src/HOL/Groups.thy
author haftmann
Sun Apr 25 08:25:34 2010 +0200 (2010-04-25)
changeset 36343 30bcceed0dc2
parent 36302 4e7f5b22dd7d
child 36348 89c54f51f55a
permissions -rw-r--r--
field_simps as named theorems
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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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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{* 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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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]: "(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]: "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] = 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]: "(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]: "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] = 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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lemma equation_minus_iff:
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  "a = - b \<longleftrightarrow> b = - a"
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proof -
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  have "- (- a) = - b \<longleftrightarrow> - a = b" by (rule neg_equal_iff_equal)
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  thus ?thesis by (simp add: eq_commute)
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qed
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lemma minus_equation_iff:
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  "- a = b \<longleftrightarrow> - b = a"
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proof -
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  have "- a = - (- b) \<longleftrightarrow> a = -b" by (rule neg_equal_iff_equal)
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  thus ?thesis by (simp add: eq_commute)
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   365
qed
obua@14738
   366
huffman@28130
   367
lemma diff_add_cancel: "a - b + b = a"
nipkow@29667
   368
by (simp add: diff_minus add_assoc)
huffman@28130
   369
huffman@28130
   370
lemma add_diff_cancel: "a + b - b = a"
nipkow@29667
   371
by (simp add: diff_minus add_assoc)
nipkow@29667
   372
nipkow@29667
   373
declare diff_minus[symmetric, algebra_simps]
huffman@28130
   374
huffman@29914
   375
lemma eq_neg_iff_add_eq_0: "a = - b \<longleftrightarrow> a + b = 0"
huffman@29914
   376
proof
huffman@29914
   377
  assume "a = - b" then show "a + b = 0" by simp
huffman@29914
   378
next
huffman@29914
   379
  assume "a + b = 0"
huffman@29914
   380
  moreover have "a + (b + - b) = (a + b) + - b"
huffman@29914
   381
    by (simp only: add_assoc)
huffman@29914
   382
  ultimately show "a = - b" by simp
huffman@29914
   383
qed
huffman@29914
   384
haftmann@25062
   385
end
haftmann@25062
   386
haftmann@25762
   387
class ab_group_add = minus + uminus + comm_monoid_add +
haftmann@25062
   388
  assumes ab_left_minus: "- a + a = 0"
haftmann@25062
   389
  assumes ab_diff_minus: "a - b = a + (- b)"
haftmann@25267
   390
begin
haftmann@25062
   391
haftmann@25267
   392
subclass group_add
haftmann@28823
   393
  proof qed (simp_all add: ab_left_minus ab_diff_minus)
haftmann@25062
   394
huffman@29904
   395
subclass cancel_comm_monoid_add
haftmann@28823
   396
proof
haftmann@25062
   397
  fix a b c :: 'a
haftmann@25062
   398
  assume "a + b = a + c"
haftmann@25062
   399
  then have "- a + a + b = - a + a + c"
haftmann@25062
   400
    unfolding add_assoc by simp
haftmann@25062
   401
  then show "b = c" by simp
haftmann@25062
   402
qed
haftmann@25062
   403
nipkow@29667
   404
lemma uminus_add_conv_diff[algebra_simps]:
haftmann@25062
   405
  "- a + b = b - a"
nipkow@29667
   406
by (simp add:diff_minus add_commute)
haftmann@25062
   407
haftmann@25062
   408
lemma minus_add_distrib [simp]:
haftmann@25062
   409
  "- (a + b) = - a + - b"
huffman@34146
   410
by (rule minus_unique) (simp add: add_ac)
haftmann@25062
   411
haftmann@25062
   412
lemma minus_diff_eq [simp]:
haftmann@25062
   413
  "- (a - b) = b - a"
nipkow@29667
   414
by (simp add: diff_minus add_commute)
haftmann@25077
   415
nipkow@29667
   416
lemma add_diff_eq[algebra_simps]: "a + (b - c) = (a + b) - c"
nipkow@29667
   417
by (simp add: diff_minus add_ac)
haftmann@25077
   418
nipkow@29667
   419
lemma diff_add_eq[algebra_simps]: "(a - b) + c = (a + c) - b"
nipkow@29667
   420
by (simp add: diff_minus add_ac)
haftmann@25077
   421
nipkow@29667
   422
lemma diff_eq_eq[algebra_simps]: "a - b = c \<longleftrightarrow> a = c + b"
nipkow@29667
   423
by (auto simp add: diff_minus add_assoc)
haftmann@25077
   424
nipkow@29667
   425
lemma eq_diff_eq[algebra_simps]: "a = c - b \<longleftrightarrow> a + b = c"
nipkow@29667
   426
by (auto simp add: diff_minus add_assoc)
haftmann@25077
   427
nipkow@29667
   428
lemma diff_diff_eq[algebra_simps]: "(a - b) - c = a - (b + c)"
nipkow@29667
   429
by (simp add: diff_minus add_ac)
haftmann@25077
   430
nipkow@29667
   431
lemma diff_diff_eq2[algebra_simps]: "a - (b - c) = (a + c) - b"
nipkow@29667
   432
by (simp add: diff_minus add_ac)
haftmann@25077
   433
haftmann@25077
   434
lemma eq_iff_diff_eq_0: "a = b \<longleftrightarrow> a - b = 0"
nipkow@29667
   435
by (simp add: algebra_simps)
haftmann@25077
   436
huffman@35216
   437
(* FIXME: duplicates right_minus_eq from class group_add *)
huffman@35216
   438
(* but only this one is declared as a simp rule. *)
blanchet@35828
   439
lemma diff_eq_0_iff_eq [simp, no_atp]: "a - b = 0 \<longleftrightarrow> a = b"
huffman@30629
   440
by (simp add: algebra_simps)
huffman@30629
   441
haftmann@25062
   442
end
obua@14738
   443
obua@14738
   444
subsection {* (Partially) Ordered Groups *} 
obua@14738
   445
haftmann@35301
   446
text {*
haftmann@35301
   447
  The theory of partially ordered groups is taken from the books:
haftmann@35301
   448
  \begin{itemize}
haftmann@35301
   449
  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
haftmann@35301
   450
  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
haftmann@35301
   451
  \end{itemize}
haftmann@35301
   452
  Most of the used notions can also be looked up in 
haftmann@35301
   453
  \begin{itemize}
haftmann@35301
   454
  \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
haftmann@35301
   455
  \item \emph{Algebra I} by van der Waerden, Springer.
