src/HOL/Groups.thy
author haftmann
Fri Jun 11 17:14:02 2010 +0200 (2010-06-11)
changeset 37407 61dd8c145da7
parent 36977 71c8973a604b
child 37884 314a88278715
permissions -rw-r--r--
declare lex_prod_def [code del]
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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 ("~~/src/Provers/Arith/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 {*
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let
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  fun tr' c = (c, fn show_sorts => fn T => fn ts =>
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    if (not o null) ts orelse T = dummyT
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      orelse not (! 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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use "~~/src/Provers/Arith/abel_cancel.ML"
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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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obua@14738
   365
text{*The next two equations can make the simplifier loop!*}
obua@14738
   366
haftmann@25062
   367
lemma equation_minus_iff:
haftmann@25062
   368
  "a = - b \<longleftrightarrow> b = - a"
obua@14738
   369
proof -
haftmann@25062
   370
  have "- (- a) = - b \<longleftrightarrow> - a = b" by (rule neg_equal_iff_equal)
haftmann@25062
   371
  thus ?thesis by (simp add: eq_commute)
haftmann@25062
   372
qed
haftmann@25062
   373
haftmann@25062
   374
lemma minus_equation_iff:
haftmann@25062
   375
  "- a = b \<longleftrightarrow> - b = a"
haftmann@25062
   376
proof -
haftmann@25062
   377
  have "- a = - (- b) \<longleftrightarrow> a = -b" by (rule neg_equal_iff_equal)
obua@14738
   378
  thus ?thesis by (simp add: eq_commute)
obua@14738
   379
qed
obua@14738
   380
huffman@28130
   381
lemma diff_add_cancel: "a - b + b = a"
nipkow@29667
   382
by (simp add: diff_minus add_assoc)
huffman@28130
   383
huffman@28130
   384
lemma add_diff_cancel: "a + b - b = a"
nipkow@29667
   385
by (simp add: diff_minus add_assoc)
nipkow@29667
   386
haftmann@36348
   387
declare diff_minus[symmetric, algebra_simps, field_simps]
huffman@28130
   388
huffman@29914
   389
lemma eq_neg_iff_add_eq_0: "a = - b \<longleftrightarrow> a + b = 0"
huffman@29914
   390
proof
huffman@29914
   391
  assume "a = - b" then show "a + b = 0" by simp
huffman@29914
   392
next
huffman@29914
   393
  assume "a + b = 0"
huffman@29914
   394
  moreover have "a + (b + - b) = (a + b) + - b"
huffman@29914
   395
    by (simp only: add_assoc)
huffman@29914
   396
  ultimately show "a = - b" by simp
huffman@29914
   397
qed
huffman@29914
   398
haftmann@25062
   399
end
haftmann@25062
   400
haftmann@25762
   401
class ab_group_add = minus + uminus + comm_monoid_add +
haftmann@25062
   402
  assumes ab_left_minus: "- a + a = 0"
haftmann@25062
   403
  assumes ab_diff_minus: "a - b = a + (- b)"
haftmann@25267
   404
begin
haftmann@25062
   405
haftmann@25267
   406
subclass group_add
haftmann@28823
   407
  proof qed (simp_all add: ab_left_minus ab_diff_minus)
haftmann@25062
   408
huffman@29904
   409
subclass cancel_comm_monoid_add
haftmann@28823
   410
proof
haftmann@25062
   411
  fix a b c :: 'a
haftmann@25062
   412
  assume "a + b = a + c"
haftmann@25062
   413
  then have "- a + a + b = - a + a + c"
haftmann@25062
   414
    unfolding add_assoc by simp
haftmann@25062
   415
  then show "b = c" by simp
haftmann@25062
   416
qed
haftmann@25062
   417
haftmann@36348
   418
lemma uminus_add_conv_diff[algebra_simps, field_simps]:
haftmann@25062
   419
  "- a + b = b - a"
nipkow@29667
   420
by (simp add:diff_minus add_commute)
haftmann@25062
   421
haftmann@25062
   422
lemma minus_add_distrib [simp]:
haftmann@25062
   423
  "- (a + b) = - a + - b"
huffman@34146
   424
by (rule minus_unique) (simp add: add_ac)
haftmann@25062
   425
haftmann@25062
   426
lemma minus_diff_eq [simp]:
haftmann@25062
   427
  "- (a - b) = b - a"
nipkow@29667
   428
by (simp add: diff_minus add_commute)
haftmann@25077
   429
haftmann@36348
   430
lemma add_diff_eq[algebra_simps, field_simps]: "a + (b - c) = (a + b) - c"
nipkow@29667
   431
by (simp add: diff_minus add_ac)
haftmann@25077
   432
haftmann@36348
   433
lemma diff_add_eq[algebra_simps, field_simps]: "(a - b) + c = (a + c) - b"
nipkow@29667
   434
by (simp add: diff_minus add_ac)
haftmann@25077
   435
haftmann@36348
   436
lemma diff_eq_eq[algebra_simps, field_simps]: "a - b = c \<longleftrightarrow> a = c + b"
nipkow@29667
   437
by (auto simp add: diff_minus add_assoc)
haftmann@25077
   438
haftmann@36348
   439
lemma eq_diff_eq[algebra_simps, field_simps]: "a = c - b \<longleftrightarrow> a + b = c"
nipkow@29667
   440
by (auto simp add: diff_minus add_assoc)
haftmann@25077
   441
haftmann@36348
   442
lemma diff_diff_eq[algebra_simps, field_simps]: "(a - b) - c = a - (b + c)"
nipkow@29667
   443
by (simp add: diff_minus add_ac)
haftmann@25077
   444
haftmann@36348
   445
lemma diff_diff_eq2[algebra_simps, field_simps]: "a - (b - c) = (a + c) - b"
nipkow@29667
   446
by (simp add: diff_minus add_ac)
haftmann@25077
   447
haftmann@25077
   448
lemma eq_iff_diff_eq_0: "a = b \<longleftrightarrow> a - b = 0"
nipkow@29667
   449
by (simp add: algebra_simps)
haftmann@25077
   450
huffman@35216
   451
(* FIXME: duplicates right_minus_eq from class group_add *)
huffman@35216
   452
(* but only this one is declared as a simp rule. *)
blanchet@35828
   453
lemma diff_eq_0_iff_eq [simp, no_atp]: "a - b = 0 \<longleftrightarrow> a = b"
huffman@30629
   454
by (simp add: algebra_simps)
huffman@30629
   455
haftmann@25062
   456
end
obua@14738
   457
obua@14738
   458
subsection {* (Partially) Ordered Groups *} 
obua@14738
   459
haftmann@35301
   460
text {*
haftmann@35301
   461
  The theory of partially ordered groups is taken from the books:
haftmann@35301
   462
  \begin{itemize}
haftmann@35301
   463
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
haftmann@35301
   464
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
haftmann@35301
   465
  \end{itemize}
haftmann@35301
   466
  Most of the used notions can also be looked up in 
haftmann@35301
   467
  \begin{itemize}
haftmann@35301
   468
  \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
haftmann@35301
   469
  \item \emph{Algebra I} by van der Waerden, Springer.
