   726
lemma neg_less_0_iff_less [simp]: "- a < 0 \<longleftrightarrow> 0 < a"
nipkow@29667
   727
by (subst neg_less_iff_less [symmetric], simp)
haftmann@25077
   728
haftmann@25077
   729
lemma neg_0_less_iff_less [simp]: "0 < - a \<longleftrightarrow> a < 0"
nipkow@29667
   730
by (subst neg_less_iff_less [symmetric], simp)
haftmann@25077
   731
haftmann@25077
   732
text{*The next several equations can make the simplifier loop!*}
haftmann@25077
   733
haftmann@25077
   734
lemma less_minus_iff: "a < - b \<longleftrightarrow> b < - a"
haftmann@25077
   735
proof -
haftmann@25077
   736
  have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
haftmann@25077
   737
  thus ?thesis by simp
haftmann@25077
   738
qed
haftmann@25077
   739
haftmann@25077
   740
lemma minus_less_iff: "- a < b \<longleftrightarrow> - b < a"
haftmann@25077
   741
proof -
haftmann@25077
   742
  have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
haftmann@25077
   743
  thus ?thesis by simp
haftmann@25077
   744
qed
haftmann@25077
   745
haftmann@25077
   746
lemma le_minus_iff: "a \<le> - b \<longleftrightarrow> b \<le> - a"
haftmann@25077
   747
proof -
haftmann@25077
   748
  have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff)
haftmann@25077
   749
  have "(- (- a) <= -b) = (b <= - a)" 
haftmann@25077
   750
    apply (auto simp only: le_less)
haftmann@25077
   751
    apply (drule mm)
haftmann@25077
   752
    apply (simp_all)
haftmann@25077
   753
    apply (drule mm[simplified], assumption)
haftmann@25077
   754
    done
haftmann@25077
   755
  then show ?thesis by simp
haftmann@25077
   756
qed
haftmann@25077
   757
haftmann@25077
   758
lemma minus_le_iff: "- a \<le> b \<longleftrightarrow> - b \<le> a"
nipkow@29667
   759
by (auto simp add: le_less minus_less_iff)
haftmann@25077
   760
haftmann@37884
   761
lemma diff_less_0_iff_less [simp, no_atp]:
haftmann@37884
   762
  "a - b < 0 \<longleftrightarrow> a < b"
haftmann@25077
   763
proof -
haftmann@37884
   764
  have "a - b < 0 \<longleftrightarrow> a + (- b) < b + (- b)" by (simp add: diff_minus)
haftmann@37884
   765
  also have "... \<longleftrightarrow> a < b" by (simp only: add_less_cancel_right)
haftmann@25077
   766
  finally show ?thesis .
haftmann@25077
   767
qed
haftmann@25077
   768
haftmann@37884
   769
lemmas less_iff_diff_less_0 = diff_less_0_iff_less [symmetric]
haftmann@37884
   770
haftmann@36348
   771
lemma diff_less_eq[algebra_simps, field_simps]: "a - b < c \<longleftrightarrow> a < c + b"
haftmann@25077
   772
apply (subst less_iff_diff_less_0 [of a])
haftmann@25077
   773
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
haftmann@25077
   774
apply (simp add: diff_minus add_ac)
haftmann@25077
   775
done
haftmann@25077
   776
haftmann@36348
   777
lemma less_diff_eq[algebra_simps, field_simps]: "a < c - b \<longleftrightarrow> a + b < c"
haftmann@36302
   778
apply (subst less_iff_diff_less_0 [of "a + b"])
haftmann@25077
   779
apply (subst less_iff_diff_less_0 [of a])
haftmann@25077
   780
apply (simp add: diff_minus add_ac)
haftmann@25077
   781
done
haftmann@25077
   782
haftmann@36348
   783
lemma diff_le_eq[algebra_simps, field_simps]: "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
nipkow@29667
   784
