src/HOL/OrderedGroup.thy
author nipkow
Thu May 24 07:27:44 2007 +0200 (2007-05-24)
changeset 23085 fd30d75a6614
parent 22997 d4f3b015b50b
child 23181 f52b555f8141
permissions -rw-r--r--
Introduced new classes monoid_add and group_add
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(*  Title:   HOL/OrderedGroup.thy
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    ID:      $Id$
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    Author:  Gertrud Bauer, Steven Obua, Lawrence C Paulson, and Markus Wenzel,
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             with contributions by Jeremy Avigad
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*)
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header {* Ordered Groups *}
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theory OrderedGroup
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imports Lattices
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uses "~~/src/Provers/Arith/abel_cancel.ML"
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begin
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text {*
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  The theory of partially ordered groups is taken from the books:
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  \begin{itemize}
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  \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979 
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  \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
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  \end{itemize}
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  Most of the used notions can also be looked up in 
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  \begin{itemize}
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  \item \url{http://www.mathworld.com} by Eric Weisstein et. al.
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  \item \emph{Algebra I} by van der Waerden, Springer.
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  \end{itemize}
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*}
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subsection {* Semigroups and Monoids *}
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class semigroup_add = plus +
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  assumes add_assoc: "(a \<^loc>+ b) \<^loc>+ c = a \<^loc>+ (b \<^loc>+ c)"
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class ab_semigroup_add = semigroup_add +
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  assumes add_commute: "a \<^loc>+ b = b \<^loc>+ a"
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lemma add_left_commute: "a + (b + c) = b + (a + (c::'a::ab_semigroup_add))"
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  by (rule mk_left_commute [of "op +", OF add_assoc add_commute])
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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: "(a \<^loc>* b) \<^loc>* c = a \<^loc>* (b \<^loc>* c)"
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class ab_semigroup_mult = semigroup_mult +
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  assumes mult_commute: "a \<^loc>* b = b \<^loc>* a"
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lemma mult_left_commute: "a * (b * c) = b * (a * (c::'a::ab_semigroup_mult))"
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  by (rule mk_left_commute [of "op *", OF mult_assoc mult_commute])
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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 [simp]: "\<^loc>0 \<^loc>+ a = a" and add_0_right [simp]: "a \<^loc>+ \<^loc>0 = a"
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class comm_monoid_add = zero + ab_semigroup_add +
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  assumes add_0: "\<^loc>0 \<^loc>+ a = a"
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instance comm_monoid_add < monoid_add
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by intro_classes (insert comm_monoid_add_class.zero_plus.add_0, simp_all add: add_commute, auto)
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class monoid_mult = one + semigroup_mult +
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  assumes mult_1_left [simp]: "\<^loc>1 \<^loc>* a  = a"
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  assumes mult_1_right [simp]: "a \<^loc>* \<^loc>1 = a"
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class comm_monoid_mult = one + ab_semigroup_mult +
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  assumes mult_1: "\<^loc>1 \<^loc>* a = a"
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instance comm_monoid_mult \<subseteq> monoid_mult
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  by intro_classes (insert mult_1, simp_all add: mult_commute, auto)
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class cancel_semigroup_add = semigroup_add +
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  assumes add_left_imp_eq: "a \<^loc>+ b = a \<^loc>+ c \<Longrightarrow> b = c"
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  assumes add_right_imp_eq: "b \<^loc>+ a = c \<^loc>+ a \<Longrightarrow> b = c"
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class cancel_ab_semigroup_add = ab_semigroup_add +
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  assumes add_imp_eq: "a \<^loc>+ b = a \<^loc>+ c \<Longrightarrow> b = c"
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instance cancel_ab_semigroup_add \<subseteq> cancel_semigroup_add
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proof intro_classes
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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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lemma add_left_cancel [simp]:
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  "a + b = a + c \<longleftrightarrow> b = (c \<Colon> 'a\<Colon>cancel_semigroup_add)"
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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 \<Colon> 'a\<Colon>cancel_semigroup_add)"
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  by (blast dest: add_right_imp_eq)
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subsection {* Groups *}
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class ab_group_add = minus + comm_monoid_add +
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  assumes ab_left_minus: "uminus a \<^loc>+ a = \<^loc>0"
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  assumes ab_diff_minus: "a \<^loc>- b = a \<^loc>+ (uminus b)"
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class group_add = minus + monoid_add +
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  assumes left_minus [simp]: "uminus a \<^loc>+ a = \<^loc>0"
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  assumes diff_minus: "a \<^loc>- b = a \<^loc>+ (uminus b)"
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instance ab_group_add < group_add
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by intro_classes (simp_all add: ab_left_minus ab_diff_minus)
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instance ab_group_add \<subseteq> cancel_ab_semigroup_add
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proof intro_classes
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  fix a b c :: 'a
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  assume "a + b = a + c"
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  then have "uminus a + a + b = uminus a + a + c" unfolding add_assoc by simp
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  then show "b = c" by simp
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qed
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lemma minus_add_cancel: "-(a::'a::group_add) + (a+b) = b"
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by(simp add:add_assoc[symmetric])
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lemma minus_zero[simp]: "-(0::'a::group_add) = 0"
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proof -
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  have "-(0::'a::group_add) = - 0 + (0+0)" by(simp only: add_0_right)
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  also have "\<dots> = 0" by(rule minus_add_cancel)
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  finally show ?thesis .
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qed
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lemma minus_minus[simp]: "- (-(a::'a::group_add)) = a"
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proof -
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  have "-(-a) = -(-a) + (-a + a)" by simp
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  also have "\<dots> = a" by(rule minus_add_cancel)
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  finally show ?thesis .
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qed
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lemma right_minus[simp]: "a + - a = (0::'a::group_add)"
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proof -
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  have "a + -a = -(-a) + -a" by simp
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  also have "\<dots> = 0" thm group_add.left_minus by(rule left_minus)
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  finally show ?thesis .
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qed
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lemma right_minus_eq: "(a - b = 0) = (a = (b::'a::group_add))"
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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 equals_zero_I: assumes "a+b = 0" shows "-a = (b::'a::group_add)"
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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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lemma diff_self[simp]: "(a::'a::group_add) - a = 0"
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by(simp add: diff_minus)
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lemma diff_0 [simp]: "(0::'a::group_add) - a = -a"