haftmann@35301
   456
  \end{itemize}
haftmann@35301
   457
*}
haftmann@35301
   458
haftmann@35028
   459
class ordered_ab_semigroup_add = order + ab_semigroup_add +
haftmann@25062
   460
  assumes add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b"
haftmann@25062
   461
begin
haftmann@24380
   462
haftmann@25062
   463
lemma add_right_mono:
haftmann@25062
   464
  "a \<le> b \<Longrightarrow> a + c \<le> b + c"
nipkow@29667
   465
by (simp add: add_commute [of _ c] add_left_mono)
obua@14738
   466
obua@14738
   467
text {* non-strict, in both arguments *}
obua@14738
   468
lemma add_mono:
haftmann@25062
   469
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d"
obua@14738
   470
  apply (erule add_right_mono [THEN order_trans])
obua@14738
   471
  apply (simp add: add_commute add_left_mono)
obua@14738
   472
  done
obua@14738
   473
haftmann@25062
   474
end
haftmann@25062
   475
haftmann@35028
   476
class ordered_cancel_ab_semigroup_add =
haftmann@35028
   477
  ordered_ab_semigroup_add + cancel_ab_semigroup_add
haftmann@25062
   478
begin
haftmann@25062
   479
obua@14738
   480
lemma add_strict_left_mono:
haftmann@25062
   481
  "a < b \<Longrightarrow> c + a < c + b"
nipkow@29667
   482
by (auto simp add: less_le add_left_mono)
obua@14738
   483
obua@14738
   484
lemma add_strict_right_mono:
haftmann@25062
   485
  "a < b \<Longrightarrow> a + c < b + c"
nipkow@29667
   486
by (simp add: add_commute [of _ c] add_strict_left_mono)
obua@14738
   487
obua@14738
   488
text{*Strict monotonicity in both arguments*}
haftmann@25062
   489
lemma add_strict_mono:
haftmann@25062
   490
  "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
haftmann@25062
   491
apply (erule add_strict_right_mono [THEN less_trans])
obua@14738
   492
apply (erule add_strict_left_mono)
obua@14738
   493
done
obua@14738
   494
obua@14738
   495
lemma add_less_le_mono:
haftmann@25062
   496
  "a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c < b + d"
haftmann@25062
   497
apply (erule add_strict_right_mono [THEN less_le_trans])
haftmann@25062
   498
apply (erule add_left_mono)
obua@14738
   499
done
obua@14738
   500
obua@14738
   501
lemma add_le_less_mono:
haftmann@25062
   502
  "a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
haftmann@25062
   503
apply (erule add_right_mono [THEN le_less_trans])
obua@14738
   504
apply (erule add_strict_left_mono) 
obua@14738
   505
done
obua@14738
   506
haftmann@25062
   507
end
haftmann@25062
   508
haftmann@35028
   509
class ordered_ab_semigroup_add_imp_le =
haftmann@35028
   510
  ordered_cancel_ab_semigroup_add +
haftmann@25062
   511
  assumes add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b"
haftmann@25062
   512
begin
haftmann@25062
   513
obua@14738
   514
lemma add_less_imp_less_left:
nipkow@29667
   515
  assumes less: "c + a < c + b" shows "a < b"
obua@14738
   516
proof -
obua@14738
   517
  from less have le: "c + a <= c + b" by (simp add: order_le_less)
obua@14738
   518
  have "a <= b" 
obua@14738
   519
    apply (insert le)
obua@14738
   520
    apply (drule add_le_imp_le_left)
obua@14738
   521
    by (insert le, drule add_le_imp_le_left, assumption)
obua@14738
   522
  moreover have "a \<noteq> b"
obua@14738
   523
  proof (rule ccontr)
obua@14738
   524
    assume "~(a \<noteq> b)"
obua@14738
   525
    then have "a = b" by simp
obua@14738
   526
    then have "c + a = c + b" by simp
obua@14738
   527
    with less show "False"by simp
obua@14738
   528
  qed
obua@14738
   529
  ultimately show "a < b" by (simp add: order_le_less)
obua@14738
   530
qed
obua@14738
   531
obua@14738
   532
lemma add_less_imp_less_right:
haftmann@25062
   533
  "a + c < b + c \<Longrightarrow> a < b"
obua@14738
   534
apply (rule add_less_imp_less_left [of c])
obua@14738
   535
apply (simp add: add_commute)  
obua@14738
   536
done
obua@14738
   537
obua@14738
   538
lemma add_less_cancel_left [simp]:
haftmann@25062
   539
  "c + a < c + b \<longleftrightarrow> a < b"
nipkow@29667
   540
by (blast intro: add_less_imp_less_left add_strict_left_mono) 
obua@14738
   541
obua@14738
   542
lemma add_less_cancel_right [simp]:
haftmann@25062
   543
  "a + c < b + c \<longleftrightarrow> a < b"
nipkow@29667
   544
by (blast intro: add_less_imp_less_right add_strict_right_mono)
obua@14738
   545
obua@14738
   546
lemma add_le_cancel_left [simp]:
haftmann@25062
   547
  "c + a \<le> c + b \<longleftrightarrow> a \<le> b"
nipkow@29667
   548
by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono) 
obua@14738
   549
obua@14738
   550
lemma add_le_cancel_right [simp]:
haftmann@25062
   551
  "a + c \<le> b + c \<longleftrightarrow> a \<le> b"
nipkow@29667
   552
by (simp add: add_commute [of a c] add_commute [of b c])
obua@14738
   553
obua@14738
   554
lemma add_le_imp_le_right:
haftmann@25062
   555
  "a + c \<le> b + c \<Longrightarrow> a \<le> b"
nipkow@29667
   556
by simp
haftmann@25062
   557
haftmann@25077
   558
lemma max_add_distrib_left:
haftmann@25077
   559
  "max x y + z = max (x + z) (y + z)"
haftmann@25077
   560
  unfolding max_def by auto
haftmann@25077
   561
haftmann@25077
   562
lemma min_add_distrib_left:
haftmann@25077
   563
  "min x y + z = min (x + z) (y + z)"
haftmann@25077
   564
  unfolding min_def by auto
haftmann@25077
   565
haftmann@25062
   566
end
haftmann@25062
   567
haftmann@25303
   568
subsection {* Support for reasoning about signs *}
haftmann@25303
   569
haftmann@35028
   570
class ordered_comm_monoid_add =
haftmann@35028
   571
  ordered_cancel_ab_semigroup_add + comm_monoid_add
haftmann@25303
   572
begin
haftmann@25303
   573
haftmann@25303
   574
lemma add_pos_nonneg:
nipkow@29667
   575
  assumes "0 < a" and "0 \<le> b" shows "0 < a + b"
haftmann@25303
   576
proof -
haftmann@25303
   577
  have "0 + 0 < a + b" 
haftmann@25303
   578
    using assms by (rule add_less_le_mono)
haftmann@25303
   579
  then show ?thesis by simp
haftmann@25303
   580
qed
haftmann@25303
   581
haftmann@25303
   582
lemma add_pos_pos:
nipkow@29667
   583
  assumes "0 < a" and "0 < b" shows "0 < a + b"
nipkow@29667
   584
by (rule add_pos_nonneg) (insert assms, auto)
haftmann@25303
   585
haftmann@25303
   586
lemma add_nonneg_pos:
nipkow@29667
   587
  assumes "0 \<le> a" and "0 < b" shows "0 < a + b"
haftmann@25303
   588
proof -
haftmann@25303
   589
  have "0 + 0 < a + b" 
haftmann@25303
   590
    using assms by (rule add_le_less_mono)
haftmann@25303
   591
  then show ?thesis by simp
haftmann@25303
   592
qed
haftmann@25303
   593
haftmann@25303
   594
lemma add_nonneg_nonneg:
nipkow@29667
   595
  assumes "0 \<le> a" and "0 \<le> b" shows "0 \<le> a + b"
haftmann@25303
   596
proof -
haftmann@25303
   597
  have "0 + 0 \<le> a + b" 
haftmann@25303
   598
    using assms by (rule add_mono)
haftmann@25303
   599
  then show ?thesis by simp
haftmann@25303
   600
qed
haftmann@25303
   601
huffman@30691
   602
lemma add_neg_nonpos:
nipkow@29667
   603
  assumes "a < 0" and "b \<le> 0" shows "a + b < 0"
haftmann@25303
   604
proof -
haftmann@25303
   605
  have "a + b < 0 + 0"
haftmann@25303
   606
    using assms by (rule add_less_le_mono)
haftmann@25303
   607