by (auto simp add: le_less diff_less_eq diff_add_cancel add_diff_cancel)
haftmann@25077
   785
haftmann@36348
   786
lemma le_diff_eq[algebra_simps, field_simps]: "a \<le> c - b \<longleftrightarrow> a + b \<le> c"
nipkow@29667
   787
by (auto simp add: le_less less_diff_eq diff_add_cancel add_diff_cancel)
haftmann@25077
   788
haftmann@37884
   789
lemma diff_le_0_iff_le [simp, no_atp]:
haftmann@37884
   790
  "a - b \<le> 0 \<longleftrightarrow> a \<le> b"
haftmann@37884
   791
  by (simp add: algebra_simps)
haftmann@37884
   792
haftmann@37884
   793
lemmas le_iff_diff_le_0 = diff_le_0_iff_le [symmetric]
haftmann@37884
   794
haftmann@37884
   795
lemma diff_eq_diff_less:
haftmann@37884
   796
  "a - b = c - d \<Longrightarrow> a < b \<longleftrightarrow> c < d"
haftmann@37884
   797
  by (auto simp only: less_iff_diff_less_0 [of a b] less_iff_diff_less_0 [of c d])
haftmann@37884
   798
haftmann@37889
   799
lemma diff_eq_diff_less_eq:
haftmann@37889
   800
  "a - b = c - d \<Longrightarrow> a \<le> b \<longleftrightarrow> c \<le> d"
haftmann@37889
   801
  by (auto simp only: le_iff_diff_le_0 [of a b] le_iff_diff_le_0 [of c d])
haftmann@25077
   802
haftmann@25077
   803
end
haftmann@25077
   804
wenzelm@37986
   805
use "Tools/abel_cancel.ML"
haftmann@37884
   806
haftmann@37889
   807
simproc_setup abel_cancel_sum
haftmann@37889
   808
  ("a + b::'a::ab_group_add" | "a - b::'a::ab_group_add") =
haftmann@37889
   809
  {* fn phi => Abel_Cancel.sum_proc *}
haftmann@37889
   810
haftmann@37889
   811
simproc_setup abel_cancel_relation
haftmann@37889
   812
  ("a < (b::'a::ordered_ab_group_add)" | "a \<le> (b::'a::ordered_ab_group_add)" | "c = (d::'b::ab_group_add)") =
haftmann@37889
   813
  {* fn phi => Abel_Cancel.rel_proc *}
haftmann@37884
   814
haftmann@35028
   815
class linordered_ab_semigroup_add =
haftmann@35028
   816
  linorder + ordered_ab_semigroup_add
haftmann@25062
   817
haftmann@35028
   818
class linordered_cancel_ab_semigroup_add =
haftmann@35028
   819
  linorder + ordered_cancel_ab_semigroup_add
haftmann@25267
   820
begin
haftmann@25062
   821
haftmann@35028
   822
subclass linordered_ab_semigroup_add ..
haftmann@25062
   823
haftmann@35028
   824
subclass ordered_ab_semigroup_add_imp_le
haftmann@28823
   825
proof
haftmann@25062
   826
  fix a b c :: 'a
haftmann@25062
   827
  assume le: "c + a <= c + b"  
haftmann@25062
   828
  show "a <= b"
haftmann@25062
   829
  proof (rule ccontr)
haftmann@25062
   830
    assume w: "~ a \<le> b"
haftmann@25062
   831
    hence "b <= a" by (simp add: linorder_not_le)
haftmann@25062
   832
    hence le2: "c + b <= c + a" by (rule add_left_mono)
haftmann@25062
   833
    have "a = b" 
haftmann@25062
   834
      apply (insert le)
haftmann@25062
   835
      apply (insert le2)
haftmann@25062
   836
      apply (drule antisym, simp_all)
haftmann@25062
   837
      done
haftmann@25062
   838
    with w show False 
haftmann@25062
   839
      by (simp add: linorder_not_le [symmetric])
haftmann@25062
   840
  qed
haftmann@25062
   841
qed
haftmann@25062
   842
haftmann@25267
   843
end
haftmann@25267
   844
haftmann@35028
   845
class linordered_ab_group_add = linorder + ordered_ab_group_add
haftmann@25267
   846
begin
haftmann@25230
   847
haftmann@35028
   848
subclass linordered_cancel_ab_semigroup_add ..