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by (simp add: diff_minus)
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lemma diff_0_right [simp]: "a - (0::'a::group_add) = a" 
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by (simp add: diff_minus)
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lemma diff_minus_eq_add [simp]: "a - - b = a + (b::'a::group_add)"
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by (simp add: diff_minus)
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lemma neg_equal_iff_equal [simp]: "(-a = -b) = (a = (b::'a::group_add))" 
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proof 
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  assume "- a = - b"
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  hence "- (- a) = - (- b)"
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    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]: "(-a = 0) = (a = (0::'a::group_add))"
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by (subst neg_equal_iff_equal [symmetric], simp)
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lemma neg_0_equal_iff_equal [simp]: "(0 = -a) = (0 = (a::'a::group_add))"
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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: "(a = - b) = (b = - (a::'a::group_add))"
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proof -
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  have "(- (-a) = - b) = (- 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: "(- a = b) = (- (b::'a::group_add) = a)"
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proof -
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  have "(- a = - (-b)) = (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_add_distrib [simp]: "- (a + b) = -a + -(b::'a::ab_group_add)"
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apply (rule equals_zero_I)
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apply (simp add: add_ac)
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done
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lemma minus_diff_eq [simp]: "- (a - b) = b - (a::'a::ab_group_add)"
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by (simp add: diff_minus add_commute)
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subsection {* (Partially) Ordered Groups *} 
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class pordered_ab_semigroup_add = order + ab_semigroup_add +
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  assumes add_left_mono: "a \<sqsubseteq> b \<Longrightarrow> c \<^loc>+ a \<sqsubseteq> c \<^loc>+ b"
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class pordered_cancel_ab_semigroup_add =
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  pordered_ab_semigroup_add + cancel_ab_semigroup_add
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class pordered_ab_semigroup_add_imp_le = pordered_cancel_ab_semigroup_add +
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  assumes add_le_imp_le_left: "c \<^loc>+ a \<sqsubseteq> c \<^loc>+ b \<Longrightarrow> a \<sqsubseteq> b"
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class pordered_ab_group_add = ab_group_add + pordered_ab_semigroup_add
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instance pordered_ab_group_add \<subseteq> pordered_ab_semigroup_add_imp_le
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proof
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  fix a b c :: 'a
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  assume "c + a \<le> c + b"
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  hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono)
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  hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc)
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  thus "a \<le> b" by simp
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qed
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class ordered_cancel_ab_semigroup_add = pordered_cancel_ab_semigroup_add + linorder
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instance ordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add_imp_le
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proof
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  fix a b c :: 'a
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  assume le: "c + a <= c + b"  
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  show "a <= b"
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  proof (rule ccontr)
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    assume w: "~ a \<le> b"
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    hence "b <= a" by (simp add: linorder_not_le)
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    hence le2: "c+b <= c+a" by (rule add_left_mono)
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    have "a = b" 
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      apply (insert le)
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      apply (insert le2)
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      apply (drule order_antisym, simp_all)
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      done
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    with w  show False 
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      by (simp add: linorder_not_le [symmetric])
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  qed
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qed
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lemma add_right_mono: "a \<le> (b::'a::pordered_ab_semigroup_add) ==> a + c \<le> b + c"
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  by (simp add: add_commute [of _ c] add_left_mono)
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text {* non-strict, in both arguments *}
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lemma add_mono:
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     "[|a \<le> b;  c \<le> d|] ==> a + c \<le> b + (d::'a::pordered_ab_semigroup_add)"
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  apply (erule add_right_mono [THEN order_trans])
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  apply (simp add: add_commute add_left_mono)
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  done
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lemma add_strict_left_mono:
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     "a < b ==> c + a < c + (b::'a::pordered_cancel_ab_semigroup_add)"
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 by (simp add: order_less_le add_left_mono) 
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lemma add_strict_right_mono:
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     "a < b ==> a + c < b + (c::'a::pordered_cancel_ab_semigroup_add)"
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 by (simp add: add_commute [of _ c] add_strict_left_mono)
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text{*Strict monotonicity in both arguments*}
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lemma add_strict_mono: "[|a<b; c<d|] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)"
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apply (erule add_strict_right_mono [THEN order_less_trans])
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apply (erule add_strict_left_mono)
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done
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lemma add_less_le_mono:
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     "[| a<b; c\<le>d |] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)"
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apply (erule add_strict_right_mono [THEN order_less_le_trans])
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apply (erule add_left_mono) 
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done
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lemma add_le_less_mono:
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     "[| a\<le>b; c<d |] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)"
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apply (erule add_right_mono [THEN order_le_less_trans])
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apply (erule add_strict_left_mono) 
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done
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lemma add_less_imp_less_left:
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      assumes less: "c + a < c + b"  shows "a < (b::'a::pordered_ab_semigroup_add_imp_le)"
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proof -
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  from less have le: "c + a <= c + b" by (simp add: order_le_less)
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  have "a <= b" 
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    apply (insert le)
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    apply (drule add_le_imp_le_left)
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    by (insert le, drule add_le_imp_le_left, assumption)
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  moreover have "a \<noteq> b"
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  proof (rule ccontr)
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    assume "~(a \<noteq> b)"
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    then have "a = b" by simp
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    then have "c + a = c + b" by simp
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    with less show "False"by simp
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  qed
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  ultimately show "a < b" by (simp add: order_le_less)
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qed
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lemma add_less_imp_less_right:
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      "a + c < b + c ==> a < (b::'a::pordered_ab_semigroup_add_imp_le)"