haftmann@35301
   470
  \end{itemize}
haftmann@35301
   471
*}
haftmann@35301
   472
haftmann@35028
   473
class ordered_ab_semigroup_add = order + ab_semigroup_add +
haftmann@25062
   474
  assumes add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b"
haftmann@25062
   475
begin
haftmann@24380
   476
haftmann@25062
   477
lemma add_right_mono:
haftmann@25062
   478
  "a \<le> b \<Longrightarrow> a + c \<le> b + c"
nipkow@29667
   479
by (simp add: add_commute [of _ c] add_left_mono)
obua@14738
   480
obua@14738
   481
text {* non-strict, in both arguments *}
obua@14738
   482
lemma add_mono:
haftmann@25062
   483
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d"
obua@14738
   484
  apply (erule add_right_mono [THEN order_trans])
obua@14738
   485
  apply (simp add: add_commute add_left_mono)
obua@14738
   486
  done
obua@14738
   487
haftmann@25062
   488
end
haftmann@25062
   489
haftmann@35028
   490
class ordered_cancel_ab_semigroup_add =
haftmann@35028
   491
  ordered_ab_semigroup_add + cancel_ab_semigroup_add
haftmann@25062
   492
begin
haftmann@25062
   493
obua@14738
   494
lemma add_strict_left_mono:
haftmann@25062
   495
  "a < b \<Longrightarrow> c + a < c + b"
nipkow@29667
   496
by (auto simp add: less_le add_left_mono)
obua@14738
   497
obua@14738
   498
lemma add_strict_right_mono:
haftmann@25062
   499
  "a < b \<Longrightarrow> a + c < b + c"
nipkow@29667
   500
by (simp add: add_commute [of _ c] add_strict_left_mono)
obua@14738
   501
obua@14738
   502
text{*Strict monotonicity in both arguments*}
haftmann@25062
   503
lemma add_strict_mono:
haftmann@25062
   504
  "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
haftmann@25062
   505
apply (erule add_strict_right_mono [THEN less_trans])
obua@14738
   506
apply (erule add_strict_left_mono)
obua@14738
   507
done
obua@14738
   508
obua@14738
   509
lemma add_less_le_mono:
haftmann@25062
   510
  "a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c < b + d"
haftmann@25062
   511
apply (erule add_strict_right_mono [THEN less_le_trans])
haftmann@25062
   512
apply (erule add_left_mono)
obua@14738
   513
done
obua@14738
   514
obua@14738
   515
lemma add_le_less_mono:
haftmann@25062
   516
  "a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
haftmann@25062
   517
apply (erule add_right_mono [THEN le_less_trans])
obua@14738
   518
apply (erule add_strict_left_mono) 
obua@14738
   519
done
obua@14738
   520
haftmann@25062
   521
end
haftmann@25062
   522
haftmann@35028
   523
class ordered_ab_semigroup_add_imp_le =
haftmann@35028
   524
  ordered_cancel_ab_semigroup_add +
haftmann@25062
   525
  assumes add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b"
haftmann@25062
   526
begin
haftmann@25062
   527
obua@14738
   528
lemma add_less_imp_less_left:
nipkow@29667
   529
  assumes less: "c + a < c + b" shows "a < b"
obua@14738
   530
proof -
obua@14738
   531
  from less have le: "c + a <= c + b" by (simp add: order_le_less)
obua@14738
   532
  have "a <= b" 
obua@14738
   533
    apply (insert le)
obua@14738
   534
    apply (drule add_le_imp_le_left)
obua@14738
   535
    by (insert le, drule add_le_imp_le_left, assumption)
obua@14738
   536
  moreover have "a \<noteq> b"
obua@14738
   537
  proof (rule ccontr)
obua@14738
   538
    assume "~(a \<noteq> b)"
obua@14738
   539
    then have "a = b" by simp
obua@14738
   540
    then have "c + a = c + b" by simp
obua@14738
   541
    with less show "False"by simp
obua@14738
   542
  qed
obua@14738
   543
  ultimately show "a < b" by (simp add: order_le_less)
obua@14738
   544
qed
obua@14738
   545
obua@14738
   546
lemma add_less_imp_less_right:
haftmann@25062
   547
  "a + c < b + c \<Longrightarrow> a < b"
obua@14738
   548
apply (rule add_less_imp_less_left [of c])
obua@14738
   549
apply (simp add: add_commute)  
obua@14738
   550
done
obua@14738
   551
obua@14738
   552
lemma add_less_cancel_left [simp]:
haftmann@25062
   553
  "c + a < c + b \<longleftrightarrow> a < b"
nipkow@29667
   554
by (blast intro: add_less_imp_less_left add_strict_left_mono) 
obua@14738
   555
obua@14738
   556
lemma add_less_cancel_right [simp]:
haftmann@25062
   557
  "a + c < b + c \<longleftrightarrow> a < b"
nipkow@29667
   558
by (blast intro: add_less_imp_less_right add_strict_right_mono)
obua@14738
   559
obua@14738
   560
lemma add_le_cancel_left [simp]:
haftmann@25062
   561
  "c + a \<le> c + b \<longleftrightarrow> a \<le> b"
nipkow@29667
   562
by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono) 
obua@14738
   563
obua@14738
   564
lemma add_le_cancel_right [simp]:
haftmann@25062
   565
  "a + c \<le> b + c \<longleftrightarrow> a \<le> b"
nipkow@29667
   566
by (simp add: add_commute [of a c] add_commute [of b c])
obua@14738
   567
obua@14738
   568
lemma add_le_imp_le_right:
haftmann@25062
   569
  "a + c \<le> b + c \<Longrightarrow> a \<le> b"
nipkow@29667
   570
by simp
haftmann@25062
   571
haftmann@25077
   572
lemma max_add_distrib_left:
haftmann@25077
   573
  "max x y + z = max (x + z) (y + z)"
haftmann@25077
   574
  unfolding max_def by auto
haftmann@25077
   575
haftmann@25077
   576
lemma min_add_distrib_left:
haftmann@25077
   577
  "min x y + z = min (x + z) (y + z)"
haftmann@25077
   578
  unfolding min_def by auto
haftmann@25077
   579
haftmann@25062
   580
end
haftmann@25062
   581
haftmann@25303
   582
subsection {* Support for reasoning about signs *}
haftmann@25303
   583
haftmann@35028
   584
class ordered_comm_monoid_add =
haftmann@35028
   585
  ordered_cancel_ab_semigroup_add + comm_monoid_add
haftmann@25303
   586
begin
haftmann@25303
   587
haftmann@25303
   588
lemma add_pos_nonneg:
nipkow@29667
   589
  assumes "0 < a" and "0 \<le> b" shows "0 < a + b"
haftmann@25303
   590
proof -
haftmann@25303
   591
  have "0 + 0 < a + b" 
haftmann@25303
   592
    using assms by (rule add_less_le_mono)
haftmann@25303
   593
  then show ?thesis by simp
haftmann@25303
   594
qed
haftmann@25303
   595
haftmann@25303
   596
lemma add_pos_pos:
nipkow@29667
   597
  assumes "0 < a" and "0 < b" shows "0 < a + b"