haftmann@25230
   849
haftmann@35036
   850
lemma neg_less_eq_nonneg [simp]:
haftmann@25303
   851
  "- a \<le> a \<longleftrightarrow> 0 \<le> a"
haftmann@25303
   852
proof
haftmann@25303
   853
  assume A: "- a \<le> a" show "0 \<le> a"
haftmann@25303
   854
  proof (rule classical)
haftmann@25303
   855
    assume "\<not> 0 \<le> a"
haftmann@25303
   856
    then have "a < 0" by auto
haftmann@25303
   857
    with A have "- a < 0" by (rule le_less_trans)
haftmann@25303
   858
    then show ?thesis by auto
haftmann@25303
   859
  qed
haftmann@25303
   860
next
haftmann@25303
   861
  assume A: "0 \<le> a" show "- a \<le> a"
haftmann@25303
   862
  proof (rule order_trans)
haftmann@25303
   863
    show "- a \<le> 0" using A by (simp add: minus_le_iff)
haftmann@25303
   864
  next
haftmann@25303
   865
    show "0 \<le> a" using A .
haftmann@25303
   866
  qed
haftmann@25303
   867
qed
haftmann@35036
   868
haftmann@35036
   869
lemma neg_less_nonneg [simp]:
haftmann@35036
   870
  "- a < a \<longleftrightarrow> 0 < a"
haftmann@35036
   871
proof
haftmann@35036
   872
  assume A: "- a < a" show "0 < a"
haftmann@35036
   873
  proof (rule classical)
haftmann@35036
   874
    assume "\<not> 0 < a"
haftmann@35036
   875
    then have "a \<le> 0" by auto
haftmann@35036
   876
    with A have "- a < 0" by (rule less_le_trans)
haftmann@35036
   877
    then show ?thesis by auto
haftmann@35036
   878
  qed
haftmann@35036
   879
next
haftmann@35036
   880
  assume A: "0 < a" show "- a < a"
haftmann@35036
   881
  proof (rule less_trans)
haftmann@35036
   882
    show "- a < 0" using A by (simp add: minus_le_iff)
haftmann@35036
   883
  next
haftmann@35036
   884
    show "0 < a" using A .
haftmann@35036
   885
  qed
haftmann@35036
   886
qed
haftmann@35036
   887
haftmann@35036
   888
lemma less_eq_neg_nonpos [simp]:
haftmann@25303
   889
  "a \<le> - a \<longleftrightarrow> a \<le> 0"
haftmann@25303
   890
proof
haftmann@25303
   891
  assume A: "a \<le> - a" show "a \<le> 0"
haftmann@25303
   892
  proof (rule classical)
haftmann@25303
   893
    assume "\<not> a \<le> 0"
haftmann@25303
   894
    then have "0 < a" by auto
haftmann@25303
   895
    then have "0 < - a" using A by (rule less_le_trans)
haftmann@25303
   896
    then show ?thesis by auto
haftmann@25303
   897
  qed
haftmann@25303
   898
next
haftmann@25303
   899
  assume A: "a \<le> 0" show "a \<le> - a"
haftmann@25303
   900
  proof (rule order_trans)
haftmann@25303
   901
    show "0 \<le> - a" using A by (simp add: minus_le_iff)
haftmann@25303
   902
  next
haftmann@25303
   903
    show "a \<le> 0" using A .