  then show ?thesis by simp
haftmann@25303
   608
qed
haftmann@25303
   609
haftmann@25303
   610
lemma add_neg_neg: 
nipkow@29667
   611
  assumes "a < 0" and "b < 0" shows "a + b < 0"
nipkow@29667
   612
by (rule add_neg_nonpos) (insert assms, auto)
haftmann@25303
   613
haftmann@25303
   614
lemma add_nonpos_neg:
nipkow@29667
   615
  assumes "a \<le> 0" and "b < 0" shows "a + b < 0"
haftmann@25303
   616
proof -
haftmann@25303
   617
  have "a + b < 0 + 0"
haftmann@25303
   618
    using assms by (rule add_le_less_mono)
haftmann@25303
   619
  then show ?thesis by simp
haftmann@25303
   620
qed
haftmann@25303
   621
haftmann@25303
   622
lemma add_nonpos_nonpos:
nipkow@29667
   623
  assumes "a \<le> 0" and "b \<le> 0" shows "a + b \<le> 0"
haftmann@25303
   624
proof -
haftmann@25303
   625
  have "a + b \<le> 0 + 0"
haftmann@25303
   626
    using assms by (rule add_mono)
haftmann@25303
   627
  then show ?thesis by simp
haftmann@25303
   628
qed
haftmann@25303
   629
huffman@30691
   630
lemmas add_sign_intros =
huffman@30691
   631
  add_pos_nonneg add_pos_pos add_nonneg_pos add_nonneg_nonneg
huffman@30691
   632
  add_neg_nonpos add_neg_neg add_nonpos_neg add_nonpos_nonpos
huffman@30691
   633
huffman@29886
   634
lemma add_nonneg_eq_0_iff:
huffman@29886
   635
  assumes x: "0 \<le> x" and y: "0 \<le> y"
huffman@29886
   636
  shows "x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
huffman@29886
   637
proof (intro iffI conjI)
huffman@29886
   638
  have "x = x + 0" by simp
huffman@29886
   639
  also have "x + 0 \<le> x + y" using y by (rule add_left_mono)
huffman@29886
   640
  also assume "x + y = 0"
huffman@29886
   641
  also have "0 \<le> x" using x .
huffman@29886
   642
  finally show "x = 0" .
huffman@29886
   643
next
huffman@29886
   644
  have "y = 0 + y" by simp
huffman@29886
   645
  also have "0 + y \<le> x + y" using x by (rule add_right_mono)
huffman@29886
   646
  also assume "x + y = 0"
huffman@29886
   647
  also have "0 \<le> y" using y .
huffman@29886
   648
  finally show "y = 0" .
huffman@29886
   649
next
huffman@29886
   650
  assume "x = 0 \<and> y = 0"
huffman@29886
   651
  then show "x + y = 0" by simp
huffman@29886
   652
qed
huffman@29886
   653
haftmann@25303
   654
end
haftmann@25303
   655
haftmann@35028
   656
class ordered_ab_group_add =
haftmann@35028
   657
  ab_group_add + ordered_ab_semigroup_add
haftmann@25062
   658
begin
haftmann@25062
   659
haftmann@35028
   660
subclass ordered_cancel_ab_semigroup_add ..
haftmann@25062
   661
haftmann@35028
   662
subclass ordered_ab_semigroup_add_imp_le
haftmann@28823
   663
proof
haftmann@25062
   664
  fix a b c :: 'a
haftmann@25062
   665
  assume "c + a \<le> c + b"
haftmann@25062
   666
  hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono)
haftmann@25062
   667
  hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc)
haftmann@25062
   668
  thus "a \<le> b" by simp
haftmann@25062
   669
qed
haftmann@25062
   670
haftmann@35028
   671
subclass ordered_comm_monoid_add ..
haftmann@25303
   672
haftmann@25077
   673
lemma max_diff_distrib_left:
haftmann@25077
   674
  shows "max x y - z = max (x - z) (y - z)"
nipkow@29667
   675
by (simp add: diff_minus, rule max_add_distrib_left) 
haftmann@25077
   676
haftmann@25077
   677
lemma min_diff_distrib_left:
haftmann@25077
   678
  shows "min x y - z = min (x - z) (y - z)"
nipkow@29667
   679
by (simp add: diff_minus, rule min_add_distrib_left) 
haftmann@25077
   680
haftmann@25077
   681
lemma le_imp_neg_le:
nipkow@29667
   682
  assumes "a \<le> b" shows "-b \<le> -a"
haftmann@25077
   683
proof -
nipkow@29667
   684
  have "-a+a \<le> -a+b" using `a \<le> b` by (rule add_left_mono) 
nipkow@29667
   685
  hence "0 \<le> -a+b" by simp
nipkow@29667
   686
  hence "0 + (-b) \<le> (-a + b) + (-b)" by (rule add_right_mono) 
nipkow@29667
   687
  thus ?thesis by (simp add: add_assoc)
haftmann@25077
   688
qed
haftmann@25077
   689
haftmann@25077
   690
lemma neg_le_iff_le [simp]: "- b \<le> - a \<longleftrightarrow> a \<le> b"
haftmann@25077
   691
proof 
haftmann@25077
   692
  assume "- b \<le> - a"
nipkow@29667
   693
  hence "- (- a) \<le> - (- b)" by (rule le_imp_neg_le)
haftmann@25077
   694
  thus "a\<le>b" by simp
haftmann@25077
   695
next
haftmann@25077
   696
  assume "a\<le>b"
haftmann@25077
   697
  thus "-b \<le> -a" by (rule le_imp_neg_le)
haftmann@25077
   698
qed
haftmann@25077
   699
haftmann@25077
   700
lemma neg_le_0_iff_le [simp]: "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"
nipkow@29667
   701
by (subst neg_le_iff_le [symmetric], simp)
haftmann@25077
   702
haftmann@25077
   703
lemma neg_0_le_iff_le [simp]: "0 \<le> - a \<longleftrightarrow> a \<le> 0"
nipkow@29667
   704
by (subst neg_le_iff_le [symmetric], simp)
haftmann@25077
   705
haftmann@25077
   706
lemma neg_less_iff_less [simp]: "- b < - a \<longleftrightarrow> a < b"
nipkow@29667
   707
by (force simp add: less_le) 
haftmann@25077
   708
haftmann@25077
   709
lemma neg_less_0_iff_less [simp]: "- a < 0 \<longleftrightarrow> 0 < a"
nipkow@29667
   710
by (subst neg_less_iff_less [symmetric], simp)
haftmann@25077
   711
haftmann@25077
   712
lemma neg_0_less_iff_less [simp]: "0 < - a \<longleftrightarrow> a < 0"
nipkow@29667
   713
by (subst neg_less_iff_less [symmetric], simp)
haftmann@25077
   714
haftmann@25077
   715
text{*The next several equations can make the simplifier loop!*}
haftmann@25077
   716
haftmann@25077
   717
lemma less_minus_iff: "a < - b \<longleftrightarrow> b < - a"
haftmann@25077
   718
proof -
haftmann@25077
   719
  have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
haftmann@25077
   720
  thus ?thesis by simp
haftmann@25077
   721
qed
haftmann@25077
   722
haftmann@25077
   723
lemma minus_less_iff: "- a < b \<longleftrightarrow> - b < a"
haftmann@25077
   724
proof -
haftmann@25077
   725
  have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
haftmann@25077
   726
  thus ?thesis by simp
haftmann@25077
   727
qed
haftmann@25077
   728
haftmann@25077
   729
lemma le_minus_iff: "a \<le> - b \<longleftrightarrow> b \<le> - a"
haftmann@25077
   730
proof -
haftmann@25077
   731
  have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff)
haftmann@25077
   732
  have "(- (- a) <= -b) = (b <= - a)" 
haftmann@25077
   733
    apply (auto simp only: le_less)
haftmann@25077
   734
    apply (drule mm)
haftmann@25077
   735
    apply (simp_all)
haftmann@25077
   736
    apply (drule mm[simplified], assumption)
haftmann@25077
   737
    done
haftmann@25077
   738
  then show ?thesis by simp
haftmann@25077
   739
qed
haftmann@25077
   740
haftmann@25077
   741
lemma minus_le_iff: "- a \<le> b \<longleftrightarrow> - b \<le> a"
nipkow@29667
   742
by (auto simp add: le_less minus_less_iff)
haftmann@25077
   743
haftmann@25077
   744
lemma less_iff_diff_less_0: "a < b \<longleftrightarrow> a - b < 0"
haftmann@25077
   745
proof -
haftmann@25077
   746
  have  "(a < b) = (a + (- b) < b + (-b))"  
haftmann@25077
   747
    by (simp only: add_less_cancel_right)
haftmann@25077
   748
  also have "... =  (a - b < 0)" by (simp add: diff_minus)
haftmann@25077
   749
  finally show ?thesis .