nipkow@29667
   598
by (rule add_pos_nonneg) (insert assms, auto)
haftmann@25303
   599
haftmann@25303
   600
lemma add_nonneg_pos:
nipkow@29667
   601
  assumes "0 \<le> a" and "0 < b" shows "0 < a + b"
haftmann@25303
   602
proof -
haftmann@25303
   603
  have "0 + 0 < a + b" 
haftmann@25303
   604
    using assms by (rule add_le_less_mono)
haftmann@25303
   605
  then show ?thesis by simp
haftmann@25303
   606
qed
haftmann@25303
   607
huffman@36977
   608
lemma add_nonneg_nonneg [simp]:
nipkow@29667
   609
  assumes "0 \<le> a" and "0 \<le> b" shows "0 \<le> a + b"
haftmann@25303
   610
proof -
haftmann@25303
   611
  have "0 + 0 \<le> a + b" 
haftmann@25303
   612
    using assms by (rule add_mono)
haftmann@25303
   613
  then show ?thesis by simp
haftmann@25303
   614
qed
haftmann@25303
   615
huffman@30691
   616
lemma add_neg_nonpos:
nipkow@29667
   617
  assumes "a < 0" and "b \<le> 0" shows "a + b < 0"
haftmann@25303
   618
proof -
haftmann@25303
   619
  have "a + b < 0 + 0"
haftmann@25303
   620
    using assms by (rule add_less_le_mono)
haftmann@25303
   621
  then show ?thesis by simp
haftmann@25303
   622
qed
haftmann@25303
   623
haftmann@25303
   624
lemma add_neg_neg: 
nipkow@29667
   625
  assumes "a < 0" and "b < 0" shows "a + b < 0"
nipkow@29667
   626
by (rule add_neg_nonpos) (insert assms, auto)
haftmann@25303
   627
haftmann@25303
   628
lemma add_nonpos_neg:
nipkow@29667
   629
  assumes "a \<le> 0" and "b < 0" shows "a + b < 0"
haftmann@25303
   630
proof -
haftmann@25303
   631
  have "a + b < 0 + 0"
haftmann@25303
   632
    using assms by (rule add_le_less_mono)
haftmann@25303
   633
  then show ?thesis by simp
haftmann@25303
   634
qed
haftmann@25303
   635
haftmann@25303
   636
lemma add_nonpos_nonpos:
nipkow@29667
   637
  assumes "a \<le> 0" and "b \<le> 0" shows "a + b \<le> 0"
haftmann@25303
   638
proof -
haftmann@25303
   639
  have "a + b \<le> 0 + 0"
haftmann@25303
   640
    using assms by (rule add_mono)
haftmann@25303
   641
  then show ?thesis by simp
haftmann@25303
   642
qed
haftmann@25303
   643
huffman@30691
   644
lemmas add_sign_intros =
huffman@30691
   645
  add_pos_nonneg add_pos_pos add_nonneg_pos add_nonneg_nonneg
huffman@30691
   646
  add_neg_nonpos add_neg_neg add_nonpos_neg add_nonpos_nonpos
huffman@30691
   647
huffman@29886
   648
lemma add_nonneg_eq_0_iff:
huffman@29886
   649
  assumes x: "0 \<le> x" and y: "0 \<le> y"
huffman@29886
   650
  shows "x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@29886
   651
proof (intro iffI conjI)
huffman@29886
   652
  have "x = x + 0" by simp
huffman@29886
   653
  also have "x + 0 \<le> x + y" using y by (rule add_left_mono)
huffman@29886
   654
  also assume "x + y = 0"
huffman@29886
   655
  also have "0 \<le> x" using x .
huffman@29886
   656
  finally show "x = 0" .
huffman@29886
   657
next
huffman@29886
   658
  have "y = 0 + y" by simp
huffman@29886
   659
  also have "0 + y \<le> x + y" using x by (rule add_right_mono)
huffman@29886
   660
  also assume "x + y = 0"
huffman@29886
   661
  also have "0 \<le> y" using y .
huffman@29886
   662
  finally show "y = 0" .
huffman@29886
   663
next
huffman@29886
   664
  assume "x = 0 \<and> y = 0"
huffman@29886
   665
  then show "x + y = 0" by simp
huffman@29886
   666
qed
huffman@29886
   667
haftmann@25303
   668
end
haftmann@25303
   669
haftmann@35028
   670
class ordered_ab_group_add =
haftmann@35028
   671
  ab_group_add + ordered_ab_semigroup_add
haftmann@25062
   672
begin
haftmann@25062
   673
haftmann@35028
   674
subclass ordered_cancel_ab_semigroup_add ..
haftmann@25062
   675
haftmann@35028
   676
subclass ordered_ab_semigroup_add_imp_le
haftmann@28823
   677
proof
haftmann@25062
   678
  fix a b c :: 'a
haftmann@25062
   679
  assume "c + a \<le> c + b"
haftmann@25062
   680
  hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono)
haftmann@25062
   681
  hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc)
haftmann@25062
   682
  thus "a \<le> b" by simp
haftmann@25062
   683
qed
haftmann@25062
   684
haftmann@35028
   685
subclass ordered_comm_monoid_add ..
haftmann@25303
   686
haftmann@25077
   687
lemma max_diff_distrib_left:
haftmann@25077
   688
  shows "max x y - z = max (x - z) (y - z)"
nipkow@29667
   689
by (simp add: diff_minus, rule max_add_distrib_left) 
haftmann@25077
   690
haftmann@25077
   691
lemma min_diff_distrib_left:
haftmann@25077
   692
  shows "min x y - z = min (x - z) (y - z)"
nipkow@29667
   693
by (simp add: diff_minus, rule min_add_distrib_left) 
haftmann@25077
   694
haftmann@25077
   695
lemma le_imp_neg_le:
nipkow@29667
   696
  assumes "a \<le> b" shows "-b \<le> -a"
haftmann@25077
   697
proof -
nipkow@29667
   698
  have "-a+a \<le> -a+b" using `a \<le> b` by (rule add_left_mono) 
nipkow@29667
   699
  hence "0 \<le> -a+b" by simp
nipkow@29667
   700
  hence "0 + (-b) \<le> (-a + b) + (-b)" by (rule add_right_mono) 
nipkow@29667
   701
  thus ?thesis by (simp add: add_assoc)
haftmann@25077
   702
qed
haftmann@25077
   703
haftmann@25077
   704
lemma neg_le_iff_le [simp]: "- b \<le> - a \<longleftrightarrow> a \<le> b"
haftmann@25077
   705
proof 
haftmann@25077
   706
  assume "- b \<le> - a"
nipkow@29667
   707
  hence "- (- a) \<le> - (- b)" by (rule le_imp_neg_le)
haftmann@25077
   708
  thus "a\<le>b" by simp
haftmann@25077
   709
next
haftmann@25077
   710
  assume "a\<le>b"
haftmann@25077
   711
  thus "-b \<le> -a" by (rule le_imp_neg_le)
haftmann@25077
   712
qed
haftmann@25077
   713
haftmann@25077
   714
lemma neg_le_0_iff_le [simp]: "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"
nipkow@29667
   715
by (subst neg_le_iff_le [symmetric], simp)
haftmann@25077
   716
haftmann@25077
   717
lemma neg_0_le_iff_le [simp]: "0 \<le> - a \<longleftrightarrow> a \<le> 0"
nipkow@29667
   718
by (subst neg_le_iff_le [symmetric], simp)
haftmann@25077
   719
haftmann@25077
   720
lemma neg_less_iff_less [simp]: "- b < - a \<longleftrightarrow> a < b"
nipkow@29667
   721
by (force simp add: less_le) 
haftmann@25077