haftmann@25303
   904
  qed
haftmann@25303
   905
qed
haftmann@25303
   906
haftmann@35036
   907
lemma equal_neg_zero [simp]:
haftmann@25303
   908
  "a = - a \<longleftrightarrow> a = 0"
haftmann@25303
   909
proof
haftmann@25303
   910
  assume "a = 0" then show "a = - a" by simp
haftmann@25303
   911
next
haftmann@25303
   912
  assume A: "a = - a" show "a = 0"
haftmann@25303
   913
  proof (cases "0 \<le> a")
haftmann@25303
   914
    case True with A have "0 \<le> - a" by auto
haftmann@25303
   915
    with le_minus_iff have "a \<le> 0" by simp
haftmann@25303
   916
    with True show ?thesis by (auto intro: order_trans)
haftmann@25303
   917
  next
haftmann@25303
   918
    case False then have B: "a \<le> 0" by auto
haftmann@25303
   919
    with A have "- a \<le> 0" by auto
haftmann@25303
   920
    with B show ?thesis by (auto intro: order_trans)
haftmann@25303
   921
  qed
haftmann@25303
   922
qed
haftmann@25303
   923
haftmann@35036
   924
lemma neg_equal_zero [simp]:
haftmann@25303
   925
  "- a = a \<longleftrightarrow> a = 0"
haftmann@35036
   926
  by (auto dest: sym)
haftmann@35036
   927
haftmann@35036
   928
lemma double_zero [simp]:
haftmann@35036
   929
  "a + a = 0 \<longleftrightarrow> a = 0"
haftmann@35036
   930
proof
haftmann@35036
   931
  assume assm: "a + a = 0"
haftmann@35036
   932
  then have a: "- a = a" by (rule minus_unique)
huffman@35216
   933
  then show "a = 0" by (simp only: neg_equal_zero)
haftmann@35036
   934
qed simp
haftmann@35036
   935
haftmann@35036
   936
lemma double_zero_sym [simp]:
haftmann@35036
   937
  "0 = a + a \<longleftrightarrow> a = 0"
haftmann@35036
   938
  by (rule, drule sym) simp_all
haftmann@35036
   939
haftmann@35036
   940
lemma zero_less_double_add_iff_zero_less_single_add [simp]:
haftmann@35036
   941
  "0 < a + a \<longleftrightarrow> 0 < a"
haftmann@35036
   942
proof
haftmann@35036
   943
  assume "0 < a + a"
haftmann@35036
   944
  then have "0 - a < a" by (simp only: diff_less_eq)
haftmann@35036
   945
  then have "- a < a" by simp
huffman@35216
   946
  then show "0 < a" by (simp only: neg_less_nonneg)
haftmann@35036
   947
next
haftmann@35036
   948
  assume "0 < a"
haftmann@35036
   949
  with this have "0 + 0 < a + a"
haftmann@35036
   950
    by (rule add_strict_mono)
haftmann@35036
   951
  then show "0 < a + a" by simp
haftmann@35036
   952
qed
haftmann@35036
   953
haftmann@35036
   954
lemma zero_le_double_add_iff_zero_le_single_add [simp]:
haftmann@35036
   955
  "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
haftmann@35036
   956
  by (auto simp add: le_less)
haftmann@35036
   957
haftmann@35036
   958
lemma double_add_less_zero_iff_single_add_less_zero [simp]:
haftmann@35036
   959
  "a + a < 0 \<longleftrightarrow> a < 0"
haftmann@35036
   960
proof -
haftmann@35036
   961
  have "\<not> a + a < 0 \<longleftrightarrow> \<not> a < 0"
haftmann@35036
   962
    by (simp add: not_less)
haftmann@35036
   963
  then show ?thesis by simp
haftmann@35036
   964
qed
haftmann@35036
   965
haftmann@35036
   966
lemma double_add_le_zero_iff_single_add_le_zero [simp]:
haftmann@35036
   967
  "a + a \<le> 0 \<longleftrightarrow> a \<le> 0" 
haftmann@35036
   968
proof -
haftmann@35036
   969
  have "\<not> a + a \<le> 0 \<longleftrightarrow> \<not> a \<le> 0"
haftmann@35036
   970
    by (simp add: not_le)
haftmann@35036