haftmann@25077
   750
qed
haftmann@25077
   751
nipkow@29667
   752
lemma diff_less_eq[algebra_simps]: "a - b < c \<longleftrightarrow> a < c + b"
haftmann@25077
   753
apply (subst less_iff_diff_less_0 [of a])
haftmann@25077
   754
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
haftmann@25077
   755
apply (simp add: diff_minus add_ac)
haftmann@25077
   756
done
haftmann@25077
   757
nipkow@29667
   758
lemma less_diff_eq[algebra_simps]: "a < c - b \<longleftrightarrow> a + b < c"
haftmann@36302
   759
apply (subst less_iff_diff_less_0 [of "a + b"])
haftmann@25077
   760
apply (subst less_iff_diff_less_0 [of a])
haftmann@25077
   761
apply (simp add: diff_minus add_ac)
haftmann@25077
   762
done
haftmann@25077
   763
nipkow@29667
   764
lemma diff_le_eq[algebra_simps]: "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
nipkow@29667
   765
by (auto simp add: le_less diff_less_eq diff_add_cancel add_diff_cancel)
haftmann@25077
   766
nipkow@29667
   767
lemma le_diff_eq[algebra_simps]: "a \<le> c - b \<longleftrightarrow> a + b \<le> c"
nipkow@29667
   768
by (auto simp add: le_less less_diff_eq diff_add_cancel add_diff_cancel)
haftmann@25077
   769
haftmann@25077
   770
lemma le_iff_diff_le_0: "a \<le> b \<longleftrightarrow> a - b \<le> 0"
nipkow@29667
   771
by (simp add: algebra_simps)
haftmann@25077
   772
nipkow@29667
   773
text{*Legacy - use @{text algebra_simps} *}
blanchet@35828
   774
lemmas group_simps[no_atp] = algebra_simps
haftmann@25230
   775
haftmann@25077
   776
end
haftmann@25077
   777
nipkow@29667
   778
text{*Legacy - use @{text algebra_simps} *}
blanchet@35828
   779
lemmas group_simps[no_atp] = algebra_simps
haftmann@25230
   780
haftmann@35028
   781
class linordered_ab_semigroup_add =
haftmann@35028
   782
  linorder + ordered_ab_semigroup_add
haftmann@25062
   783
haftmann@35028
   784
class linordered_cancel_ab_semigroup_add =
haftmann@35028
   785
  linorder + ordered_cancel_ab_semigroup_add
haftmann@25267
   786
begin
haftmann@25062
   787
haftmann@35028
   788
subclass linordered_ab_semigroup_add ..
haftmann@25062
   789
haftmann@35028
   790
subclass ordered_ab_semigroup_add_imp_le
haftmann@28823
   791
proof
haftmann@25062
   792
  fix a b c :: 'a
haftmann@25062
   793
  assume le: "c + a <= c + b"  
haftmann@25062
   794
  show "a <= b"
haftmann@25062
   795
  proof (rule ccontr)
haftmann@25062
   796
    assume w: "~ a \<le> b"
haftmann@25062
   797
    hence "b <= a" by (simp add: linorder_not_le)
haftmann@25062
   798
    hence le2: "c + b <= c + a" by (rule add_left_mono)
haftmann@25062
   799
    have "a = b" 
haftmann@25062
   800
      apply (insert le)
haftmann@25062
   801
      apply (insert le2)
haftmann@25062
   802
      apply (drule antisym, simp_all)
haftmann@25062
   803
      done
haftmann@25062
   804
    with w show False 
haftmann@25062
   805
      by (simp add: linorder_not_le [symmetric])
haftmann@25062
   806
  qed
haftmann@25062
   807
qed
haftmann@25062
   808
haftmann@25267
   809
end
haftmann@25267
   810
haftmann@35028
   811
class linordered_ab_group_add = linorder + ordered_ab_group_add
haftmann@25267
   812
begin
haftmann@25230
   813
haftmann@35028
   814
subclass linordered_cancel_ab_semigroup_add ..
haftmann@25230
   815
haftmann@35036
   816
lemma neg_less_eq_nonneg [simp]:
haftmann@25303
   817
  "- a \<le> a \<longleftrightarrow> 0 \<le> a"
haftmann@25303
   818
proof
haftmann@25303
   819
  assume A: "- a \<le> a" show "0 \<le> a"
haftmann@25303
   820
  proof (rule classical)
haftmann@25303
   821
    assume "\<not> 0 \<le> a"
haftmann@25303
   822
    then have "a < 0" by auto
haftmann@25303
   823
    with A have "- a < 0" by (rule le_less_trans)
haftmann@25303
   824
    then show ?thesis by auto
haftmann@25303
   825
  qed
haftmann@25303
   826
next
haftmann@25303
   827
  assume A: "0 \<le> a" show "- a \<le> a"
haftmann@25303
   828
  proof (rule order_trans)
haftmann@25303
   829
    show "- a \<le> 0" using A by (simp add: minus_le_iff)
haftmann@25303
   830
  next
haftmann@25303
   831
    show "0 \<le> a" using A .
haftmann@25303
   832
  qed
haftmann@25303
   833
qed
haftmann@35036
   834
haftmann@35036
   835
lemma neg_less_nonneg [simp]:
haftmann@35036
   836
  "- a < a \<longleftrightarrow> 0 < a"
haftmann@35036
   837
proof
haftmann@35036
   838
  assume A: "- a < a" show "0 < a"
haftmann@35036
   839
  proof (rule classical)
haftmann@35036
   840
    assume "\<not> 0 < a"
haftmann@35036
   841
    then have "a \<le> 0" by auto
haftmann@35036
   842
    with A have "- a < 0" by (rule less_le_trans)
haftmann@35036
   843
    then show ?thesis by auto
haftmann@35036
   844
  qed
haftmann@35036
   845
next
haftmann@35036
   846
  assume A: "0 < a" show "- a < a"
haftmann@35036
   847
  proof (rule less_trans)
haftmann@35036
   848
    show "- a < 0" using A by (simp add: minus_le_iff)
haftmann@35036
   849
  next
haftmann@35036
   850
    show "0 < a" using A .