   722
haftmann@25077
   723
lemma neg_less_0_iff_less [simp]: "- a < 0 \<longleftrightarrow> 0 < a"
nipkow@29667
   724
by (subst neg_less_iff_less [symmetric], simp)
haftmann@25077
   725
haftmann@25077
   726
lemma neg_0_less_iff_less [simp]: "0 < - a \<longleftrightarrow> a < 0"
nipkow@29667
   727
by (subst neg_less_iff_less [symmetric], simp)
haftmann@25077
   728
haftmann@25077
   729
text{*The next several equations can make the simplifier loop!*}
haftmann@25077
   730
haftmann@25077
   731
lemma less_minus_iff: "a < - b \<longleftrightarrow> b < - a"
haftmann@25077
   732
proof -
haftmann@25077
   733
  have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
haftmann@25077
   734
  thus ?thesis by simp
haftmann@25077
   735
qed
haftmann@25077
   736
haftmann@25077
   737
lemma minus_less_iff: "- a < b \<longleftrightarrow> - b < a"
haftmann@25077
   738
proof -
haftmann@25077
   739
  have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
haftmann@25077
   740
  thus ?thesis by simp
haftmann@25077
   741
qed
haftmann@25077
   742
haftmann@25077
   743
lemma le_minus_iff: "a \<le> - b \<longleftrightarrow> b \<le> - a"
haftmann@25077
   744
proof -
haftmann@25077
   745
  have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff)
haftmann@25077
   746
  have "(- (- a) <= -b) = (b <= - a)" 
haftmann@25077
   747
    apply (auto simp only: le_less)
haftmann@25077
   748
    apply (drule mm)
haftmann@25077
   749
    apply (simp_all)
haftmann@25077
   750
    apply (drule mm[simplified], assumption)
haftmann@25077
   751
    done
haftmann@25077
   752
  then show ?thesis by simp
haftmann@25077
   753
qed
haftmann@25077
   754
haftmann@25077
   755
lemma minus_le_iff: "- a \<le> b \<longleftrightarrow> - b \<le> a"
nipkow@29667
   756
by (auto simp add: le_less minus_less_iff)
haftmann@25077
   757
haftmann@25077
   758
lemma less_iff_diff_less_0: "a < b \<longleftrightarrow> a - b < 0"
haftmann@25077
   759
proof -
haftmann@25077
   760
  have  "(a < b) = (a + (- b) < b + (-b))"  
haftmann@25077
   761
    by (simp only: add_less_cancel_right)
haftmann@25077
   762
  also have "... =  (a - b < 0)" by (simp add: diff_minus)
haftmann@25077
   763
  finally show ?thesis .
haftmann@25077
   764
qed
haftmann@25077
   765
haftmann@36348
   766
lemma diff_less_eq[algebra_simps, field_simps]: "a - b < c \<longleftrightarrow> a < c + b"
haftmann@25077
   767
apply (subst less_iff_diff_less_0 [of a])
haftmann@25077
   768
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
haftmann@25077
   769
apply (simp add: diff_minus add_ac)
haftmann@25077
   770
done
haftmann@25077
   771
haftmann@36348
   772
lemma less_diff_eq[algebra_simps, field_simps]: "a < c - b \<longleftrightarrow> a + b < c"
haftmann@36302
   773
apply (subst less_iff_diff_less_0 [of "a + b"])
haftmann@25077
   774
apply (subst less_iff_diff_less_0 [of a])
haftmann@25077
   775
apply (simp add: diff_minus add_ac)
haftmann@25077
   776
done
haftmann@25077
   777
haftmann@36348
   778
lemma diff_le_eq[algebra_simps, field_simps]: "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
nipkow@29667
   779
by (auto simp add: le_less diff_less_eq diff_add_cancel add_diff_cancel)
haftmann@25077
   780
haftmann@36348
   781
lemma le_diff_eq[algebra_simps, field_simps]: "a \<le> c - b \<longleftrightarrow> a + b \<le> c"
nipkow@29667
   782
by (auto simp add: le_less less_diff_eq diff_add_cancel add_diff_cancel)
haftmann@25077
   783
haftmann@25077
   784
lemma le_iff_diff_le_0: "a \<le> b \<longleftrightarrow> a - b \<le> 0"
nipkow@29667
   785
by (simp add: algebra_simps)
haftmann@25077
   786
haftmann@25077
   787
end
haftmann@25077
   788
haftmann@35028
   789
class linordered_ab_semigroup_add =
haftmann@35028
   790
  linorder + ordered_ab_semigroup_add
haftmann@25062
   791
haftmann@35028
   792
class linordered_cancel_ab_semigroup_add =
haftmann@35028
   793
  linorder + ordered_cancel_ab_semigroup_add
haftmann@25267
   794
begin
haftmann@25062
   795
haftmann@35028
   796
subclass linordered_ab_semigroup_add ..
haftmann@25062
   797
haftmann@35028
   798
subclass ordered_ab_semigroup_add_imp_le
haftmann@28823
   799
proof
haftmann@25062
   800
  fix a b c :: 'a
haftmann@25062
   801
  assume le: "c + a <= c + b"  
haftmann@25062
   802
  show "a <= b"
haftmann@25062
   803
  proof (rule ccontr)
haftmann@25062
   804
    assume w: "~ a \<le> b"
haftmann@25062
   805
    hence "b <= a" by (simp add: linorder_not_le)
haftmann@25062
   806
    hence le2: "c + b <= c + a" by (rule add_left_mono)
haftmann@25062
   807
    have "a = b" 
haftmann@25062
   808
      apply (insert le)
haftmann@25062
   809
      apply (insert le2)
haftmann@25062
   810
      apply (drule antisym, simp_all)
haftmann@25062
   811
      done
haftmann@25062
   812
    with w show False 
haftmann@25062
   813
      by (simp add: linorder_not_le [symmetric])
haftmann@25062
   814
  qed
haftmann@25062
   815
qed
haftmann@25062
   816
haftmann@25267
   817
end
haftmann@25267
   818
haftmann@35028
   819
class linordered_ab_group_add = linorder + ordered_ab_group_add
haftmann@25267
   820
begin
haftmann@25230
   821
haftmann@35028
   822
subclass linordered_cancel_ab_semigroup_add ..
haftmann@25230
   823
haftmann@35036
   824
lemma neg_less_eq_nonneg [simp]:
haftmann@25303
   825
  "- a \<le> a \<longleftrightarrow> 0 \<le> a"
haftmann@25303
   826
proof
haftmann@25303
   827
  assume A: "- a \<le> a" show "0 \<le> a"
haftmann@25303
   828
  proof (rule classical)
haftmann@25303
   829
    assume "\<not> 0 \<le> a"
haftmann@25303
   830
    then have "a < 0" by auto
haftmann@25303
   831
    with A have "- a < 0" by (rule le_less_trans)
haftmann@25303
   832
    then show ?thesis by auto
haftmann@25303
   833
  qed
haftmann@25303
   834
next
haftmann@25303
   835
  assume A: "0 \<le> a" show "- a \<le> a"
haftmann@25303
   836
  proof (rule order_trans)
haftmann@25303
   837
    show "- a \<le> 0" using A by (simp add: minus_le_iff)
haftmann@25303
   838
  next
haftmann@25303
   839
    show "0 \<le> a" using A .