   971
  then show ?thesis by simp
haftmann@35036
   972
qed
haftmann@35036
   973
haftmann@35036
   974
lemma le_minus_self_iff:
haftmann@35036
   975
  "a \<le> - a \<longleftrightarrow> a \<le> 0"
haftmann@35036
   976
proof -
haftmann@35036
   977
  from add_le_cancel_left [of "- a" "a + a" 0]
haftmann@35036
   978
  have "a \<le> - a \<longleftrightarrow> a + a \<le> 0" 
haftmann@35036
   979
    by (simp add: add_assoc [symmetric])
haftmann@35036
   980
  thus ?thesis by simp
haftmann@35036
   981
qed
haftmann@35036
   982
haftmann@35036
   983
lemma minus_le_self_iff:
haftmann@35036
   984
  "- a \<le> a \<longleftrightarrow> 0 \<le> a"
haftmann@35036
   985
proof -
haftmann@35036
   986
  from add_le_cancel_left [of "- a" 0 "a + a"]
haftmann@35036
   987
  have "- a \<le> a \<longleftrightarrow> 0 \<le> a + a" 
haftmann@35036
   988
    by (simp add: add_assoc [symmetric])
haftmann@35036
   989
  thus ?thesis by simp
haftmann@35036
   990
qed
haftmann@35036
   991
haftmann@35036
   992
lemma minus_max_eq_min:
haftmann@35036
   993
  "- max x y = min (-x) (-y)"
haftmann@35036
   994
  by (auto simp add: max_def min_def)
haftmann@35036
   995
haftmann@35036
   996
lemma minus_min_eq_max:
haftmann@35036
   997
  "- min x y = max (-x) (-y)"
haftmann@35036
   998
  by (auto simp add: max_def min_def)
haftmann@25303
   999
haftmann@25267
  1000
end
haftmann@25267
  1001
haftmann@36302
  1002
context ordered_comm_monoid_add
haftmann@36302
  1003
begin
obua@14738
  1004
paulson@15234
  1005
lemma add_increasing:
haftmann@36302
  1006
  "0 \<le> a \<Longrightarrow> b \<le> c \<Longrightarrow> b \<le> a + c"
haftmann@36302
  1007
  by (insert add_mono [of 0 a b c], simp)
obua@14738
  1008
nipkow@15539
  1009
lemma add_increasing2:
haftmann@36302
  1010
  "0 \<le> c \<Longrightarrow> b \<le> a \<Longrightarrow> b \<le> a + c"
haftmann@36302
  1011
  by (simp add: add_increasing add_commute [of a])
nipkow@15539
  1012
paulson@15234
  1013
lemma add_strict_increasing:
haftmann@36302
  1014
  "0 < a \<Longrightarrow> b \<le> c \<Longrightarrow> b < a + c"
haftmann@36302
  1015
  by (insert add_less_le_mono [of 0 a b c], simp)
paulson@15234
  1016
paulson@15234
  1017
lemma add_strict_increasing2:
haftmann@36302
  1018
  "0 \<le> a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@36302
  1019
  by (insert add_le_less_mono [of 0 a b c], simp)
haftmann@36302
  1020
haftmann@36302
  1021
end
paulson@15234
  1022
haftmann@35092
  1023
class abs =
haftmann@35092
  1024
  fixes abs :: "'a \<Rightarrow> 'a"
haftmann@35092
  1025
begin
haftmann@35092
  1026
haftmann@35092
  1027
notation (xsymbols)
haftmann@35092
  1028
  abs  ("\<bar>_\<bar>")
haftmann@35092
  1029
haftmann@35092
  1030
notation (HTML output)
haftmann@35092
  1031
  abs  ("\<bar>_\<bar>")
haftmann@35092
  1032
haftmann@35092
  1033
end
haftmann@35092
  1034
haftmann@35092
  1035
class sgn =
haftmann@35092
  1036
  fixes sgn :: "'a \<Rightarrow> 'a"
haftmann@35092
  1037
haftmann@35092
  1038
class abs_if = minus + uminus + ord + zero + abs +
haftmann@35092
  1039
  assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"
haftmann@35092
  1040
haftmann@35092
  1041
class sgn_if = minus + uminus + zero + one + ord + sgn +
haftmann@35092
  1042
  assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
haftmann@35092
  1043
begin
haftmann@35092
  1044