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apply (rule add_less_imp_less_left [of c])
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apply (simp add: add_commute)  
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done
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lemma add_less_cancel_left [simp]:
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    "(c+a < c+b) = (a < (b::'a::pordered_ab_semigroup_add_imp_le))"
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by (blast intro: add_less_imp_less_left add_strict_left_mono) 
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obua@14738
   317
lemma add_less_cancel_right [simp]:
obua@14738
   318
    "(a+c < b+c) = (a < (b::'a::pordered_ab_semigroup_add_imp_le))"
obua@14738
   319
by (blast intro: add_less_imp_less_right add_strict_right_mono)
obua@14738
   320
obua@14738
   321
lemma add_le_cancel_left [simp]:
obua@14738
   322
    "(c+a \<le> c+b) = (a \<le> (b::'a::pordered_ab_semigroup_add_imp_le))"
obua@14738
   323
by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono) 
obua@14738
   324
obua@14738
   325
lemma add_le_cancel_right [simp]:
obua@14738
   326
    "(a+c \<le> b+c) = (a \<le> (b::'a::pordered_ab_semigroup_add_imp_le))"
obua@14738
   327
by (simp add: add_commute[of a c] add_commute[of b c])
obua@14738
   328
obua@14738
   329
lemma add_le_imp_le_right:
obua@14738
   330
      "a + c \<le> b + c ==> a \<le> (b::'a::pordered_ab_semigroup_add_imp_le)"
obua@14738
   331
by simp
obua@14738
   332
paulson@15234
   333
lemma add_increasing:
paulson@15234
   334
  fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
paulson@15234
   335
  shows  "[|0\<le>a; b\<le>c|] ==> b \<le> a + c"
obua@14738
   336
by (insert add_mono [of 0 a b c], simp)
obua@14738
   337
nipkow@15539
   338
lemma add_increasing2:
nipkow@15539
   339
  fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
nipkow@15539
   340
  shows  "[|0\<le>c; b\<le>a|] ==> b \<le> a + c"
nipkow@15539
   341
by (simp add:add_increasing add_commute[of a])
nipkow@15539
   342
paulson@15234
   343
lemma add_strict_increasing:
paulson@15234
   344
  fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
paulson@15234
   345
  shows "[|0<a; b\<le>c|] ==> b < a + c"
paulson@15234
   346
by (insert add_less_le_mono [of 0 a b c], simp)
paulson@15234
   347
paulson@15234
   348
lemma add_strict_increasing2:
paulson@15234
   349
  fixes c :: "'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add}"
paulson@15234
   350
  shows "[|0\<le>a; b<c|] ==> b < a + c"
paulson@15234
   351
by (insert add_le_less_mono [of 0 a b c], simp)
paulson@15234
   352
paulson@19527
   353
lemma max_add_distrib_left:
paulson@19527
   354
  fixes z :: "'a::pordered_ab_semigroup_add_imp_le"
paulson@19527
   355
  shows  "(max x y) + z = max (x+z) (y+z)"
paulson@19527
   356
by (rule max_of_mono [THEN sym], rule add_le_cancel_right)
paulson@19527
   357
paulson@19527
   358
lemma min_add_distrib_left:
paulson@19527
   359
  fixes z :: "'a::pordered_ab_semigroup_add_imp_le"
paulson@19527
   360
  shows  "(min x y) + z = min (x+z) (y+z)"
paulson@19527
   361
by (rule min_of_mono [THEN sym], rule add_le_cancel_right)
paulson@19527
   362
paulson@19527
   363
lemma max_diff_distrib_left:
paulson@19527
   364
  fixes z :: "'a::pordered_ab_group_add"
paulson@19527
   365
  shows  "(max x y) - z = max (x-z) (y-z)"
paulson@19527
   366
by (simp add: diff_minus, rule max_add_distrib_left) 
paulson@19527
   367
paulson@19527
   368
lemma min_diff_distrib_left:
paulson@19527
   369
  fixes z :: "'a::pordered_ab_group_add"
paulson@19527
   370
  shows  "(min x y) - z = min (x-z) (y-z)"
paulson@19527
   371
by (simp add: diff_minus, rule min_add_distrib_left) 
paulson@19527
   372
paulson@15234
   373
obua@14738
   374
subsection {* Ordering Rules for Unary Minus *}
obua@14738
   375
obua@14738
   376
lemma le_imp_neg_le:
obua@14738
   377
      assumes "a \<le> (b::'a::{pordered_ab_semigroup_add_imp_le, ab_group_add})" shows "-b \<le> -a"
obua@14738
   378
proof -
obua@14738
   379
  have "-a+a \<le> -a+b"
obua@14738
   380
    by (rule add_left_mono) 
obua@14738
   381
  hence "0 \<le> -a+b"
obua@14738
   382
    by simp
obua@14738
   383
  hence "0 + (-b) \<le> (-a + b) + (-b)"
obua@14738
   384
    by (rule add_right_mono) 
obua@14738
   385
  thus ?thesis
obua@14738
   386
    by (simp add: add_assoc)
obua@14738
   387
qed
obua@14738
   388
obua@14738
   389
lemma neg_le_iff_le [simp]: "(-b \<le> -a) = (a \<le> (b::'a::pordered_ab_group_add))"
obua@14738
   390
proof 
obua@14738
   391
  assume "- b \<le> - a"
obua@14738
   392
  hence "- (- a) \<le> - (- b)"
obua@14738
   393
    by (rule le_imp_neg_le)
obua@14738
   394
  thus "a\<le>b" by simp
obua@14738
   395
next
obua@14738
   396
  assume "a\<le>b"
obua@14738
   397
  thus "-b \<le> -a" by (rule le_imp_neg_le)
obua@14738
   398
qed
obua@14738
   399
obua@14738
   400
lemma neg_le_0_iff_le [simp]: "(-a \<le> 0) = (0 \<le> (a::'a::pordered_ab_group_add))"
obua@14738
   401
by (subst neg_le_iff_le [symmetric], simp)
obua@14738
   402
obua@14738
   403
lemma neg_0_le_iff_le [simp]: "(0 \<le> -a) = (a \<le> (0::'a::pordered_ab_group_add))"
obua@14738
   404
by (subst neg_le_iff_le [symmetric], simp)
obua@14738
   405
obua@14738
   406
lemma neg_less_iff_less [simp]: "(-b < -a) = (a < (b::'a::pordered_ab_group_add))"
obua@14738
   407
by (force simp add: order_less_le) 
obua@14738
   408
obua@14738
   409
lemma neg_less_0_iff_less [simp]: "(-a < 0) = (0 < (a::'a::pordered_ab_group_add))"
obua@14738
   410
by (subst neg_less_iff_less [symmetric], simp)
obua@14738
   411
obua@14738
   412
lemma neg_0_less_iff_less [simp]: "(0 < -a) = (a < (0::'a::pordered_ab_group_add))"
obua@14738
   413
by (subst neg_less_iff_less [symmetric], simp)
obua@14738
   414
obua@14738
   415
text{*The next several equations can make the simplifier loop!*}
obua@14738
   416
obua@14738
   417
lemma less_minus_iff: "(a < - b) = (b < - (a::'a::pordered_ab_group_add))"
obua@14738
   418
proof -
obua@14738
   419
  have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
obua@14738
   420
  thus ?thesis by simp
obua@14738
   421
qed
obua@14738
   422
obua@14738
   423
lemma minus_less_iff: "(- a < b) = (- b < (a::'a::pordered_ab_group_add))"
obua@14738
   424
proof -
obua@14738
   425
  have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
obua@14738
   426
  thus ?thesis by simp
obua@14738
   427
qed
obua@14738
   428
obua@14738
   429
lemma le_minus_iff: "(a \<le> - b) = (b \<le> - (a::'a::pordered_ab_group_add))"
obua@14738
   430
proof -
obua@14738
   431
  have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff)
obua@14738
   432
  have "(- (- a) <= -b) = (b <= - a)" 
obua@14738
   433
    apply (auto simp only: order_le_less)
obua@14738
   434
    apply (drule mm)
obua@14738
   435
    apply (simp_all)
obua@14738
   436
    apply (drule mm[simplified], assumption)
obua@14738
   437
    done
obua@14738
   438
  then show ?thesis by simp
obua@14738
   439
qed
obua@14738
   440
obua@14738
   441
lemma minus_le_iff: "(- a \<le> b) = (- b \<le> (a::'a::pordered_ab_group_add))"
obua@14738
   442
by (auto simp add: order_le_less minus_less_iff)
obua@14738
   443
obua@14738
   444
lemma add_diff_eq: "a + (b - c) = (a + b) - (c::'a::ab_group_add)"
obua@14738
   445
by (simp add: diff_minus add_ac)
obua@14738
   446
obua@14738
   447
lemma diff_add_eq: "(a - b) + c = (a + c) - (b::'a::ab_group_add)"
obua@14738
   448
by (simp add: diff_minus add_ac)
obua@14738
   449
obua@14738
   450
lemma diff_eq_eq: "(a-b = c) = (a = c + (b::'a::ab_group_add))"
obua@14738
   451
by (auto simp add: diff_minus add_assoc)
obua@14738
   452
obua@14738
   453
lemma eq_diff_eq: "(a = c-b) = (a + (b::'a::ab_group_add) = c)"
obua@14738
   454
by (auto simp add: diff_minus add_assoc)
obua@14738
   455
obua@14738
   456
lemma diff_diff_eq: "(a - b) - c = a - (b + (c::'a::ab_group_add))"
obua@14738
   457
by (simp add: diff_minus add_ac)
obua@14738
   458
obua@14738
   459
lemma diff_diff_eq2: "a - (b - c) = (a + c) - (b::'a::ab_group_add)"
obua@14738
   460
by (simp add: diff_minus add_ac)
obua@14738
   461
obua@14738
   462
lemma diff_add_cancel: "a - b + b = (a::'a::ab_group_add)"
obua@14738
   463
by (simp add: diff_minus add_ac)
obua@14738
   464
obua@14738
   465
lemma add_diff_cancel: "a + b - b = (a::'a::ab_group_add)"
obua@14738
   466
by (simp add: diff_minus add_ac)
obua@14738
   467
obua@14754
   468
text{*Further subtraction laws*}
obua@14738
   469
obua@14738
   470
lemma less_iff_diff_less_0: "(a < b) = (a - b < (0::'a::pordered_ab_group_add))"
obua@14738
   471
proof -
obua@14738
   472
  have  "(a < b) = (a + (- b) < b + (-b))"  
obua@14738
   473
    by (simp only: add_less_cancel_right)
obua@14738
   474
  also have "... =  (a - b < 0)" by (simp add: diff_minus)
obua@14738
   475
  finally show ?thesis .
obua@14738
   476
qed
obua@14738
   477
obua@14738
   478
lemma diff_less_eq: "(a-b < c) = (a < c + (b::'a::pordered_ab_group_add))"
paulson@15481
   479
apply (subst less_iff_diff_less_0 [of a])
obua@14738
   480
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
obua@14738
   481
apply (simp add: diff_minus add_ac)
obua@14738
   482
done
obua@14738
   483
obua@14738
   484
lemma less_diff_eq: "(a < c-b) = (a + (b::'a::pordered_ab_group_add) < c)"
paulson@15481
   485
apply (subst less_iff_diff_less_0 [of "a+b"])
paulson@15481
   486
apply (subst less_iff_diff_less_0 [of a])
obua@14738
   487
apply (simp add: diff_minus add_ac)
obua@14738
   488
done
obua@14738
   489
obua@14738
   490
lemma diff_le_eq: "(a-b \<le> c) = (a \<le> c + (b::'a::pordered_ab_group_add))"
obua@14738
   491
by (auto simp add: order_le_less diff_less_eq diff_add_cancel add_diff_cancel)
obua@14738
   492
obua@14738
   493
lemma le_diff_eq: "(a \<le> c-b) = (a + (b::'a::pordered_ab_group_add) \<le> c)"
obua@14738
   494
by (auto simp add: order_le_less less_diff_eq diff_add_cancel add_diff_cancel)
obua@14738
   495
obua@14738
   496
text{*This list of rewrites simplifies (in)equalities by bringing subtractions
obua@14738
   497
  to the top and then moving negative terms to the other side.