haftmann@25303
   840
  qed
haftmann@25303
   841
qed
haftmann@35036
   842
haftmann@35036
   843
lemma neg_less_nonneg [simp]:
haftmann@35036
   844
  "- a < a \<longleftrightarrow> 0 < a"
haftmann@35036
   845
proof
haftmann@35036
   846
  assume A: "- a < a" show "0 < a"
haftmann@35036
   847
  proof (rule classical)
haftmann@35036
   848
    assume "\<not> 0 < a"
haftmann@35036
   849
    then have "a \<le> 0" by auto
haftmann@35036
   850
    with A have "- a < 0" by (rule less_le_trans)
haftmann@35036
   851
    then show ?thesis by auto
haftmann@35036
   852
  qed
haftmann@35036
   853
next
haftmann@35036
   854
  assume A: "0 < a" show "- a < a"
haftmann@35036
   855
  proof (rule less_trans)
haftmann@35036
   856
    show "- a < 0" using A by (simp add: minus_le_iff)
haftmann@35036
   857
  next
haftmann@35036
   858
    show "0 < a" using A .
haftmann@35036
   859
  qed
haftmann@35036
   860
qed
haftmann@35036
   861
haftmann@35036
   862
lemma less_eq_neg_nonpos [simp]:
haftmann@25303
   863
  "a \<le> - a \<longleftrightarrow> a \<le> 0"
haftmann@25303
   864
proof
haftmann@25303
   865
  assume A: "a \<le> - a" show "a \<le> 0"
haftmann@25303
   866
  proof (rule classical)
haftmann@25303
   867
    assume "\<not> a \<le> 0"
haftmann@25303
   868
    then have "0 < a" by auto
haftmann@25303
   869
    then have "0 < - a" using A by (rule less_le_trans)
haftmann@25303
   870
    then show ?thesis by auto
haftmann@25303
   871
  qed
haftmann@25303
   872
next
haftmann@25303
   873
  assume A: "a \<le> 0" show "a \<le> - a"
haftmann@25303
   874
  proof (rule order_trans)
haftmann@25303
   875
    show "0 \<le> - a" using A by (simp add: minus_le_iff)
haftmann@25303
   876
  next
haftmann@25303
   877
    show "a \<le> 0" using A .
haftmann@25303
   878
  qed
haftmann@25303
   879
qed
haftmann@25303
   880
haftmann@35036
   881
lemma equal_neg_zero [simp]:
haftmann@25303
   882
  "a = - a \<longleftrightarrow> a = 0"
haftmann@25303
   883
proof
haftmann@25303
   884
  assume "a = 0" then show "a = - a" by simp
haftmann@25303
   885
next
haftmann@25303
   886
  assume A: "a = - a" show "a = 0"
haftmann@25303
   887
  proof (cases "0 \<le> a")
haftmann@25303
   888
    case True with A have "0 \<le> - a" by auto
haftmann@25303
   889
    with le_minus_iff have "a \<le> 0" by simp
haftmann@25303
   890
    with True show ?thesis by (auto intro: order_trans)
haftmann@25303
   891
  next
haftmann@25303
   892
    case False then have B: "a \<le> 0" by auto
haftmann@25303
   893
    with A have "- a \<le> 0" by auto
haftmann@25303
   894
    with B show ?thesis by (auto intro: order_trans)
haftmann@25303
   895
  qed
haftmann@25303
   896
qed
haftmann@25303
   897
haftmann@35036
   898
lemma neg_equal_zero [simp]:
haftmann@25303
   899
  "- a = a \<longleftrightarrow> a = 0"
haftmann@35036
   900
  by (auto dest: sym)
haftmann@35036
   901
haftmann@35036
   902
lemma double_zero [simp]:
haftmann@35036
   903
  "a + a = 0 \<longleftrightarrow> a = 0"
haftmann@35036
   904
proof
haftmann@35036
   905
  assume assm: "a + a = 0"
haftmann@35036
   906
  then have a: "- a = a" by (rule minus_unique)
huffman@35216
   907
  then show "a = 0" by (simp only: neg_equal_zero)
haftmann@35036
   908
qed simp
haftmann@35036
   909
haftmann@35036
   910
lemma double_zero_sym [simp]:
haftmann@35036
   911
  "0 = a + a \<longleftrightarrow> a = 0"
haftmann@35036
   912
  by (rule, drule sym) simp_all
haftmann@35036
   913
haftmann@35036
   914
lemma zero_less_double_add_iff_zero_less_single_add [simp]:
haftmann@35036
   915
  "0 < a + a \<longleftrightarrow> 0 < a"
haftmann@35036
   916
proof
haftmann@35036
   917
  assume "0 < a + a"
haftmann@35036
   918
  then have "0 - a < a" by (simp only: diff_less_eq)
haftmann@35036
   919
  then have "- a < a" by simp
huffman@35216
   920
  then show "0 < a" by (simp only: neg_less_nonneg)
haftmann@35036
   921
next
haftmann@35036
   922
  assume "0 < a"
haftmann@35036
   923
  with this have "0 + 0 < a + a"
haftmann@35036
   924
    by (rule add_strict_mono)
haftmann@35036
   925
  then show "0 < a + a" by simp
haftmann@35036
   926
qed
haftmann@35036
   927
haftmann@35036
   928
lemma zero_le_double_add_iff_zero_le_single_add [simp]:
haftmann@35036
   929
  "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
haftmann@35036
   930
  by (auto simp add: le_less)
haftmann@35036
   931
haftmann@35036
   932
lemma double_add_less_zero_iff_single_add_less_zero [simp]:
haftmann@35036
   933
  "a + a < 0 \<longleftrightarrow> a < 0"
haftmann@35036
   934
proof -
haftmann@35036
   935
  have "\<not> a + a < 0 \<longleftrightarrow> \<not> a < 0"
haftmann@35036
   936
    by (simp add: not_less)
haftmann@35036
   937
  then show ?thesis by simp
haftmann@35036
   938
qed
haftmann@35036
   939
haftmann@35036
   940
lemma double_add_le_zero_iff_single_add_le_zero [simp]:
haftmann@35036
   941
  "a + a \<le> 0 \<longleftrightarrow> a \<le> 0" 
haftmann@35036
   942
proof -
haftmann@35036
   943
  have "\<not> a + a \<le> 0 \<longleftrightarrow> \<not> a \<le> 0"
haftmann@35036
   944
    by (simp add: not_le)
haftmann@35036
   945
  then show ?thesis by simp
haftmann@35036
   946
qed
haftmann@35036
   947
haftmann@35036
   948
lemma le_minus_self_iff:
haftmann@35036
   949
  "a \<le> - a \<longleftrightarrow> a \<le> 0"
haftmann@35036
   950
proof -
haftmann@35036
   951
  from add_le_cancel_left [of "- a" "a + a" 0]
haftmann@35036
   952
  have "a \<le> - a \<longleftrightarrow> a + a \<le> 0" 
haftmann@35036
   953
    by (simp add: add_assoc [symmetric])
haftmann@35036
   954
  thus ?thesis by simp
haftmann@35036
   955
qed
haftmann@35036
   956
haftmann@35036
   957
lemma minus_le_self_iff:
haftmann@35036
   958
  "- a \<le> a \<longleftrightarrow> 0 \<le> a"
haftmann@35036
   959
proof -
haftmann@35036
   960
  from add_le_cancel_left [of "- a" 0 "a + a"]
haftmann@35036
   961
  have "- a \<le> a \<longleftrightarrow> 0 \<le> a + a" 
haftmann@35036
   962
    by (simp add: add_assoc [symmetric])
haftmann@35036
   963
  thus ?thesis by simp
haftmann@35036
   964
qed
haftmann@35036
   965
haftmann@35036
   966
lemma minus_max_eq_min:
haftmann@35036
   967
  "- max x y = min (-x) (-y)"
haftmann@35036
   968
  by (auto simp add: max_def min_def)
haftmann@35036
   969
haftmann@35036
   970
lemma minus_min_eq_max:
haftmann@35036
   971
  "- min x y = max (-x) (-y)"
haftmann@35036
   972