haftmann@35036
   851
  qed
haftmann@35036
   852
qed
haftmann@35036
   853
haftmann@35036
   854
lemma less_eq_neg_nonpos [simp]:
haftmann@25303
   855
  "a \<le> - a \<longleftrightarrow> a \<le> 0"
haftmann@25303
   856
proof
haftmann@25303
   857
  assume A: "a \<le> - a" show "a \<le> 0"
haftmann@25303
   858
  proof (rule classical)
haftmann@25303
   859
    assume "\<not> a \<le> 0"
haftmann@25303
   860
    then have "0 < a" by auto
haftmann@25303
   861
    then have "0 < - a" using A by (rule less_le_trans)
haftmann@25303
   862
    then show ?thesis by auto
haftmann@25303
   863
  qed
haftmann@25303
   864
next
haftmann@25303
   865
  assume A: "a \<le> 0" show "a \<le> - a"
haftmann@25303
   866
  proof (rule order_trans)
haftmann@25303
   867
    show "0 \<le> - a" using A by (simp add: minus_le_iff)
haftmann@25303
   868
  next
haftmann@25303
   869
    show "a \<le> 0" using A .
haftmann@25303
   870
  qed
haftmann@25303
   871
qed
haftmann@25303
   872
haftmann@35036
   873
lemma equal_neg_zero [simp]:
haftmann@25303
   874
  "a = - a \<longleftrightarrow> a = 0"
haftmann@25303
   875
proof
haftmann@25303
   876
  assume "a = 0" then show "a = - a" by simp
haftmann@25303
   877
next
haftmann@25303
   878
  assume A: "a = - a" show "a = 0"
haftmann@25303
   879
  proof (cases "0 \<le> a")
haftmann@25303
   880
    case True with A have "0 \<le> - a" by auto
haftmann@25303
   881
    with le_minus_iff have "a \<le> 0" by simp
haftmann@25303
   882
    with True show ?thesis by (auto intro: order_trans)
haftmann@25303
   883
  next
haftmann@25303
   884
    case False then have B: "a \<le> 0" by auto
haftmann@25303
   885
    with A have "- a \<le> 0" by auto
haftmann@25303
   886
    with B show ?thesis by (auto intro: order_trans)
haftmann@25303
   887
  qed
haftmann@25303
   888
qed
haftmann@25303
   889
haftmann@35036
   890
lemma neg_equal_zero [simp]:
haftmann@25303
   891
  "- a = a \<longleftrightarrow> a = 0"
haftmann@35036
   892
  by (auto dest: sym)
haftmann@35036
   893
haftmann@35036
   894
lemma double_zero [simp]:
haftmann@35036
   895
  "a + a = 0 \<longleftrightarrow> a = 0"
haftmann@35036
   896
proof
haftmann@35036
   897
  assume assm: "a + a = 0"
haftmann@35036
   898
  then have a: "- a = a" by (rule minus_unique)
huffman@35216
   899
  then show "a = 0" by (simp only: neg_equal_zero)
haftmann@35036
   900
qed simp
haftmann@35036
   901
haftmann@35036
   902
lemma double_zero_sym [simp]:
haftmann@35036
   903
  "0 = a + a \<longleftrightarrow> a = 0"
haftmann@35036
   904
  by (rule, drule sym) simp_all
haftmann@35036
   905
haftmann@35036
   906
lemma zero_less_double_add_iff_zero_less_single_add [simp]:
haftmann@35036
   907
  "0 < a + a \<longleftrightarrow> 0 < a"
haftmann@35036
   908
proof
haftmann@35036
   909
  assume "0 < a + a"
haftmann@35036
   910
  then have "0 - a < a" by (simp only: diff_less_eq)
haftmann@35036
   911
  then have "- a < a" by simp
huffman@35216
   912
  then show "0 < a" by (simp only: neg_less_nonneg)
haftmann@35036
   913
next
haftmann@35036
   914
  assume "0 < a"
haftmann@35036
   915
  with this have "0 + 0 < a + a"
haftmann@35036
   916
    by (rule add_strict_mono)
haftmann@35036
   917
  then show "0 < a + a" by simp
haftmann@35036
   918
qed
haftmann@35036
   919
haftmann@35036
   920
lemma zero_le_double_add_iff_zero_le_single_add [simp]:
haftmann@35036
   921
  "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"
haftmann@35036
   922
  by (auto simp add: le_less)
haftmann@35036
   923
haftmann@35036
   924
lemma double_add_less_zero_iff_single_add_less_zero [simp]:
haftmann@35036
   925
  "a + a < 0 \<longleftrightarrow> a < 0"
haftmann@35036
   926
proof -
haftmann@35036
   927
  have "\<not> a + a < 0 \<longleftrightarrow> \<not> a < 0"
haftmann@35036
   928
    by (simp add: not_less)
haftmann@35036
   929
  then show ?thesis by simp
haftmann@35036
   930
qed
haftmann@35036
   931
haftmann@35036
   932
lemma double_add_le_zero_iff_single_add_le_zero [simp]:
haftmann@35036
   933
  "a + a \<le> 0 \<longleftrightarrow> a \<le> 0" 
haftmann@35036
   934
proof -
haftmann@35036
   935
  have "\<not> a + a \<le> 0 \<longleftrightarrow> \<not> a \<le> 0"
haftmann@35036
   936
    by (simp add: not_le)
haftmann@35036
   937
  then show ?thesis by simp
haftmann@35036
   938
qed
haftmann@35036
   939
haftmann@35036
   940
lemma le_minus_self_iff:
haftmann@35036
   941
  "a \<le> - a \<longleftrightarrow> a \<le> 0"
haftmann@35036
   942
proof -
haftmann@35036
   943
  from add_le_cancel_left [of "- a" "a + a" 0]
haftmann@35036
   944
  have "a \<le> - a \<longleftrightarrow> a + a \<le> 0" 
haftmann@35036
   945
    by (simp add: add_assoc [symmetric])
haftmann@35036
   946
  thus ?thesis by simp
haftmann@35036
   947
qed
haftmann@35036
   948
haftmann@35036
   949
lemma minus_le_self_iff:
haftmann@35036
   950
  "- a \<le> a \<longleftrightarrow> 0 \<le> a"
haftmann@35036
   951
proof -
haftmann@35036
   952
  from add_le_cancel_left [of "- a" 0 "a + a"]
haftmann@35036
   953
  have "- a \<le> a \<longleftrightarrow> 0 \<le> a + a" 
haftmann@35036
   954
    by (simp add: add_assoc [symmetric])
haftmann@35036
   955
  thus ?thesis by simp
haftmann@35036
   956
qed
haftmann@35036
   957
haftmann@35036
   958
lemma minus_max_eq_min:
haftmann@35036
   959
  "- max x y = min (-x) (-y)"
haftmann@35036
   960
  by (auto simp add: max_def min_def)
haftmann@35036
   961
haftmann@35036
   962
lemma minus_min_eq_max:
haftmann@35036
   963
  "- min x y = max (-x) (-y)"
haftmann@35036
   964
  by (auto simp add: max_def min_def)
haftmann@25303
   965
haftmann@25267
   966
end
haftmann@25267
   967
haftmann@36302
   968
context ordered_comm_monoid_add
haftmann@36302
   969
begin
obua@14738
   970
paulson@15234
   971
lemma add_increasing:
haftmann@36302
   972
  "0 \<le> a \<Longrightarrow> b \<le> c \<Longrightarrow> b \<le> a + c"
haftmann@36302
   973
  by (insert add_mono [of 0 a b c], simp)
obua@14738
   974
nipkow@15539