obua@14738
   498
  Use with @{text add_ac}*}
obua@14738
   499
lemmas compare_rls =
obua@14738
   500
       diff_minus [symmetric]
obua@14738
   501
       add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
obua@14738
   502
       diff_less_eq less_diff_eq diff_le_eq le_diff_eq
obua@14738
   503
       diff_eq_eq eq_diff_eq
obua@14738
   504
avigad@16775
   505
subsection {* Support for reasoning about signs *}
avigad@16775
   506
avigad@16775
   507
lemma add_pos_pos: "0 < 
avigad@16775
   508
    (x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) 
avigad@16775
   509
      ==> 0 < y ==> 0 < x + y"
avigad@16775
   510
apply (subgoal_tac "0 + 0 < x + y")
avigad@16775
   511
apply simp
avigad@16775
   512
apply (erule add_less_le_mono)
avigad@16775
   513
apply (erule order_less_imp_le)
avigad@16775
   514
done
avigad@16775
   515
avigad@16775
   516
lemma add_pos_nonneg: "0 < 
avigad@16775
   517
    (x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) 
avigad@16775
   518
      ==> 0 <= y ==> 0 < x + y"
avigad@16775
   519
apply (subgoal_tac "0 + 0 < x + y")
avigad@16775
   520
apply simp
avigad@16775
   521
apply (erule add_less_le_mono, assumption)
avigad@16775
   522
done
avigad@16775
   523
avigad@16775
   524
lemma add_nonneg_pos: "0 <= 
avigad@16775
   525
    (x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) 
avigad@16775
   526
      ==> 0 < y ==> 0 < x + y"
avigad@16775
   527
apply (subgoal_tac "0 + 0 < x + y")
avigad@16775
   528
apply simp
avigad@16775
   529
apply (erule add_le_less_mono, assumption)
avigad@16775
   530
done
avigad@16775
   531
avigad@16775
   532
lemma add_nonneg_nonneg: "0 <= 
avigad@16775
   533
    (x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) 
avigad@16775
   534
      ==> 0 <= y ==> 0 <= x + y"
avigad@16775
   535
apply (subgoal_tac "0 + 0 <= x + y")
avigad@16775
   536
apply simp
avigad@16775
   537
apply (erule add_mono, assumption)
avigad@16775
   538
done
avigad@16775
   539
avigad@16775
   540
lemma add_neg_neg: "(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add})
avigad@16775
   541
    < 0 ==> y < 0 ==> x + y < 0"
avigad@16775
   542
apply (subgoal_tac "x + y < 0 + 0")
avigad@16775
   543
apply simp
avigad@16775
   544
apply (erule add_less_le_mono)
avigad@16775
   545
apply (erule order_less_imp_le)
avigad@16775
   546
done
avigad@16775
   547
avigad@16775
   548
lemma add_neg_nonpos: 
avigad@16775
   549
    "(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) < 0 
avigad@16775
   550
      ==> y <= 0 ==> x + y < 0"
avigad@16775
   551
apply (subgoal_tac "x + y < 0 + 0")
avigad@16775
   552
apply simp
avigad@16775
   553
apply (erule add_less_le_mono, assumption)
avigad@16775
   554
done
avigad@16775
   555
avigad@16775
   556
lemma add_nonpos_neg: 
avigad@16775
   557
    "(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) <= 0 
avigad@16775
   558
      ==> y < 0 ==> x + y < 0"
avigad@16775
   559
apply (subgoal_tac "x + y < 0 + 0")
avigad@16775
   560
apply simp
avigad@16775
   561
apply (erule add_le_less_mono, assumption)
avigad@16775
   562
done
avigad@16775
   563
avigad@16775
   564
lemma add_nonpos_nonpos: 
avigad@16775
   565
    "(x::'a::{comm_monoid_add,pordered_cancel_ab_semigroup_add}) <= 0 
avigad@16775
   566
      ==> y <= 0 ==> x + y <= 0"
avigad@16775
   567
apply (subgoal_tac "x + y <= 0 + 0")
avigad@16775
   568
apply simp
avigad@16775
   569
apply (erule add_mono, assumption)
avigad@16775
   570
done
obua@14738
   571
obua@14738
   572
subsection{*Lemmas for the @{text cancel_numerals} simproc*}
obua@14738
   573
obua@14738
   574
lemma eq_iff_diff_eq_0: "(a = b) = (a-b = (0::'a::ab_group_add))"
obua@14738
   575
by (simp add: compare_rls)
obua@14738
   576
obua@14738
   577
lemma le_iff_diff_le_0: "(a \<le> b) = (a-b \<le> (0::'a::pordered_ab_group_add))"
obua@14738
   578
by (simp add: compare_rls)
obua@14738
   579
haftmann@22452
   580
obua@14738
   581
subsection {* Lattice Ordered (Abelian) Groups *}
obua@14738
   582
haftmann@22452
   583
class lordered_ab_group_meet = pordered_ab_group_add + lower_semilattice
haftmann@22452
   584
haftmann@22452
   585
class lordered_ab_group_join = pordered_ab_group_add + upper_semilattice
obua@14738
   586
haftmann@22452
   587
class lordered_ab_group = pordered_ab_group_add + lattice
obua@14738
   588
haftmann@22452
   589
instance lordered_ab_group \<subseteq> lordered_ab_group_meet by default
haftmann@22452
   590
instance lordered_ab_group \<subseteq> lordered_ab_group_join by default
haftmann@22452
   591
haftmann@22452
   592
lemma add_inf_distrib_left: "a + inf b c = inf (a + b) (a + (c::'a::{pordered_ab_group_add, lower_semilattice}))"
obua@14738
   593
apply (rule order_antisym)
haftmann@22422
   594
apply (simp_all add: le_infI)
obua@14738
   595
apply (rule add_le_imp_le_left [of "-a"])
obua@14738
   596
apply (simp only: add_assoc[symmetric], simp)
nipkow@21312
   597
apply rule
nipkow@21312
   598
apply (rule add_le_imp_le_left[of "a"], simp only: add_assoc[symmetric], simp)+
obua@14738
   599
done
obua@14738
   600
haftmann@22452
   601
lemma add_sup_distrib_left: "a + sup b c = sup (a + b) (a+ (c::'a::{pordered_ab_group_add, upper_semilattice}))" 
obua@14738
   602
apply (rule order_antisym)
obua@14738
   603
apply (rule add_le_imp_le_left [of "-a"])
obua@14738
   604
apply (simp only: add_assoc[symmetric], simp)
nipkow@21312
   605
apply rule
nipkow@21312
   606
apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp)+
haftmann@22422
   607
apply (rule le_supI)
nipkow@21312
   608
apply (simp_all)
obua@14738
   609
done
obua@14738
   610
haftmann@22452
   611
lemma add_inf_distrib_right: "inf a b + (c::'a::lordered_ab_group) = inf (a+c) (b+c)"
obua@14738
   612
proof -
haftmann@22452
   613
  have "c + inf a b = inf (c+a) (c+b)" by (simp add: add_inf_distrib_left)
obua@14738
   614
  thus ?thesis by (simp add: add_commute)
obua@14738
   615
qed
obua@14738
   616
haftmann@22452
   617
lemma add_sup_distrib_right: "sup a b + (c::'a::lordered_ab_group) = sup (a+c) (b+c)"
obua@14738
   618
proof -
haftmann@22452
   619
  have "c + sup a b = sup (c+a) (c+b)" by (simp add: add_sup_distrib_left)
obua@14738
   620
  thus ?thesis by (simp add: add_commute)
obua@14738
   621
qed
obua@14738
   622
haftmann@22422
   623
lemmas add_sup_inf_distribs = add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left
obua@14738
   624
haftmann@22452
   625