  by (auto simp add: max_def min_def)
haftmann@25303
   973
haftmann@25267
   974
end
haftmann@25267
   975
haftmann@36302
   976
context ordered_comm_monoid_add
haftmann@36302
   977
begin
obua@14738
   978
paulson@15234
   979
lemma add_increasing:
haftmann@36302
   980
  "0 \<le> a \<Longrightarrow> b \<le> c \<Longrightarrow> b \<le> a + c"
haftmann@36302
   981
  by (insert add_mono [of 0 a b c], simp)
obua@14738
   982
nipkow@15539
   983
lemma add_increasing2:
haftmann@36302
   984
  "0 \<le> c \<Longrightarrow> b \<le> a \<Longrightarrow> b \<le> a + c"
haftmann@36302
   985
  by (simp add: add_increasing add_commute [of a])
nipkow@15539
   986
paulson@15234
   987
lemma add_strict_increasing:
haftmann@36302
   988
  "0 < a \<Longrightarrow> b \<le> c \<Longrightarrow> b < a + c"
haftmann@36302
   989
  by (insert add_less_le_mono [of 0 a b c], simp)
paulson@15234
   990
paulson@15234
   991
lemma add_strict_increasing2:
haftmann@36302
   992
  "0 \<le> a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@36302
   993
  by (insert add_le_less_mono [of 0 a b c], simp)
haftmann@36302
   994
haftmann@36302
   995
end
paulson@15234
   996
haftmann@35092
   997
class abs =
haftmann@35092
   998
  fixes abs :: "'a \<Rightarrow> 'a"
haftmann@35092
   999
begin
haftmann@35092
  1000
haftmann@35092
  1001
notation (xsymbols)
haftmann@35092
  1002
  abs  ("\<bar>_\<bar>")
haftmann@35092
  1003
haftmann@35092
  1004
notation (HTML output)
haftmann@35092
  1005
  abs  ("\<bar>_\<bar>")
haftmann@35092
  1006
haftmann@35092
  1007
end
haftmann@35092
  1008
haftmann@35092
  1009
class sgn =
haftmann@35092
  1010
  fixes sgn :: "'a \<Rightarrow> 'a"
haftmann@35092
  1011
haftmann@35092
  1012
class abs_if = minus + uminus + ord + zero + abs +
haftmann@35092
  1013
  assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"
haftmann@35092
  1014
haftmann@35092
  1015
class sgn_if = minus + uminus + zero + one + ord + sgn +
haftmann@35092
  1016
  assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
haftmann@35092
  1017
begin
haftmann@35092
  1018
haftmann@35092
  1019
lemma sgn0 [simp]: "sgn 0 = 0"
haftmann@35092
  1020
  by (simp add:sgn_if)
haftmann@35092
  1021
haftmann@35092
  1022
end
obua@14738
  1023
haftmann@35028
  1024
class ordered_ab_group_add_abs = ordered_ab_group_add + abs +
haftmann@25303
  1025
  assumes abs_ge_zero [simp]: "\<bar>a\<bar> \<ge> 0"
haftmann@25303
  1026
    and abs_ge_self: "a \<le> \<bar>a\<bar>"
haftmann@25303
  1027
    and abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
haftmann@25303
  1028
    and abs_minus_cancel [simp]: "\<bar>-a\<bar> = \<bar>a\<bar>"
haftmann@25303
  1029
    and abs_triangle_ineq: "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25303
  1030
begin
haftmann@25303
  1031
haftmann@25307
  1032
lemma abs_minus_le_zero: "- \<bar>a\<bar> \<le> 0"
haftmann@25307
  1033
  unfolding neg_le_0_iff_le by simp
haftmann@25307
  1034
haftmann@25307
  1035
lemma abs_of_nonneg [simp]:
nipkow@29667
  1036
  assumes nonneg: "0 \<le> a" shows "\<bar>a\<bar> = a"
haftmann@25307
  1037
proof (rule antisym)
haftmann@25307
  1038
  from nonneg le_imp_neg_le have "- a \<le> 0" by simp
haftmann@25307
  1039
  from this nonneg have "- a \<le> a" by (rule order_trans)
haftmann@25307
  1040
  then show "\<bar>a\<bar> \<le> a" by (auto intro: abs_leI)
haftmann@25307
  1041
qed (rule abs_ge_self)
haftmann@25307
  1042
haftmann@25307
  1043
lemma abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
nipkow@29667
  1044
by (rule antisym)
haftmann@36302
  1045
   (auto intro!: abs_ge_self abs_leI order_trans [of "- \<bar>a\<bar>" 0 "\<bar>a\<bar>"])
haftmann@25307
  1046
haftmann@25307
  1047
lemma abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
haftmann@25307
  1048
proof -
haftmann@25307
  1049
  have "\<bar>a\<bar> = 0 \<Longrightarrow> a = 0"
haftmann@25307
  1050
  proof (rule antisym)
haftmann@25307
  1051
    assume zero: "\<bar>a\<bar> = 0"
haftmann@25307
  1052
    with abs_ge_self show "a \<le> 0" by auto
haftmann@25307
  1053
    from zero have "\<bar>-a\<bar> = 0" by simp
haftmann@36302
  1054
    with abs_ge_self [of "- a"] have "- a \<le> 0" by auto
haftmann@25307
  1055
    with neg_le_0_iff_le show "0 \<le> a" by auto
haftmann@25307
  1056
  qed
haftmann@25307
  1057
  then show ?thesis by auto
haftmann@25307
  1058
qed
haftmann@25307
  1059
haftmann@25303
  1060
lemma abs_zero [simp]: "\<bar>0\<bar> = 0"
nipkow@29667
  1061
by simp
avigad@16775
  1062
blanchet@35828
  1063
lemma abs_0_eq [simp, no_atp]: "0 = \<bar>a\<bar> \<longleftrightarrow> a = 0"
haftmann@25303
  1064
proof -
haftmann@25303
  1065
  have "0 = \<bar>a\<bar> \<longleftrightarrow> \<bar>a\<bar> = 0" by (simp only: eq_ac)
haftmann@25303
  1066
  thus ?thesis by simp
haftmann@25303
  1067
qed
haftmann@25303
  1068
haftmann@25303
  1069
lemma abs_le_zero_iff [simp]: "\<bar>a\<bar> \<le> 0 \<longleftrightarrow> a = 0" 
haftmann@25303
  1070
proof
haftmann@25303
  1071
  assume "\<bar>a\<bar> \<le> 0"
haftmann@25303
  1072
  then have "\<bar>a\<bar> = 0" by (rule antisym) simp
haftmann@25303
  1073
  thus "a = 0" by simp
haftmann@25303
  1074
next
haftmann@25303
  1075
  assume "a = 0"
haftmann@25303
  1076
  thus "\<bar>a\<bar> \<le> 0" by simp
haftmann@25303
  1077
qed
haftmann@25303
  1078
haftmann@25303
  1079
lemma zero_less_abs_iff [simp]: "0 < \<bar>a\<bar> \<longleftrightarrow> a \<noteq> 0"
nipkow@29667
  1080
by (simp add: less_le)
haftmann@25303
  1081
haftmann@25303
  1082
lemma abs_not_less_zero [simp]: "\<not> \<bar>a\<bar> < 0"
haftmann@25303
  1083
proof -
haftmann@25303
  1084
  have a: "\<And>x y. x \<le> y \<Longrightarrow> \<not> y < x" by auto
haftmann@25303
  1085
  show ?thesis by (simp add: a)
haftmann@25303
  1086
qed
avigad@16775
  1087
haftmann@25303
  1088
lemma abs_ge_minus_self: "- a \<le> \<bar>a\<bar>"
haftmann@25303
  1089
proof -
haftmann@25303
  1090
  have "- a \<le> \<bar>-a\<bar>" by (rule abs_ge_self)
haftmann@25303
  1091
  then show ?thesis by simp
haftmann@25303
  1092
qed
haftmann@25303
  1093
haftmann@25303
  1094
lemma abs_minus_commute: 
haftmann@25303
  1095
  "\<bar>a - b\<bar> = \<bar>b - a\<bar>"
haftmann@25303
  1096
proof -
haftmann@25303
  1097
  have "\<bar>a - b\<bar> = \<bar>- (a - b)\<bar>" by (simp only: abs_minus_cancel)
haftmann@25303
  1098
  also have "... = \<bar>b - a\<bar>" by simp
haftmann@25303
  1099
  finally show ?thesis .