   975
lemma add_increasing2:
haftmann@36302
   976
  "0 \<le> c \<Longrightarrow> b \<le> a \<Longrightarrow> b \<le> a + c"
haftmann@36302
   977
  by (simp add: add_increasing add_commute [of a])
nipkow@15539
   978
paulson@15234
   979
lemma add_strict_increasing:
haftmann@36302
   980
  "0 < a \<Longrightarrow> b \<le> c \<Longrightarrow> b < a + c"
haftmann@36302
   981
  by (insert add_less_le_mono [of 0 a b c], simp)
paulson@15234
   982
paulson@15234
   983
lemma add_strict_increasing2:
haftmann@36302
   984
  "0 \<le> a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
haftmann@36302
   985
  by (insert add_le_less_mono [of 0 a b c], simp)
haftmann@36302
   986
haftmann@36302
   987
end
paulson@15234
   988
haftmann@35092
   989
class abs =
haftmann@35092
   990
  fixes abs :: "'a \<Rightarrow> 'a"
haftmann@35092
   991
begin
haftmann@35092
   992
haftmann@35092
   993
notation (xsymbols)
haftmann@35092
   994
  abs  ("\<bar>_\<bar>")
haftmann@35092
   995
haftmann@35092
   996
notation (HTML output)
haftmann@35092
   997
  abs  ("\<bar>_\<bar>")
haftmann@35092
   998
haftmann@35092
   999
end
haftmann@35092
  1000
haftmann@35092
  1001
class sgn =
haftmann@35092
  1002
  fixes sgn :: "'a \<Rightarrow> 'a"
haftmann@35092
  1003
haftmann@35092
  1004
class abs_if = minus + uminus + ord + zero + abs +
haftmann@35092
  1005
  assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"
haftmann@35092
  1006
haftmann@35092
  1007
class sgn_if = minus + uminus + zero + one + ord + sgn +
haftmann@35092
  1008
  assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
haftmann@35092
  1009
begin
haftmann@35092
  1010
haftmann@35092
  1011
lemma sgn0 [simp]: "sgn 0 = 0"
haftmann@35092
  1012
  by (simp add:sgn_if)
haftmann@35092
  1013
haftmann@35092
  1014
end
obua@14738
  1015
haftmann@35028
  1016
class ordered_ab_group_add_abs = ordered_ab_group_add + abs +
haftmann@25303
  1017
  assumes abs_ge_zero [simp]: "\<bar>a\<bar> \<ge> 0"
haftmann@25303
  1018
    and abs_ge_self: "a \<le> \<bar>a\<bar>"
haftmann@25303
  1019
    and abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
haftmann@25303
  1020
    and abs_minus_cancel [simp]: "\<bar>-a\<bar> = \<bar>a\<bar>"
haftmann@25303
  1021
    and abs_triangle_ineq: "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25303
  1022
begin
haftmann@25303
  1023
haftmann@25307
  1024
lemma abs_minus_le_zero: "- \<bar>a\<bar> \<le> 0"
haftmann@25307
  1025
  unfolding neg_le_0_iff_le by simp
haftmann@25307
  1026
haftmann@25307
  1027
lemma abs_of_nonneg [simp]:
nipkow@29667
  1028
  assumes nonneg: "0 \<le> a" shows "\<bar>a\<bar> = a"
haftmann@25307
  1029
proof (rule antisym)
haftmann@25307
  1030
  from nonneg le_imp_neg_le have "- a \<le> 0" by simp
haftmann@25307
  1031
  from this nonneg have "- a \<le> a" by (rule order_trans)
haftmann@25307
  1032
  then show "\<bar>a\<bar> \<le> a" by (auto intro: abs_leI)
haftmann@25307
  1033
qed (rule abs_ge_self)
haftmann@25307
  1034
haftmann@25307
  1035
lemma abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
nipkow@29667
  1036
by (rule antisym)
haftmann@36302
  1037
   (auto intro!: abs_ge_self abs_leI order_trans [of "- \<bar>a\<bar>" 0 "\<bar>a\<bar>"])
haftmann@25307
  1038
haftmann@25307
  1039
lemma abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
haftmann@25307
  1040
proof -
haftmann@25307
  1041
  have "\<bar>a\<bar> = 0 \<Longrightarrow> a = 0"
haftmann@25307
  1042
  proof (rule antisym)
haftmann@25307
  1043
    assume zero: "\<bar>a\<bar> = 0"
haftmann@25307
  1044
    with abs_ge_self show "a \<le> 0" by auto
haftmann@25307
  1045
    from zero have "\<bar>-a\<bar> = 0" by simp
haftmann@36302
  1046
    with abs_ge_self [of "- a"] have "- a \<le> 0" by auto
haftmann@25307
  1047
    with neg_le_0_iff_le show "0 \<le> a" by auto
haftmann@25307
  1048
  qed
haftmann@25307
  1049
  then show ?thesis by auto
haftmann@25307
  1050
qed
haftmann@25307
  1051
haftmann@25303
  1052
lemma abs_zero [simp]: "\<bar>0\<bar> = 0"
nipkow@29667
  1053
by simp
avigad@16775
  1054
blanchet@35828
  1055
lemma abs_0_eq [simp, no_atp]: "0 = \<bar>a\<bar> \<longleftrightarrow> a = 0"
haftmann@25303
  1056
proof -
haftmann@25303
  1057
  have "0 = \<bar>a\<bar> \<longleftrightarrow> \<bar>a\<bar> = 0" by (simp only: eq_ac)
haftmann@25303
  1058
  thus ?thesis by simp
haftmann@25303
  1059
qed
haftmann@25303
  1060
haftmann@25303
  1061
lemma abs_le_zero_iff [simp]: "\<bar>a\<bar> \<le> 0 \<longleftrightarrow> a = 0" 
haftmann@25303
  1062
proof
haftmann@25303
  1063
  assume "\<bar>a\<bar> \<le> 0"
haftmann@25303
  1064
  then have "\<bar>a\<bar> = 0" by (rule antisym) simp
haftmann@25303
  1065
  thus "a = 0" by simp
haftmann@25303
  1066
next
haftmann@25303
  1067
  assume "a = 0"
haftmann@25303
  1068
  thus "\<bar>a\<bar> \<le> 0" by simp
haftmann@25303
  1069
qed
haftmann@25303
  1070
haftmann@25303
  1071
lemma zero_less_abs_iff [simp]: "0 < \<bar>a\<bar> \<longleftrightarrow> a \<noteq> 0"
nipkow@29667
  1072
by (simp add: less_le)
haftmann@25303
  1073
haftmann@25303
  1074
lemma abs_not_less_zero [simp]: "\<not> \<bar>a\<bar> < 0"
haftmann@25303
  1075
proof -
haftmann@25303
  1076
  have a: "\<And>x y. x \<le> y \<Longrightarrow> \<not> y < x" by auto
haftmann@25303
  1077
  show ?thesis by (simp add: a)
haftmann@25303
  1078
qed
avigad@16775
  1079
haftmann@25303
  1080
lemma abs_ge_minus_self: "- a \<le> \<bar>a\<bar>"
haftmann@25303
  1081
proof -
haftmann@25303
  1082
  have "- a \<le> \<bar>-a\<bar>" by (rule abs_ge_self)
haftmann@25303
  1083
  then show ?thesis by simp
haftmann@25303
  1084
qed
haftmann@25303
  1085
haftmann@25303
  1086
lemma abs_minus_commute: 
haftmann@25303
  1087
  "\<bar>a - b\<bar> = \<bar>b - a\<bar>"
haftmann@25303
  1088
proof -
haftmann@25303
  1089
  have "\<bar>a - b\<bar> = \<bar>- (a - b)\<bar>" by (simp only: abs_minus_cancel)
haftmann@25303
  1090
  also have "... = \<bar>b - a\<bar>" by simp
haftmann@25303
  1091
  finally show ?thesis .