lemma inf_eq_neg_sup: "inf a (b\<Colon>'a\<Colon>lordered_ab_group) = - sup (-a) (-b)"
haftmann@22452
   626
proof (rule inf_unique)
haftmann@22452
   627
  fix a b :: 'a
haftmann@22452
   628
  show "- sup (-a) (-b) \<le> a" by (rule add_le_imp_le_right [of _ "sup (-a) (-b)"])
haftmann@22452
   629
    (simp, simp add: add_sup_distrib_left)
haftmann@22452
   630
next
haftmann@22452
   631
  fix a b :: 'a
haftmann@22452
   632
  show "- sup (-a) (-b) \<le> b" by (rule add_le_imp_le_right [of _ "sup (-a) (-b)"])
haftmann@22452
   633
    (simp, simp add: add_sup_distrib_left)
haftmann@22452
   634
next
haftmann@22452
   635
  fix a b c :: 'a
haftmann@22452
   636
  assume "a \<le> b" "a \<le> c"
haftmann@22452
   637
  then show "a \<le> - sup (-b) (-c)" by (subst neg_le_iff_le [symmetric])
haftmann@22452
   638
    (simp add: le_supI)
haftmann@22452
   639
qed
haftmann@22452
   640
  
haftmann@22452
   641
lemma sup_eq_neg_inf: "sup a (b\<Colon>'a\<Colon>lordered_ab_group) = - inf (-a) (-b)"
haftmann@22452
   642
proof (rule sup_unique)
haftmann@22452
   643
  fix a b :: 'a
haftmann@22452
   644
  show "a \<le> - inf (-a) (-b)" by (rule add_le_imp_le_right [of _ "inf (-a) (-b)"])
haftmann@22452
   645
    (simp, simp add: add_inf_distrib_left)
haftmann@22452
   646
next
haftmann@22452
   647
  fix a b :: 'a
haftmann@22452
   648
  show "b \<le> - inf (-a) (-b)" by (rule add_le_imp_le_right [of _ "inf (-a) (-b)"])
haftmann@22452
   649
    (simp, simp add: add_inf_distrib_left)
haftmann@22452
   650
next
haftmann@22452
   651
  fix a b c :: 'a
haftmann@22452
   652
  assume "a \<le> c" "b \<le> c"
haftmann@22452
   653
  then show "- inf (-a) (-b) \<le> c" by (subst neg_le_iff_le [symmetric])
haftmann@22452
   654
    (simp add: le_infI)
haftmann@22452
   655
qed
obua@14738
   656
haftmann@22452
   657
lemma add_eq_inf_sup: "a + b = sup a b + inf a (b\<Colon>'a\<Colon>lordered_ab_group)"
obua@14738
   658
proof -
haftmann@22422
   659
  have "0 = - inf 0 (a-b) + inf (a-b) 0" by (simp add: inf_commute)
haftmann@22422
   660
  hence "0 = sup 0 (b-a) + inf (a-b) 0" by (simp add: inf_eq_neg_sup)
haftmann@22422
   661
  hence "0 = (-a + sup a b) + (inf a b + (-b))"
haftmann@22422
   662
    apply (simp add: add_sup_distrib_left add_inf_distrib_right)
obua@14738
   663
    by (simp add: diff_minus add_commute)
obua@14738
   664
  thus ?thesis
obua@14738
   665
    apply (simp add: compare_rls)
haftmann@22422
   666
    apply (subst add_left_cancel[symmetric, of "a+b" "sup a b + inf a b" "-a"])
obua@14738
   667
    apply (simp only: add_assoc, simp add: add_assoc[symmetric])
obua@14738
   668
    done
obua@14738
   669
qed
obua@14738
   670
obua@14738
   671
subsection {* Positive Part, Negative Part, Absolute Value *}
obua@14738
   672
haftmann@22422
   673
definition
haftmann@22422
   674
  nprt :: "'a \<Rightarrow> ('a::lordered_ab_group)" where
haftmann@22422
   675
  "nprt x = inf x 0"
haftmann@22422
   676
haftmann@22422
   677
definition
haftmann@22422
   678
  pprt :: "'a \<Rightarrow> ('a::lordered_ab_group)" where
haftmann@22422
   679
  "pprt x = sup x 0"
obua@14738
   680
obua@14738
   681
lemma prts: "a = pprt a + nprt a"
haftmann@22422
   682
by (simp add: pprt_def nprt_def add_eq_inf_sup[symmetric])
obua@14738
   683
obua@14738
   684
lemma zero_le_pprt[simp]: "0 \<le> pprt a"
nipkow@21312
   685
by (simp add: pprt_def)
obua@14738
   686
obua@14738
   687
lemma nprt_le_zero[simp]: "nprt a \<le> 0"
nipkow@21312
   688
by (simp add: nprt_def)
obua@14738
   689
obua@14738
   690
lemma le_eq_neg: "(a \<le> -b) = (a + b \<le> (0::_::lordered_ab_group))" (is "?l = ?r")
obua@14738
   691
proof -
obua@14738
   692
  have a: "?l \<longrightarrow> ?r"
obua@14738
   693
    apply (auto)
obua@14738
   694
    apply (rule add_le_imp_le_right[of _ "-b" _])
obua@14738
   695
    apply (simp add: add_assoc)
obua@14738
   696
    done
obua@14738
   697
  have b: "?r \<longrightarrow> ?l"
obua@14738
   698
    apply (auto)
obua@14738
   699
    apply (rule add_le_imp_le_right[of _ "b" _])
obua@14738
   700
    apply (simp)
obua@14738
   701
    done
obua@14738
   702
  from a b show ?thesis by blast
obua@14738
   703
qed
obua@14738
   704
obua@15580
   705
lemma pprt_0[simp]: "pprt 0 = 0" by (simp add: pprt_def)
obua@15580
   706
lemma nprt_0[simp]: "nprt 0 = 0" by (simp add: nprt_def)
obua@15580
   707
obua@15580
   708
lemma pprt_eq_id[simp]: "0 <= x \<Longrightarrow> pprt x = x"
haftmann@22422
   709
  by (simp add: pprt_def le_iff_sup sup_aci)
obua@15580
   710
obua@15580
   711
lemma nprt_eq_id[simp]: "x <= 0 \<Longrightarrow> nprt x = x"
haftmann@22422
   712
  by (simp add: nprt_def le_iff_inf inf_aci)
obua@15580
   713
obua@15580
   714
lemma pprt_eq_0[simp]: "x <= 0 \<Longrightarrow> pprt x = 0"
haftmann@22422
   715
  by (simp add: pprt_def le_iff_sup sup_aci)
obua@15580
   716
obua@15580
   717
lemma nprt_eq_0[simp]: "0 <= x \<Longrightarrow> nprt x = 0"
haftmann@22422
   718
  by (simp add: nprt_def le_iff_inf inf_aci)
obua@15580
   719
haftmann@22422
   720
lemma sup_0_imp_0: "sup a (-a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)"
obua@14738
   721
proof -
obua@14738
   722
  {
obua@14738
   723
    fix a::'a
haftmann@22422
   724
    assume hyp: "sup a (-a) = 0"
haftmann@22422
   725
    hence "sup a (-a) + a = a" by (simp)
haftmann@22422
   726
    hence "sup (a+a) 0 = a" by (simp add: add_sup_distrib_right) 
haftmann@22422
   727
    hence "sup (a+a) 0 <= a" by (simp)
haftmann@22422
   728
    hence "0 <= a" by (blast intro: order_trans inf_sup_ord)
obua@14738
   729
  }
obua@14738
   730
  note p = this
haftmann@22422
   731
  assume hyp:"sup a (-a) = 0"
haftmann@22422
   732
  hence hyp2:"sup (-a) (-(-a)) = 0" by (simp add: sup_commute)
obua@14738
   733
  from p[OF hyp] p[OF hyp2] show "a = 0" by simp
obua@14738
   734
qed
obua@14738
   735
haftmann@22422
   736
lemma inf_0_imp_0: "inf a (-a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)"
haftmann@22422
   737
apply (simp add: inf_eq_neg_sup)
haftmann@22422
   738
apply (simp add: sup_commute)
haftmann@22422
   739
apply (erule sup_0_imp_0)
paulson@15481
   740
done
obua@14738
   741
haftmann@22422
   742
lemma inf_0_eq_0[simp]: "(inf a (-a) = 0) = (a = (0::'a::lordered_ab_group))"
haftmann@22422
   743
by (auto, erule inf_0_imp_0)
obua@14738
   744
haftmann@22422
   745
lemma sup_0_eq_0[simp]: "(sup a (-a) = 0) = (a = (0::'a::lordered_ab_group))"
haftmann@22422
   746
by (auto, erule sup_0_imp_0)