haftmann@35092
  1045
lemma sgn0 [simp]: "sgn 0 = 0"
haftmann@35092
  1046
  by (simp add:sgn_if)
haftmann@35092
  1047
haftmann@35092
  1048
end
obua@14738
  1049
haftmann@35028
  1050
class ordered_ab_group_add_abs = ordered_ab_group_add + abs +
haftmann@25303
  1051
  assumes abs_ge_zero [simp]: "\<bar>a\<bar> \<ge> 0"
haftmann@25303
  1052
    and abs_ge_self: "a \<le> \<bar>a\<bar>"
haftmann@25303
  1053
    and abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
haftmann@25303
  1054
    and abs_minus_cancel [simp]: "\<bar>-a\<bar> = \<bar>a\<bar>"
haftmann@25303
  1055
    and abs_triangle_ineq: "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25303
  1056
begin
haftmann@25303
  1057
haftmann@25307
  1058
lemma abs_minus_le_zero: "- \<bar>a\<bar> \<le> 0"
haftmann@25307
  1059
  unfolding neg_le_0_iff_le by simp
haftmann@25307
  1060
haftmann@25307
  1061
lemma abs_of_nonneg [simp]:
nipkow@29667
  1062
  assumes nonneg: "0 \<le> a" shows "\<bar>a\<bar> = a"
haftmann@25307
  1063
proof (rule antisym)
haftmann@25307
  1064
  from nonneg le_imp_neg_le have "- a \<le> 0" by simp
haftmann@25307
  1065
  from this nonneg have "- a \<le> a" by (rule order_trans)
haftmann@25307
  1066
  then show "\<bar>a\<bar> \<le> a" by (auto intro: abs_leI)
haftmann@25307
  1067
qed (rule abs_ge_self)
haftmann@25307
  1068
haftmann@25307
  1069
lemma abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
nipkow@29667
  1070
by (rule antisym)
haftmann@36302
  1071
   (auto intro!: abs_ge_self abs_leI order_trans [of "- \<bar>a\<bar>" 0 "\<bar>a\<bar>"])
haftmann@25307
  1072
haftmann@25307
  1073
lemma abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
haftmann@25307
  1074
proof -
haftmann@25307
  1075
  have "\<bar>a\<bar> = 0 \<Longrightarrow> a = 0"
haftmann@25307
  1076
  proof (rule antisym)
haftmann@25307
  1077
    assume zero: "\<bar>a\<bar> = 0"
haftmann@25307
  1078
    with abs_ge_self show "a \<le> 0" by auto
haftmann@25307
  1079
    from zero have "\<bar>-a\<bar> = 0" by simp
haftmann@36302
  1080
    with abs_ge_self [of "- a"] have "- a \<le> 0" by auto
haftmann@25307
  1081
    with neg_le_0_iff_le show "0 \<le> a" by auto
haftmann@25307
  1082
  qed
haftmann@25307
  1083
  then show ?thesis by auto
haftmann@25307
  1084
qed
haftmann@25307
  1085
haftmann@25303
  1086
lemma abs_zero [simp]: "\<bar>0\<bar> = 0"
nipkow@29667
  1087
by simp
avigad@16775
  1088
blanchet@35828
  1089
lemma abs_0_eq [simp, no_atp]: "0 = \<bar>a\<bar> \<longleftrightarrow> a = 0"
haftmann@25303
  1090
proof -
haftmann@25303
  1091
  have "0 = \<bar>a\<bar> \<longleftrightarrow> \<bar>a\<bar> = 0" by (simp only: eq_ac)
haftmann@25303
  1092
  thus ?thesis by simp
haftmann@25303
  1093
qed
haftmann@25303
  1094
haftmann@25303
  1095
lemma abs_le_zero_iff [simp]: "\<bar>a\<bar> \<le> 0 \<longleftrightarrow> a = 0" 
haftmann@25303
  1096
proof
haftmann@25303
  1097
  assume "\<bar>a\<bar> \<le> 0"
haftmann@25303
  1098
  then have "\<bar>a\<bar> = 0" by (rule antisym) simp
haftmann@25303
  1099
  thus "a = 0" by simp
haftmann@25303
  1100
next
haftmann@25303
  1101
  assume "a = 0"
haftmann@25303
  1102
  thus "\<bar>a\<bar> \<le> 0" by simp
haftmann@25303
  1103
qed
haftmann@25303
  1104
haftmann@25303
  1105
lemma zero_less_abs_iff [simp]: "0 < \<bar>a\<bar> \<longleftrightarrow> a \<noteq> 0"
nipkow@29667
  1106