haftmann@25303
  1092
qed
haftmann@25303
  1093
haftmann@25303
  1094
lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a"
nipkow@29667
  1095
by (rule abs_of_nonneg, rule less_imp_le)
avigad@16775
  1096
haftmann@25303
  1097
lemma abs_of_nonpos [simp]:
nipkow@29667
  1098
  assumes "a \<le> 0" shows "\<bar>a\<bar> = - a"
haftmann@25303
  1099
proof -
haftmann@25303
  1100
  let ?b = "- a"
haftmann@25303
  1101
  have "- ?b \<le> 0 \<Longrightarrow> \<bar>- ?b\<bar> = - (- ?b)"
haftmann@25303
  1102
  unfolding abs_minus_cancel [of "?b"]
haftmann@25303
  1103
  unfolding neg_le_0_iff_le [of "?b"]
haftmann@25303
  1104
  unfolding minus_minus by (erule abs_of_nonneg)
haftmann@25303
  1105
  then show ?thesis using assms by auto
haftmann@25303
  1106
qed
haftmann@25303
  1107
  
haftmann@25303
  1108
lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a"
nipkow@29667
  1109
by (rule abs_of_nonpos, rule less_imp_le)
haftmann@25303
  1110
haftmann@25303
  1111
lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b"
nipkow@29667
  1112
by (insert abs_ge_self, blast intro: order_trans)
haftmann@25303
  1113
haftmann@25303
  1114
lemma abs_le_D2: "\<bar>a\<bar> \<le> b \<Longrightarrow> - a \<le> b"
haftmann@36302
  1115
by (insert abs_le_D1 [of "- a"], simp)
haftmann@25303
  1116
haftmann@25303
  1117
lemma abs_le_iff: "\<bar>a\<bar> \<le> b \<longleftrightarrow> a \<le> b \<and> - a \<le> b"
nipkow@29667
  1118
by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
haftmann@25303
  1119
haftmann@25303
  1120
lemma abs_triangle_ineq2: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>a - b\<bar>"
haftmann@36302
  1121
proof -
haftmann@36302
  1122
  have "\<bar>a\<bar> = \<bar>b + (a - b)\<bar>"
haftmann@36302
  1123
    by (simp add: algebra_simps add_diff_cancel)
haftmann@36302
  1124
  then have "\<bar>a\<bar> \<le> \<bar>b\<bar> + \<bar>a - b\<bar>"
haftmann@36302
  1125
    by (simp add: abs_triangle_ineq)
haftmann@36302
  1126
  then show ?thesis
haftmann@36302
  1127
    by (simp add: algebra_simps)
haftmann@36302
  1128
qed
haftmann@36302
  1129
haftmann@36302
  1130
lemma abs_triangle_ineq2_sym: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>b - a\<bar>"
haftmann@36302
  1131
  by (simp only: abs_minus_commute [of b] abs_triangle_ineq2)
avigad@16775
  1132
haftmann@25303
  1133
lemma abs_triangle_ineq3: "\<bar>\<bar>a\<bar> - \<bar>b\<bar>\<bar> \<le> \<bar>a - b\<bar>"
haftmann@36302
  1134
  by (simp add: abs_le_iff abs_triangle_ineq2 abs_triangle_ineq2_sym)
avigad@16775
  1135
haftmann@25303
  1136
lemma abs_triangle_ineq4: "\<bar>a - b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
haftmann@25303
  1137
proof -
haftmann@36302
  1138
  have "\<bar>a - b\<bar> = \<bar>a + - b\<bar>" by (subst diff_minus, rule refl)
haftmann@36302
  1139
  also have "... \<le> \<bar>a\<bar> + \<bar>- b\<bar>" by (rule abs_triangle_ineq)
nipkow@29667
  1140
  finally show ?thesis by simp
haftmann@25303
  1141
qed
avigad@16775
  1142
haftmann@25303
  1143
lemma abs_diff_triangle_ineq: "\<bar>a + b - (c + d)\<bar> \<le> \<bar>a - c\<bar> + \<bar>b - d\<bar>"
haftmann@25303
  1144
proof -
haftmann@25303
  1145
  have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: diff_minus add_ac)
haftmann@25303
  1146
  also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq)
haftmann@25303
  1147
  finally show ?thesis .
haftmann@25303
  1148
qed
avigad@16775
  1149
haftmann@25303
  1150
lemma abs_add_abs [simp]:
haftmann@25303
  1151
  "\<bar>\<bar>a\<bar> + \<bar>b\<bar>\<bar> = \<bar>a\<bar> + \<bar>b\<bar>" (is "?L = ?R")
haftmann@25303
  1152
proof (rule antisym)
haftmann@25303
  1153
  show "?L \<ge> ?R" by(rule abs_ge_self)
haftmann@25303
  1154
next
haftmann@25303
  1155
  have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq)
haftmann@25303
  1156
  also have "\<dots> = ?R" by simp
haftmann@25303
  1157
  finally show "?L \<le> ?R" .