haftmann@25303
  1100
qed
haftmann@25303
  1101
haftmann@25303
  1102
lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a"
nipkow@29667
  1103
by (rule abs_of_nonneg, rule less_imp_le)
avigad@16775
  1104
haftmann@25303
  1105
lemma abs_of_nonpos [simp]:
nipkow@29667
  1106
  assumes "a \<le> 0" shows "\<bar>a\<bar> = - a"
haftmann@25303
  1107
proof -
haftmann@25303
  1108
  let ?b = "- a"
haftmann@25303
  1109
  have "- ?b \<le> 0 \<Longrightarrow> \<bar>- ?b\<bar> = - (- ?b)"
haftmann@25303
  1110
  unfolding abs_minus_cancel [of "?b"]
haftmann@25303
  1111
  unfolding neg_le_0_iff_le [of "?b"]
haftmann@25303
  1112
  unfolding minus_minus by (erule abs_of_nonneg)
haftmann@25303
  1113
  then show ?thesis using assms by auto
haftmann@25303
  1114
qed
haftmann@25303
  1115
  
haftmann@25303
  1116
lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a"
nipkow@29667
  1117
by (rule abs_of_nonpos, rule less_imp_le)
haftmann@25303
  1118
haftmann@25303
  1119
lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b"
nipkow@29667
  1120
by (insert abs_ge_self, blast intro: order_trans)
haftmann@25303
  1121
haftmann@25303
  1122
lemma abs_le_D2: "\<bar>a\<bar> \<le> b \<Longrightarrow> - a \<le> b"
haftmann@36302
  1123
by (insert abs_le_D1 [of "- a"], simp)
haftmann@25303
  1124
haftmann@25303
  1125
lemma abs_le_iff: "\<bar>a\<bar> \<le> b \<longleftrightarrow> a \<le> b \<and> - a \<le> b"
nipkow@29667
  1126
by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
haftmann@25303
  1127
haftmann@25303
  1128
lemma abs_triangle_ineq2: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>a - b\<bar>"
haftmann@36302
  1129
proof -
haftmann@36302
  1130
  have "\<bar>a\<bar> = \<bar>b + (a - b)\<bar>"
haftmann@36302
  1131
    by (simp add: algebra_simps add_diff_cancel)
haftmann@36302
  1132
  then have "\<bar>a\<bar> \<le> \<bar>b\<bar> + \<bar>a - b\<bar>"
haftmann@36302
  1133
    by (simp add: abs_triangle_ineq)
haftmann@36302
  1134
  then show ?thesis
haftmann@36302
  1135
    by (simp add: algebra_simps)
haftmann@36302
  1136
qed
haftmann@36302
  1137
haftmann@36302
  1138
lemma abs_triangle_ineq2_sym: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>b - a\<bar>"
haftmann@36302
  1139
  by (simp only: abs_minus_commute [of b] abs_triangle_ineq2)
avigad@16775
  1140
haftmann@25303
  1141
lemma abs_triangle_ineq3: "\<bar>\<bar>a\<bar> - \<bar>b\<bar>\<bar> \<le> \<bar>a - b\<bar>"
haftmann@36302
  1142
  by (simp add: abs_le_iff abs_triangle_ineq2 abs_triangle_ineq2_sym)
avigad@16775
  1143
haftmann@25303
  1144
lemma abs_triangle_ineq4: "\<bar>a - b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25303
  1145
proof -
haftmann@36302
  1146
  have "\<bar>a - b\<bar> = \<bar>a + - b\<bar>" by (subst diff_minus, rule refl)
haftmann@36302
  1147
  also have "... \<le> \<bar>a\<bar> + \<bar>- b\<bar>" by (rule abs_triangle_ineq)
nipkow@29667
  1148
  finally show ?thesis by simp
haftmann@25303
  1149
qed
avigad@16775
  1150
haftmann@25303
  1151
lemma abs_diff_triangle_ineq: "\<bar>a + b - (c + d)\<bar> \<le> \<bar>a - c\<bar> + \<bar>b - d\<bar>"
haftmann@25303
  1152
proof -
haftmann@25303
  1153
  have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: diff_minus add_ac)
haftmann@25303
  1154
  also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq)
haftmann@25303
  1155
  finally show ?thesis .
haftmann@25303
  1156
qed
avigad@16775
  1157
haftmann@25303
  1158
lemma abs_add_abs [simp]:
haftmann@25303
  1159
  "\<bar>\<bar>a\<bar> + \<bar>b\<bar>\<bar> = \<bar>a\<bar> + \<bar>b\<bar>" (is "?L = ?R")
haftmann@25303
  1160
proof (rule antisym)
haftmann@25303
  1161
  show "?L \<ge> ?R" by(rule abs_ge_self)
haftmann@25303
  1162
next
haftmann@25303
  1163
  have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq)
haftmann@25303
  1164
  also have "\<dots> = ?R" by simp
haftmann@25303
  1165
  finally show "?L \<le> ?R" .