obua@14738
   747
obua@14738
   748
lemma zero_le_double_add_iff_zero_le_single_add[simp]: "(0 \<le> a + a) = (0 \<le> (a::'a::lordered_ab_group))"
obua@14738
   749
proof
obua@14738
   750
  assume "0 <= a + a"
haftmann@22422
   751
  hence a:"inf (a+a) 0 = 0" by (simp add: le_iff_inf inf_commute)
haftmann@22422
   752
  have "(inf a 0)+(inf a 0) = inf (inf (a+a) 0) a" (is "?l=_") by (simp add: add_sup_inf_distribs inf_aci)
haftmann@22422
   753
  hence "?l = 0 + inf a 0" by (simp add: a, simp add: inf_commute)
haftmann@22422
   754
  hence "inf a 0 = 0" by (simp only: add_right_cancel)
haftmann@22422
   755
  then show "0 <= a" by (simp add: le_iff_inf inf_commute)    
obua@14738
   756
next  
obua@14738
   757
  assume a: "0 <= a"
obua@14738
   758
  show "0 <= a + a" by (simp add: add_mono[OF a a, simplified])
obua@14738
   759
qed
obua@14738
   760
obua@14738
   761
lemma double_add_le_zero_iff_single_add_le_zero[simp]: "(a + a <= 0) = ((a::'a::lordered_ab_group) <= 0)" 
obua@14738
   762
proof -
obua@14738
   763
  have "(a + a <= 0) = (0 <= -(a+a))" by (subst le_minus_iff, simp)
obua@14738
   764
  moreover have "\<dots> = (a <= 0)" by (simp add: zero_le_double_add_iff_zero_le_single_add)
obua@14738
   765
  ultimately show ?thesis by blast
obua@14738
   766
qed
obua@14738
   767
obua@14738
   768
lemma double_add_less_zero_iff_single_less_zero[simp]: "(a+a<0) = ((a::'a::{pordered_ab_group_add,linorder}) < 0)" (is ?s)
obua@14738
   769
proof cases
obua@14738
   770
  assume a: "a < 0"
obua@14738
   771
  thus ?s by (simp add:  add_strict_mono[OF a a, simplified])
obua@14738
   772
next
obua@14738
   773
  assume "~(a < 0)" 
obua@14738
   774
  hence a:"0 <= a" by (simp)
obua@14738
   775
  hence "0 <= a+a" by (simp add: add_mono[OF a a, simplified])
obua@14738
   776
  hence "~(a+a < 0)" by simp
obua@14738
   777
  with a show ?thesis by simp 
obua@14738
   778
qed
obua@14738
   779
haftmann@22452
   780
class lordered_ab_group_abs = lordered_ab_group +
haftmann@22452
   781
  assumes abs_lattice: "abs x = sup x (uminus x)"
obua@14738
   782
obua@14738
   783
lemma abs_zero[simp]: "abs 0 = (0::'a::lordered_ab_group_abs)"
obua@14738
   784
by (simp add: abs_lattice)
obua@14738
   785
obua@14738
   786
lemma abs_eq_0[simp]: "(abs a = 0) = (a = (0::'a::lordered_ab_group_abs))"
obua@14738
   787
by (simp add: abs_lattice)
obua@14738
   788
obua@14738
   789
lemma abs_0_eq[simp]: "(0 = abs a) = (a = (0::'a::lordered_ab_group_abs))"
obua@14738
   790
proof -
obua@14738
   791
  have "(0 = abs a) = (abs a = 0)" by (simp only: eq_ac)
obua@14738
   792
  thus ?thesis by simp
obua@14738
   793
qed
obua@14738
   794
haftmann@22422
   795
lemma neg_inf_eq_sup[simp]: "- inf a (b::_::lordered_ab_group) = sup (-a) (-b)"
haftmann@22422
   796
by (simp add: inf_eq_neg_sup)
obua@14738
   797
haftmann@22422
   798
lemma neg_sup_eq_inf[simp]: "- sup a (b::_::lordered_ab_group) = inf (-a) (-b)"
haftmann@22422
   799
by (simp del: neg_inf_eq_sup add: sup_eq_neg_inf)
obua@14738
   800
haftmann@22422
   801
lemma sup_eq_if: "sup a (-a) = (if a < 0 then -a else (a::'a::{lordered_ab_group, linorder}))"
obua@14738
   802
proof -
obua@14738
   803
  note b = add_le_cancel_right[of a a "-a",symmetric,simplified]
obua@14738
   804
  have c: "a + a = 0 \<Longrightarrow> -a = a" by (rule add_right_imp_eq[of _ a], simp)
haftmann@22452
   805
  show ?thesis by (auto simp add: max_def b linorder_not_less sup_max)
obua@14738
   806
qed
obua@14738
   807
obua@14738
   808
lemma abs_if_lattice: "\<bar>a\<bar> = (if a < 0 then -a else (a::'a::{lordered_ab_group_abs, linorder}))"
obua@14738
   809
proof -
haftmann@22422
   810
  show ?thesis by (simp add: abs_lattice sup_eq_if)
obua@14738
   811
qed
obua@14738
   812
obua@14738
   813
lemma abs_ge_zero[simp]: "0 \<le> abs (a::'a::lordered_ab_group_abs)"
obua@14738
   814
proof -
nipkow@21312
   815
  have a:"a <= abs a" and b:"-a <= abs a" by (auto simp add: abs_lattice)
obua@14738
   816
  show ?thesis by (rule add_mono[OF a b, simplified])
obua@14738
   817
qed
obua@14738
   818
  
obua@14738
   819
lemma abs_le_zero_iff [simp]: "(abs a \<le> (0::'a::lordered_ab_group_abs)) = (a = 0)" 
obua@14738
   820
proof
obua@14738
   821
  assume "abs a <= 0"
obua@14738
   822
  hence "abs a = 0" by (auto dest: order_antisym)
obua@14738
   823
  thus "a = 0" by simp
obua@14738
   824
next
obua@14738
   825
  assume "a = 0"
obua@14738
   826
  thus "abs a <= 0" by simp
obua@14738
   827
qed
obua@14738
   828
obua@14738
   829
lemma zero_less_abs_iff [simp]: "(0 < abs a) = (a \<noteq> (0::'a::lordered_ab_group_abs))"
obua@14738
   830
by (simp add: order_less_le)
obua@14738
   831
obua@14738
   832
lemma abs_not_less_zero [simp]: "~ abs a < (0::'a::lordered_ab_group_abs)"
obua@14738
   833
proof -
obua@14738
   834
  have a:"!! x (y::_::order). x <= y \<Longrightarrow> ~(y < x)" by auto
obua@14738
   835
  show ?thesis by (simp add: a)
obua@14738
   836
qed
obua@14738
   837
obua@14738
   838
lemma abs_ge_self: "a \<le> abs (a::'a::lordered_ab_group_abs)"
nipkow@21312
   839
by (simp add: abs_lattice)
obua@14738
   840
obua@14738
   841
lemma abs_ge_minus_self: "-a \<le> abs (a::'a::lordered_ab_group_abs)"
nipkow@21312
   842
by (simp add: abs_lattice)
obua@14738
   843
obua@14738
   844
lemma abs_prts: "abs (a::_::lordered_ab_group_abs) = pprt a - nprt a"
obua@14738
   845
apply (simp add: pprt_def nprt_def diff_minus)
haftmann@22422
   846
apply (simp add: add_sup_inf_distribs sup_aci abs_lattice[symmetric])
haftmann@22422
   847
apply (subst sup_absorb2, auto)
obua@14738
   848
done
obua@14738
   849
obua@14738
   850
lemma abs_minus_cancel [simp]: "abs (-a) = abs(a::'a::lordered_ab_group_abs)"
haftmann@22422
   851
by (simp add: abs_lattice sup_commute)
obua@14738
   852
obua@14738
   853
lemma abs_idempotent [simp]: "abs (abs a) = abs (a::'a::lordered_ab_group_abs)"
obua@14738
   854
apply (simp add: abs_lattice[of "abs a"])
haftmann@22422
   855
apply (subst sup_absorb1)
obua@14738
   856
apply (rule order_trans[of _ 0])
obua@14738
   857
by auto
obua@14738
   858
paulson@15093
   859
lemma abs_minus_commute: 
paulson@15093
   860
  fixes a :: "'a::lordered_ab_group_abs"
paulson@15093
   861
  shows "abs (a-b) = abs(b-a)"
paulson@15093
   862
proof -
paulson@15093
   863
  have "abs (a-b) = abs (- (a-b))" by (simp only: abs_minus_cancel)
paulson@15093
   864
  also have "... = abs(b-a)" by simp
paulson@15093
   865
  finally show ?thesis .