by (simp add: less_le)
haftmann@25303
  1107
haftmann@25303
  1108
lemma abs_not_less_zero [simp]: "\<not> \<bar>a\<bar> < 0"
haftmann@25303
  1109
proof -
haftmann@25303
  1110
  have a: "\<And>x y. x \<le> y \<Longrightarrow> \<not> y < x" by auto
haftmann@25303
  1111
  show ?thesis by (simp add: a)
haftmann@25303
  1112
qed
avigad@16775
  1113
haftmann@25303
  1114
lemma abs_ge_minus_self: "- a \<le> \<bar>a\<bar>"
haftmann@25303
  1115
proof -
haftmann@25303
  1116
  have "- a \<le> \<bar>-a\<bar>" by (rule abs_ge_self)
haftmann@25303
  1117
  then show ?thesis by simp
haftmann@25303
  1118
qed
haftmann@25303
  1119
haftmann@25303
  1120
lemma abs_minus_commute: 
haftmann@25303
  1121
  "\<bar>a - b\<bar> = \<bar>b - a\<bar>"
haftmann@25303
  1122
proof -
haftmann@25303
  1123
  have "\<bar>a - b\<bar> = \<bar>- (a - b)\<bar>" by (simp only: abs_minus_cancel)
haftmann@25303
  1124
  also have "... = \<bar>b - a\<bar>" by simp
haftmann@25303
  1125
  finally show ?thesis .
haftmann@25303
  1126
qed
haftmann@25303
  1127
haftmann@25303
  1128
lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a"
nipkow@29667
  1129
by (rule abs_of_nonneg, rule less_imp_le)
avigad@16775
  1130
haftmann@25303
  1131
lemma abs_of_nonpos [simp]:
nipkow@29667
  1132
  assumes "a \<le> 0" shows "\<bar>a\<bar> = - a"
haftmann@25303
  1133
proof -
haftmann@25303
  1134
  let ?b = "- a"
haftmann@25303
  1135
  have "- ?b \<le> 0 \<Longrightarrow> \<bar>- ?b\<bar> = - (- ?b)"
haftmann@25303
  1136
  unfolding abs_minus_cancel [of "?b"]
haftmann@25303
  1137
  unfolding neg_le_0_iff_le [of "?b"]
haftmann@25303
  1138
  unfolding minus_minus by (erule abs_of_nonneg)
haftmann@25303
  1139
  then show ?thesis using assms by auto
haftmann@25303
  1140
qed
haftmann@25303
  1141
  
haftmann@25303
  1142
lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a"
nipkow@29667
  1143
by (rule abs_of_nonpos, rule less_imp_le)
haftmann@25303
  1144
haftmann@25303
  1145
lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b"
nipkow@29667
  1146
by (insert abs_ge_self, blast intro: order_trans)
haftmann@25303
  1147
haftmann@25303
  1148
lemma abs_le_D2: "\<bar>a\<bar> \<le> b \<Longrightarrow> - a \<le> b"
haftmann@36302
  1149
by (insert abs_le_D1 [of "- a"], simp)
haftmann@25303
  1150
haftmann@25303
  1151
lemma abs_le_iff: "\<bar>a\<bar> \<le> b \<longleftrightarrow> a \<le> b \<and> - a \<le> b"
nipkow@29667
  1152
by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
haftmann@25303
  1153
haftmann@25303
  1154
lemma abs_triangle_ineq2: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>a - b\<bar>"
haftmann@36302
  1155
proof -
haftmann@36302
  1156
  have "\<bar>a\<bar> = \<bar>b + (a - b)\<bar>"
haftmann@36302
  1157
    by (simp add: algebra_simps add_diff_cancel)
haftmann@36302
  1158
  then have "\<bar>a\<bar> \<le> \<bar>b\<bar> + \<bar>a - b\<bar>"
haftmann@36302
  1159
    by (simp add: abs_triangle_ineq)
haftmann@36302
  1160
  then show ?thesis
haftmann@36302
  1161
    by (simp add: algebra_simps)
haftmann@36302
  1162
qed
haftmann@36302
  1163
haftmann@36302
  1164
lemma abs_triangle_ineq2_sym: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>b - a\<bar>"
haftmann@36302
  1165
  by (simp only: abs_minus_commute [of b] abs_triangle_ineq2)
avigad@16775
  1166
haftmann@25303
  1167