haftmann@25303
  1166
qed
haftmann@25303
  1167
haftmann@25303
  1168
end
obua@14738
  1169
obua@14754
  1170
text {* Needed for abelian cancellation simprocs: *}
obua@14754
  1171
obua@14754
  1172
lemma add_cancel_21: "((x::'a::ab_group_add) + (y + z) = y + u) = (x + z = u)"
obua@14754
  1173
apply (subst add_left_commute)
obua@14754
  1174
apply (subst add_left_cancel)
obua@14754
  1175
apply simp
obua@14754
  1176
done
obua@14754
  1177
obua@14754
  1178
lemma add_cancel_end: "(x + (y + z) = y) = (x = - (z::'a::ab_group_add))"
obua@14754
  1179
apply (subst add_cancel_21[of _ _ _ 0, simplified])
obua@14754
  1180
apply (simp add: add_right_cancel[symmetric, of "x" "-z" "z", simplified])
obua@14754
  1181
done
obua@14754
  1182
haftmann@35028
  1183
lemma less_eqI: "(x::'a::ordered_ab_group_add) - y = x' - y' \<Longrightarrow> (x < y) = (x' < y')"
obua@14754
  1184
by (simp add: less_iff_diff_less_0[of x y] less_iff_diff_less_0[of x' y'])
obua@14754
  1185
haftmann@35028
  1186
lemma le_eqI: "(x::'a::ordered_ab_group_add) - y = x' - y' \<Longrightarrow> (y <= x) = (y' <= x')"
obua@14754
  1187
apply (simp add: le_iff_diff_le_0[of y x] le_iff_diff_le_0[of  y' x'])
obua@14754
  1188
apply (simp add: neg_le_iff_le[symmetric, of "y-x" 0] neg_le_iff_le[symmetric, of "y'-x'" 0])
obua@14754
  1189
done
obua@14754
  1190
obua@14754
  1191
lemma eq_eqI: "(x::'a::ab_group_add) - y = x' - y' \<Longrightarrow> (x = y) = (x' = y')"
huffman@30629
  1192
by (simp only: eq_iff_diff_eq_0[of x y] eq_iff_diff_eq_0[of x' y'])
obua@14754
  1193
obua@14754
  1194
lemma diff_def: "(x::'a::ab_group_add) - y == x + (-y)"
obua@14754
  1195
by (simp add: diff_minus)
obua@14754
  1196
haftmann@25090
  1197
lemma le_add_right_mono: 
obua@15178
  1198
  assumes 
haftmann@35028
  1199
  "a <= b + (c::'a::ordered_ab_group_add)"
obua@15178
  1200
  "c <= d"    
obua@15178
  1201
  shows "a <= b + d"
obua@15178
  1202
  apply (rule_tac order_trans[where y = "b+c"])
obua@15178
  1203
  apply (simp_all add: prems)
obua@15178
  1204
  done
obua@15178
  1205
obua@15178
  1206
haftmann@25090
  1207
subsection {* Tools setup *}
haftmann@25090
  1208
blanchet@35828
  1209
lemma add_mono_thms_linordered_semiring [no_atp]:
haftmann@35028
  1210
  fixes i j k :: "'a\<Colon>ordered_ab_semigroup_add"
haftmann@25077
  1211
  shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1212
    and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1213
    and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1214
    and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"
haftmann@25077
  1215
by (rule add_mono, clarify+)+
haftmann@25077
  1216
blanchet@35828
  1217
lemma add_mono_thms_linordered_field [no_atp]:
haftmann@35028
  1218
  fixes i j k :: "'a\<Colon>ordered_cancel_ab_semigroup_add"
haftmann@25077
  1219
  shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1220
    and "i = j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1221
    and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1222
    and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1223
    and "i < j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1224
by (auto intro: add_strict_right_mono add_strict_left_mono
haftmann@25077
  1225
  add_less_le_mono add_le_less_mono add_strict_mono)
haftmann@25077
  1226
paulson@17085
  1227
text{*Simplification of @{term "x-y < 0"}, etc.*}
blanchet@35828
  1228
lemmas diff_less_0_iff_less [simp, no_atp] = less_iff_diff_less_0 [symmetric]
blanchet@35828
  1229
lemmas diff_le_0_iff_le [simp, no_atp] = le_iff_diff_le_0 [symmetric]
paulson@17085
  1230
haftmann@22482
  1231
ML {*
wenzelm@27250
  1232
structure ab_group_add_cancel = Abel_Cancel
wenzelm@27250
  1233
(
haftmann@22482
  1234
haftmann@22482
  1235
(* term order for abelian groups *)
haftmann@22482
  1236
haftmann@22482
  1237
fun agrp_ord (Const (a, _)) = find_index (fn a' => a = a')
haftmann@35267
  1238
      [@{const_name Groups.zero}, @{const_name Groups.plus},
haftmann@35267
  1239
        @{const_name Groups.uminus}, @{const_name Groups.minus}]
haftmann@22482
  1240
  | agrp_ord _ = ~1;
haftmann@22482
  1241
wenzelm@35408
  1242
fun termless_agrp (a, b) = (Term_Ord.term_lpo agrp_ord (a, b) = LESS);
haftmann@22482
  1243
haftmann@22482
  1244
local
haftmann@22482
  1245
  val ac1 = mk_meta_eq @{thm add_assoc};
haftmann@22482
  1246
  val ac2 = mk_meta_eq @{thm add_commute};
haftmann@22482
  1247
  val ac3 = mk_meta_eq @{thm add_left_commute};
haftmann@35267
  1248
  fun solve_add_ac thy _ (_ $ (Const (@{const_name Groups.plus},_) $ _ $ _) $ _) =
haftmann@22482
  1249
        SOME ac1
haftmann@35267
  1250
    | solve_add_ac thy _ (_ $ x $ (Const (@{const_name Groups.plus},_) $ y $ z)) =
haftmann@22482
  1251
        if termless_agrp (y, x) then SOME ac3 else NONE
haftmann@22482
  1252
    | solve_add_ac thy _ (_ $ x $ y) =
haftmann@22482
  1253
        if termless_agrp (y, x) then SOME ac2 else NONE
haftmann@22482
  1254
    | solve_add_ac thy _ _ = NONE
haftmann@22482
  1255
in
wenzelm@32010
  1256
  val add_ac_proc = Simplifier.simproc @{theory}
haftmann@22482
  1257
    "add_ac_proc" ["x + y::'a::ab_semigroup_add"] solve_add_ac;
haftmann@22482
  1258
end;
haftmann@22482
  1259
wenzelm@27250
  1260
val eq_reflection = @{thm eq_reflection};
wenzelm@27250
  1261
  
wenzelm@27250
  1262
val T = @{typ "'a::ab_group_add"};
wenzelm@27250
  1263
haftmann@22482
  1264
val cancel_ss = HOL_basic_ss settermless termless_agrp
haftmann@22482
  1265
  addsimprocs [add_ac_proc] addsimps
nipkow@23085
  1266
  [@{thm add_0_left}, @{thm add_0_right}, @{thm diff_def},
haftmann@22482
  1267
   @{thm minus_add_distrib}, @{thm minus_minus}, @{thm minus_zero},
haftmann@22482
  1268
   @{thm right_minus}, @{thm left_minus}, @{thm add_minus_cancel},
haftmann@22482
  1269
   @{thm minus_add_cancel}];
wenzelm@27250
  1270
wenzelm@27250
  1271
val sum_pats = [@{cterm "x + y::'a::ab_group_add"}, @{cterm "x - y::'a::ab_group_add"}];
haftmann@22482
  1272
  
haftmann@22548
  1273
val eqI_rules = [@{thm less_eqI}, @{thm le_eqI}, @{thm eq_eqI}];
haftmann@22482
  1274
haftmann@22482
  1275
val dest_eqI = 
wenzelm@35364
  1276
  fst o HOLogic.dest_bin @{const_name "op ="} HOLogic.boolT o HOLogic.dest_Trueprop o concl_of;
haftmann@22482
  1277
wenzelm@27250
  1278
);
haftmann@22482
  1279
*}
haftmann@22482
  1280
wenzelm@26480
  1281
ML {*
haftmann@22482
  1282
  Addsimprocs [ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv];
haftmann@22482
  1283
*}
paulson@17085
  1284
haftmann@33364
  1285
code_modulename SML
haftmann@35050
  1286
  Groups Arith
haftmann@33364
  1287
haftmann@33364
  1288
code_modulename OCaml
haftmann@35050
  1289
  Groups Arith
haftmann@33364
  1290
haftmann@33364
  1291
code_modulename Haskell
haftmann@35050
  1292
  Groups Arith
haftmann@33364
  1293
obua@14738
  1294
end