paulson@15093
   866
qed
paulson@15093
   867
obua@14738
   868
lemma zero_le_iff_zero_nprt: "(0 \<le> a) = (nprt a = 0)"
haftmann@22422
   869
by (simp add: le_iff_inf nprt_def inf_commute)
obua@14738
   870
obua@14738
   871
lemma le_zero_iff_zero_pprt: "(a \<le> 0) = (pprt a = 0)"
haftmann@22422
   872
by (simp add: le_iff_sup pprt_def sup_commute)
obua@14738
   873
obua@14738
   874
lemma le_zero_iff_pprt_id: "(0 \<le> a) = (pprt a = a)"
haftmann@22422
   875
by (simp add: le_iff_sup pprt_def sup_commute)
obua@14738
   876
obua@14738
   877
lemma zero_le_iff_nprt_id: "(a \<le> 0) = (nprt a = a)"
haftmann@22422
   878
by (simp add: le_iff_inf nprt_def inf_commute)
obua@14738
   879
obua@15580
   880
lemma pprt_mono[simp]: "(a::_::lordered_ab_group) <= b \<Longrightarrow> pprt a <= pprt b"
haftmann@22422
   881
  by (simp add: le_iff_sup pprt_def sup_aci)
obua@15580
   882
obua@15580
   883
lemma nprt_mono[simp]: "(a::_::lordered_ab_group) <= b \<Longrightarrow> nprt a <= nprt b"
haftmann@22422
   884
  by (simp add: le_iff_inf nprt_def inf_aci)
obua@15580
   885
obua@19404
   886
lemma pprt_neg: "pprt (-x) = - nprt x"
obua@19404
   887
  by (simp add: pprt_def nprt_def)
obua@19404
   888
obua@19404
   889
lemma nprt_neg: "nprt (-x) = - pprt x"
obua@19404
   890
  by (simp add: pprt_def nprt_def)
obua@19404
   891
obua@14738
   892
lemma iff2imp: "(A=B) \<Longrightarrow> (A \<Longrightarrow> B)"
obua@14738
   893
by (simp)
obua@14738
   894
avigad@16775
   895
lemma abs_of_nonneg [simp]: "0 \<le> a \<Longrightarrow> abs a = (a::'a::lordered_ab_group_abs)"
obua@14738
   896
by (simp add: iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_pprt_id] abs_prts)
obua@14738
   897
avigad@16775
   898
lemma abs_of_pos: "0 < (x::'a::lordered_ab_group_abs) ==> abs x = x";
avigad@16775
   899
by (rule abs_of_nonneg, rule order_less_imp_le);
avigad@16775
   900
avigad@16775
   901
lemma abs_of_nonpos [simp]: "a \<le> 0 \<Longrightarrow> abs a = -(a::'a::lordered_ab_group_abs)"
obua@14738
   902
by (simp add: iff2imp[OF le_zero_iff_zero_pprt] iff2imp[OF zero_le_iff_nprt_id] abs_prts)
obua@14738
   903
avigad@16775
   904
lemma abs_of_neg: "(x::'a::lordered_ab_group_abs) <  0 ==> 
avigad@16775
   905
  abs x = - x"
avigad@16775
   906
by (rule abs_of_nonpos, rule order_less_imp_le)
avigad@16775
   907
obua@14738
   908
lemma abs_leI: "[|a \<le> b; -a \<le> b|] ==> abs a \<le> (b::'a::lordered_ab_group_abs)"
haftmann@22422
   909
by (simp add: abs_lattice le_supI)
obua@14738
   910
obua@14738
   911
lemma le_minus_self_iff: "(a \<le> -a) = (a \<le> (0::'a::lordered_ab_group))"
obua@14738
   912
proof -
obua@14738
   913
  from add_le_cancel_left[of "-a" "a+a" "0"] have "(a <= -a) = (a+a <= 0)" 
obua@14738
   914
    by (simp add: add_assoc[symmetric])
obua@14738
   915
  thus ?thesis by simp
obua@14738
   916
qed
obua@14738
   917
obua@14738
   918
lemma minus_le_self_iff: "(-a \<le> a) = (0 \<le> (a::'a::lordered_ab_group))"
obua@14738
   919
proof -
obua@14738
   920
  from add_le_cancel_left[of "-a" "0" "a+a"] have "(-a <= a) = (0 <= a+a)" 
obua@14738
   921
    by (simp add: add_assoc[symmetric])
obua@14738
   922
  thus ?thesis by simp
obua@14738
   923
qed
obua@14738
   924
obua@14738
   925
lemma abs_le_D1: "abs a \<le> b ==> a \<le> (b::'a::lordered_ab_group_abs)"
obua@14738
   926
by (insert abs_ge_self, blast intro: order_trans)
obua@14738
   927
obua@14738
   928
lemma abs_le_D2: "abs a \<le> b ==> -a \<le> (b::'a::lordered_ab_group_abs)"
obua@14738
   929
by (insert abs_le_D1 [of "-a"], simp)
obua@14738
   930
obua@14738
   931
lemma abs_le_iff: "(abs a \<le> b) = (a \<le> b & -a \<le> (b::'a::lordered_ab_group_abs))"
obua@14738
   932
by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
obua@14738
   933
nipkow@15539
   934
lemma abs_triangle_ineq: "abs(a+b) \<le> abs a + abs(b::'a::lordered_ab_group_abs)"
obua@14738
   935
proof -
haftmann@22422
   936
  have g:"abs a + abs b = sup (a+b) (sup (-a-b) (sup (-a+b) (a + (-b))))" (is "_=sup ?m ?n")
haftmann@22422
   937
    by (simp add: abs_lattice add_sup_inf_distribs sup_aci diff_minus)
haftmann@22422
   938
  have a:"a+b <= sup ?m ?n" by (simp)
nipkow@21312
   939
  have b:"-a-b <= ?n" by (simp) 
haftmann@22422
   940
  have c:"?n <= sup ?m ?n" by (simp)
haftmann@22422
   941
  from b c have d: "-a-b <= sup ?m ?n" by(rule order_trans)
obua@14738
   942
  have e:"-a-b = -(a+b)" by (simp add: diff_minus)
haftmann@22422
   943
  from a d e have "abs(a+b) <= sup ?m ?n" 
obua@14738
   944
    by (drule_tac abs_leI, auto)
obua@14738
   945
  with g[symmetric] show ?thesis by simp
obua@14738
   946
qed
obua@14738
   947
avigad@16775
   948
lemma abs_triangle_ineq2: "abs (a::'a::lordered_ab_group_abs) - 
avigad@16775
   949
    abs b <= abs (a - b)"
avigad@16775
   950
  apply (simp add: compare_rls)
avigad@16775
   951
  apply (subgoal_tac "abs a = abs (a - b + b)")
avigad@16775
   952
  apply (erule ssubst)
avigad@16775
   953
  apply (rule abs_triangle_ineq)
avigad@16775
   954
  apply (rule arg_cong);back;
avigad@16775
   955
  apply (simp add: compare_rls)
avigad@16775
   956
done
avigad@16775
   957
avigad@16775
   958
lemma abs_triangle_ineq3: 
avigad@16775
   959
    "abs(abs (a::'a::lordered_ab_group_abs) - abs b) <= abs (a - b)"
avigad@16775
   960
  apply (subst abs_le_iff)
avigad@16775
   961
  apply auto
avigad@16775
   962
  apply (rule abs_triangle_ineq2)
avigad@16775
   963
  apply (subst abs_minus_commute)
avigad@16775
   964
  apply (rule abs_triangle_ineq2)
avigad@16775
   965
done
avigad@16775
   966
avigad@16775
   967
lemma abs_triangle_ineq4: "abs ((a::'a::lordered_ab_group_abs) - b) <= 
avigad@16775
   968
    abs a + abs b"
avigad@16775
   969
proof -;
avigad@16775
   970
  have "abs(a - b) = abs(a + - b)"
avigad@16775
   971
    by (subst diff_minus, rule refl)
avigad@16775
   972
  also have "... <= abs a + abs (- b)"
avigad@16775
   973
    by (rule abs_triangle_ineq)
avigad@16775
   974
  finally show ?thesis
avigad@16775
   975
    by simp
avigad@16775
   976
qed
avigad@16775
   977
obua@14738
   978
lemma abs_diff_triangle_ineq:
obua@14738
   979
     "\<bar>(a::'a::lordered_ab_group_abs) + b - (c+d)\<bar> \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>"
obua@14738
   980
proof -
obua@14738
   981
  have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: diff_minus add_ac)
obua@14738
   982
  also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq)
obua@14738
   983
  finally show ?thesis .
obua@14738
   984
qed
obua@14738
   985
nipkow@15539
   986
lemma abs_add_abs[simp]:
nipkow@15539
   987
fixes a:: "'a::{lordered_ab_group_abs}"
nipkow@15539
   988
shows "abs(abs a + abs b) = abs a + abs b" (is "?L = ?R")
nipkow@15539
   989
proof (rule order_antisym)
nipkow@15539
   990
  show "?L \<ge> ?R" by(rule abs_ge_self)
nipkow@15539
   991
next
nipkow@15539
   992
  have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq)
nipkow@15539
   993
  also have "\<dots> = ?R" by simp
nipkow@15539
   994
  finally show "?L \<le> ?R" .