lemma abs_triangle_ineq3: "\<bar>\<bar>a\<bar> - \<bar>b\<bar>\<bar> \<le> \<bar>a - b\<bar>"
haftmann@36302
  1168
  by (simp add: abs_le_iff abs_triangle_ineq2 abs_triangle_ineq2_sym)
avigad@16775
  1169
haftmann@25303
  1170
lemma abs_triangle_ineq4: "\<bar>a - b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25303
  1171
proof -
haftmann@36302
  1172
  have "\<bar>a - b\<bar> = \<bar>a + - b\<bar>" by (subst diff_minus, rule refl)
haftmann@36302
  1173
  also have "... \<le> \<bar>a\<bar> + \<bar>- b\<bar>" by (rule abs_triangle_ineq)
nipkow@29667
  1174
  finally show ?thesis by simp
haftmann@25303
  1175
qed
avigad@16775
  1176
haftmann@25303
  1177
lemma abs_diff_triangle_ineq: "\<bar>a + b - (c + d)\<bar> \<le> \<bar>a - c\<bar> + \<bar>b - d\<bar>"
haftmann@25303
  1178
proof -
haftmann@25303
  1179
  have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: diff_minus add_ac)
haftmann@25303
  1180
  also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq)
haftmann@25303
  1181
  finally show ?thesis .
haftmann@25303
  1182
qed
avigad@16775
  1183
haftmann@25303
  1184
lemma abs_add_abs [simp]:
haftmann@25303
  1185
  "\<bar>\<bar>a\<bar> + \<bar>b\<bar>\<bar> = \<bar>a\<bar> + \<bar>b\<bar>" (is "?L = ?R")
haftmann@25303
  1186
proof (rule antisym)
haftmann@25303
  1187
  show "?L \<ge> ?R" by(rule abs_ge_self)
haftmann@25303
  1188
next
haftmann@25303
  1189
  have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq)
haftmann@25303
  1190
  also have "\<dots> = ?R" by simp
haftmann@25303
  1191
  finally show "?L \<le> ?R" .
haftmann@25303
  1192
qed
haftmann@25303
  1193
haftmann@25303
  1194
end
obua@14738
  1195
obua@15178
  1196
haftmann@25090
  1197
subsection {* Tools setup *}
haftmann@25090
  1198
blanchet@35828
  1199
lemma add_mono_thms_linordered_semiring [no_atp]:
haftmann@35028
  1200
  fixes i j k :: "'a\<Colon>ordered_ab_semigroup_add"
haftmann@25077
  1201
  shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1202
    and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1203
    and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1204
    and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"
haftmann@25077
  1205
by (rule add_mono, clarify+)+
haftmann@25077
  1206
blanchet@35828
  1207
lemma add_mono_thms_linordered_field [no_atp]:
haftmann@35028
  1208
  fixes i j k :: "'a\<Colon>ordered_cancel_ab_semigroup_add"
haftmann@25077
  1209
  shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1210
    and "i = j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1211
    and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1212
    and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1213
    and "i < j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1214
by (auto intro: add_strict_right_mono add_strict_left_mono
haftmann@25077
  1215
  add_less_le_mono add_le_less_mono add_strict_mono)
haftmann@25077
  1216
haftmann@33364
  1217
code_modulename SML
haftmann@35050
  1218
  Groups Arith
haftmann@33364
  1219
haftmann@33364
  1220
code_modulename OCaml
haftmann@35050
  1221
  Groups Arith
haftmann@33364
  1222
haftmann@33364
  1223
code_modulename Haskell
haftmann@35050
  1224
  Groups Arith
haftmann@33364
  1225
haftmann@37889
  1226
haftmann@37889
  1227
text {* Legacy *}
haftmann@37889
  1228
haftmann@37889
  1229
lemmas diff_def = diff_minus
haftmann@37889
  1230
obua@14738
  1231
end