haftmann@25303
  1158
qed
haftmann@25303
  1159
haftmann@25303
  1160
end
obua@14738
  1161
obua@14754
  1162
text {* Needed for abelian cancellation simprocs: *}
obua@14754
  1163
obua@14754
  1164
lemma add_cancel_21: "((x::'a::ab_group_add) + (y + z) = y + u) = (x + z = u)"
obua@14754
  1165
apply (subst add_left_commute)
obua@14754
  1166
apply (subst add_left_cancel)
obua@14754
  1167
apply simp
obua@14754
  1168
done
obua@14754
  1169
obua@14754
  1170
lemma add_cancel_end: "(x + (y + z) = y) = (x = - (z::'a::ab_group_add))"
obua@14754
  1171
apply (subst add_cancel_21[of _ _ _ 0, simplified])
obua@14754
  1172
apply (simp add: add_right_cancel[symmetric, of "x" "-z" "z", simplified])
obua@14754
  1173
done
obua@14754
  1174
haftmann@35028
  1175
lemma less_eqI: "(x::'a::ordered_ab_group_add) - y = x' - y' \<Longrightarrow> (x < y) = (x' < y')"
obua@14754
  1176
by (simp add: less_iff_diff_less_0[of x y] less_iff_diff_less_0[of x' y'])
obua@14754
  1177
haftmann@35028
  1178
lemma le_eqI: "(x::'a::ordered_ab_group_add) - y = x' - y' \<Longrightarrow> (y <= x) = (y' <= x')"
obua@14754
  1179
apply (simp add: le_iff_diff_le_0[of y x] le_iff_diff_le_0[of  y' x'])
obua@14754
  1180
apply (simp add: neg_le_iff_le[symmetric, of "y-x" 0] neg_le_iff_le[symmetric, of "y'-x'" 0])
obua@14754
  1181
done
obua@14754
  1182
obua@14754
  1183
lemma eq_eqI: "(x::'a::ab_group_add) - y = x' - y' \<Longrightarrow> (x = y) = (x' = y')"
huffman@30629
  1184
by (simp only: eq_iff_diff_eq_0[of x y] eq_iff_diff_eq_0[of x' y'])
obua@14754
  1185
obua@14754
  1186
lemma diff_def: "(x::'a::ab_group_add) - y == x + (-y)"
obua@14754
  1187
by (simp add: diff_minus)
obua@14754
  1188
haftmann@25090
  1189
lemma le_add_right_mono: 
obua@15178
  1190
  assumes 
haftmann@35028
  1191
  "a <= b + (c::'a::ordered_ab_group_add)"
obua@15178
  1192
  "c <= d"    
obua@15178
  1193
  shows "a <= b + d"
obua@15178
  1194
  apply (rule_tac order_trans[where y = "b+c"])
obua@15178
  1195
  apply (simp_all add: prems)
obua@15178
  1196
  done
obua@15178
  1197
obua@15178
  1198
haftmann@25090
  1199
subsection {* Tools setup *}
haftmann@25090
  1200
blanchet@35828
  1201
lemma add_mono_thms_linordered_semiring [no_atp]:
haftmann@35028
  1202
  fixes i j k :: "'a\<Colon>ordered_ab_semigroup_add"
haftmann@25077
  1203
  shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1204
    and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1205
    and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"
haftmann@25077
  1206
    and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"
haftmann@25077
  1207
by (rule add_mono, clarify+)+
haftmann@25077
  1208
blanchet@35828
  1209
lemma add_mono_thms_linordered_field [no_atp]:
haftmann@35028
  1210
  fixes i j k :: "'a\<Colon>ordered_cancel_ab_semigroup_add"
haftmann@25077
  1211
  shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1212
    and "i = j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1213
    and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1214
    and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1215
    and "i < j \<and> k < l \<Longrightarrow> i + k < j + l"
haftmann@25077
  1216
by (auto intro: add_strict_right_mono add_strict_left_mono
haftmann@25077
  1217
  add_less_le_mono add_le_less_mono add_strict_mono)
haftmann@25077
  1218
paulson@17085
  1219
text{*Simplification of @{term "x-y < 0"}, etc.*}
blanchet@35828
  1220
lemmas diff_less_0_iff_less [simp, no_atp] = less_iff_diff_less_0 [symmetric]
blanchet@35828
  1221
lemmas diff_le_0_iff_le [simp, no_atp] = le_iff_diff_le_0 [symmetric]
paulson@17085
  1222
haftmann@22482
  1223
ML {*
wenzelm@27250
  1224
structure ab_group_add_cancel = Abel_Cancel
wenzelm@27250
  1225
(
haftmann@22482
  1226
haftmann@22482
  1227
(* term order for abelian groups *)
haftmann@22482
  1228
haftmann@22482
  1229
fun agrp_ord (Const (a, _)) = find_index (fn a' => a = a')
haftmann@35267
  1230
      [@{const_name Groups.zero}, @{const_name Groups.plus},
haftmann@35267
  1231
        @{const_name Groups.uminus}, @{const_name Groups.minus}]
haftmann@22482
  1232
  | agrp_ord _ = ~1;
haftmann@22482
  1233
wenzelm@35408
  1234
fun termless_agrp (a, b) = (Term_Ord.term_lpo agrp_ord (a, b) = LESS);
haftmann@22482
  1235
haftmann@22482
  1236
local
haftmann@22482
  1237
  val ac1 = mk_meta_eq @{thm add_assoc};
haftmann@22482
  1238
  val ac2 = mk_meta_eq @{thm add_commute};
haftmann@22482
  1239
  val ac3 = mk_meta_eq @{thm add_left_commute};
haftmann@35267
  1240
  fun solve_add_ac thy _ (_ $ (Const (@{const_name Groups.plus},_) $ _ $ _) $ _) =
haftmann@22482
  1241
        SOME ac1
haftmann@35267
  1242
    | solve_add_ac thy _ (_ $ x $ (Const (@{const_name Groups.plus},_) $ y $ z)) =
haftmann@22482
  1243
        if termless_agrp (y, x) then SOME ac3 else NONE
haftmann@22482
  1244
    | solve_add_ac thy _ (_ $ x $ y) =
haftmann@22482
  1245
        if termless_agrp (y, x) then SOME ac2 else NONE
haftmann@22482
  1246
    | solve_add_ac thy _ _ = NONE
haftmann@22482
  1247
in
wenzelm@32010
  1248
  val add_ac_proc = Simplifier.simproc @{theory}
haftmann@22482
  1249
    "add_ac_proc" ["x + y::'a::ab_semigroup_add"] solve_add_ac;
haftmann@22482
  1250
end;
haftmann@22482
  1251
wenzelm@27250
  1252
val eq_reflection = @{thm eq_reflection};
wenzelm@27250
  1253
  
wenzelm@27250
  1254
val T = @{typ "'a::ab_group_add"};
wenzelm@27250
  1255
haftmann@22482
  1256
val cancel_ss = HOL_basic_ss settermless termless_agrp
haftmann@22482
  1257
  addsimprocs [add_ac_proc] addsimps
nipkow@23085
  1258
  [@{thm add_0_left}, @{thm add_0_right}, @{thm diff_def},
haftmann@22482
  1259
   @{thm minus_add_distrib}, @{thm minus_minus}, @{thm minus_zero},
haftmann@22482
  1260
   @{thm right_minus}, @{thm left_minus}, @{thm add_minus_cancel},
haftmann@22482
  1261
   @{thm minus_add_cancel}];
wenzelm@27250
  1262
wenzelm@27250
  1263
val sum_pats = [@{cterm "x + y::'a::ab_group_add"}, @{cterm "x - y::'a::ab_group_add"}];
haftmann@22482
  1264
  
haftmann@22548
  1265
val eqI_rules = [@{thm less_eqI}, @{thm le_eqI}, @{thm eq_eqI}];
haftmann@22482
  1266
haftmann@22482
  1267
val dest_eqI = 
wenzelm@35364
  1268
  fst o HOLogic.dest_bin @{const_name "op ="} HOLogic.boolT o HOLogic.dest_Trueprop o concl_of;
haftmann@22482
  1269
wenzelm@27250
  1270
);
haftmann@22482
  1271
*}
haftmann@22482
  1272
wenzelm@26480
  1273
ML {*
haftmann@22482
  1274
  Addsimprocs [ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv];
haftmann@22482
  1275
*}
paulson@17085
  1276
haftmann@33364
  1277
code_modulename SML
haftmann@35050
  1278
  Groups Arith
haftmann@33364
  1279
haftmann@33364
  1280
code_modulename OCaml
haftmann@35050
  1281
  Groups Arith
haftmann@33364
  1282
haftmann@33364
  1283
code_modulename Haskell
haftmann@35050
  1284
  Groups Arith
haftmann@33364
  1285
obua@14738
  1286
end