nipkow@15539
   995
qed
nipkow@15539
   996
obua@14754
   997
text {* Needed for abelian cancellation simprocs: *}
obua@14754
   998
obua@14754
   999
lemma add_cancel_21: "((x::'a::ab_group_add) + (y + z) = y + u) = (x + z = u)"
obua@14754
  1000
apply (subst add_left_commute)
obua@14754
  1001
apply (subst add_left_cancel)
obua@14754
  1002
apply simp
obua@14754
  1003
done
obua@14754
  1004
obua@14754
  1005
lemma add_cancel_end: "(x + (y + z) = y) = (x = - (z::'a::ab_group_add))"
obua@14754
  1006
apply (subst add_cancel_21[of _ _ _ 0, simplified])
obua@14754
  1007
apply (simp add: add_right_cancel[symmetric, of "x" "-z" "z", simplified])
obua@14754
  1008
done
obua@14754
  1009
obua@14754
  1010
lemma less_eqI: "(x::'a::pordered_ab_group_add) - y = x' - y' \<Longrightarrow> (x < y) = (x' < y')"
obua@14754
  1011
by (simp add: less_iff_diff_less_0[of x y] less_iff_diff_less_0[of x' y'])
obua@14754
  1012
obua@14754
  1013
lemma le_eqI: "(x::'a::pordered_ab_group_add) - y = x' - y' \<Longrightarrow> (y <= x) = (y' <= x')"
obua@14754
  1014
apply (simp add: le_iff_diff_le_0[of y x] le_iff_diff_le_0[of  y' x'])
obua@14754
  1015
apply (simp add: neg_le_iff_le[symmetric, of "y-x" 0] neg_le_iff_le[symmetric, of "y'-x'" 0])
obua@14754
  1016
done
obua@14754
  1017
obua@14754
  1018
lemma eq_eqI: "(x::'a::ab_group_add) - y = x' - y' \<Longrightarrow> (x = y) = (x' = y')"
obua@14754
  1019
by (simp add: eq_iff_diff_eq_0[of x y] eq_iff_diff_eq_0[of x' y'])
obua@14754
  1020
obua@14754
  1021
lemma diff_def: "(x::'a::ab_group_add) - y == x + (-y)"
obua@14754
  1022
by (simp add: diff_minus)
obua@14754
  1023
obua@14754
  1024
lemma add_minus_cancel: "(a::'a::ab_group_add) + (-a + b) = b"
obua@14754
  1025
by (simp add: add_assoc[symmetric])
obua@14754
  1026
obua@15178
  1027
lemma  le_add_right_mono: 
obua@15178
  1028
  assumes 
obua@15178
  1029
  "a <= b + (c::'a::pordered_ab_group_add)"
obua@15178
  1030
  "c <= d"    
obua@15178
  1031
  shows "a <= b + d"
obua@15178
  1032
  apply (rule_tac order_trans[where y = "b+c"])
obua@15178
  1033
  apply (simp_all add: prems)
obua@15178
  1034
  done
obua@15178
  1035
obua@15178
  1036
lemmas group_eq_simps =
obua@15178
  1037
  mult_ac
obua@15178
  1038
  add_ac
obua@15178
  1039
  add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
obua@15178
  1040
  diff_eq_eq eq_diff_eq
obua@15178
  1041
obua@15178
  1042
lemma estimate_by_abs:
obua@15178
  1043
"a + b <= (c::'a::lordered_ab_group_abs) \<Longrightarrow> a <= c + abs b" 
obua@15178
  1044
proof -
obua@15178
  1045
  assume 1: "a+b <= c"
obua@15178
  1046
  have 2: "a <= c+(-b)"
obua@15178
  1047
    apply (insert 1)
obua@15178
  1048
    apply (drule_tac add_right_mono[where c="-b"])
obua@15178
  1049
    apply (simp add: group_eq_simps)
obua@15178
  1050
    done
obua@15178
  1051
  have 3: "(-b) <= abs b" by (rule abs_ge_minus_self)
obua@15178
  1052
  show ?thesis by (rule le_add_right_mono[OF 2 3])
obua@15178
  1053
qed
obua@15178
  1054
haftmann@22482
  1055
haftmann@22482
  1056
subsection {* Tools setup *}
haftmann@22482
  1057
paulson@17085
  1058
text{*Simplification of @{term "x-y < 0"}, etc.*}
paulson@17085
  1059
lemmas diff_less_0_iff_less = less_iff_diff_less_0 [symmetric]
paulson@17085
  1060
lemmas diff_eq_0_iff_eq = eq_iff_diff_eq_0 [symmetric]
paulson@17085
  1061
lemmas diff_le_0_iff_le = le_iff_diff_le_0 [symmetric]
paulson@17085
  1062
declare diff_less_0_iff_less [simp]
paulson@17085
  1063
declare diff_eq_0_iff_eq [simp]
paulson@17085
  1064
declare diff_le_0_iff_le [simp]
paulson@17085
  1065
haftmann@22482
  1066
ML {*
haftmann@22482
  1067
structure ab_group_add_cancel = Abel_Cancel(
haftmann@22482
  1068
struct
haftmann@22482
  1069
haftmann@22482
  1070
(* term order for abelian groups *)
haftmann@22482
  1071
haftmann@22482
  1072
fun agrp_ord (Const (a, _)) = find_index (fn a' => a = a')
haftmann@22997
  1073
      [@{const_name HOL.zero}, @{const_name HOL.plus},
haftmann@22997
  1074
        @{const_name HOL.uminus}, @{const_name HOL.minus}]
haftmann@22482
  1075
  | agrp_ord _ = ~1;
haftmann@22482
  1076
haftmann@22482
  1077
fun termless_agrp (a, b) = (Term.term_lpo agrp_ord (a, b) = LESS);
haftmann@22482
  1078
haftmann@22482
  1079
local
haftmann@22482
  1080
  val ac1 = mk_meta_eq @{thm add_assoc};
haftmann@22482
  1081
  val ac2 = mk_meta_eq @{thm add_commute};
haftmann@22482
  1082
  val ac3 = mk_meta_eq @{thm add_left_commute};
haftmann@22997
  1083
  fun solve_add_ac thy _ (_ $ (Const (@{const_name HOL.plus},_) $ _ $ _) $ _) =
haftmann@22482
  1084
        SOME ac1
haftmann@22997
  1085
    | solve_add_ac thy _ (_ $ x $ (Const (@{const_name HOL.plus},_) $ y $ z)) =
haftmann@22482
  1086
        if termless_agrp (y, x) then SOME ac3 else NONE
haftmann@22482
  1087
    | solve_add_ac thy _ (_ $ x $ y) =
haftmann@22482
  1088
        if termless_agrp (y, x) then SOME ac2 else NONE
haftmann@22482
  1089
    | solve_add_ac thy _ _ = NONE
haftmann@22482
  1090
in
haftmann@22482
  1091
  val add_ac_proc = Simplifier.simproc @{theory}
haftmann@22482
  1092
    "add_ac_proc" ["x + y::'a::ab_semigroup_add"] solve_add_ac;
haftmann@22482
  1093
end;
haftmann@22482
  1094
haftmann@22482
  1095
val cancel_ss = HOL_basic_ss settermless termless_agrp
haftmann@22482
  1096
  addsimprocs [add_ac_proc] addsimps
nipkow@23085
  1097
  [@{thm add_0_left}, @{thm add_0_right}, @{thm diff_def},
haftmann@22482
  1098
   @{thm minus_add_distrib}, @{thm minus_minus}, @{thm minus_zero},
haftmann@22482
  1099
   @{thm right_minus}, @{thm left_minus}, @{thm add_minus_cancel},
haftmann@22482
  1100
   @{thm minus_add_cancel}];
haftmann@22482
  1101
  
haftmann@22548
  1102
val eq_reflection = @{thm eq_reflection};
haftmann@22482
  1103
  
haftmann@22548
  1104
val thy_ref = Theory.self_ref @{theory};
haftmann@22482
  1105
haftmann@22548
  1106
val T = TFree("'a", ["OrderedGroup.ab_group_add"]);
haftmann@22482
  1107
haftmann@22548
  1108
val eqI_rules = [@{thm less_eqI}, @{thm le_eqI}, @{thm eq_eqI}];
haftmann@22482
  1109
haftmann@22482
  1110
val dest_eqI = 
haftmann@22482
  1111
  fst o HOLogic.dest_bin "op =" HOLogic.boolT o HOLogic.dest_Trueprop o concl_of;
haftmann@22482
  1112
haftmann@22482
  1113
end);
haftmann@22482
  1114
*}
haftmann@22482
  1115
haftmann@22482
  1116
ML_setup {*
haftmann@22482
  1117
  Addsimprocs [ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv];
haftmann@22482
  1118
*}
paulson@17085
  1119
obua@14738
  1120
end