--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/OrderedGroup.thy Tue May 11 20:11:08 2004 +0200
@@ -0,0 +1,950 @@
+(* Title: HOL/Group.thy
+ ID: $Id$
+ Author: Gertrud Bauer and Markus Wenzel, TU Muenchen
+ Lawrence C Paulson, University of Cambridge
+ Revised and decoupled from Ring_and_Field.thy
+ by Steven Obua, TU Muenchen, in May 2004
+ License: GPL (GNU GENERAL PUBLIC LICENSE)
+*)
+
+header {* Ordered Groups *}
+
+theory OrderedGroup = Inductive + LOrder:
+
+text {*
+ The theory of partially ordered groups is taken from the books:
+ \begin{itemize}
+ \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979
+ \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
+ \end{itemize}
+ Most of the used notions can also be looked up in
+ \begin{itemize}
+ \item \emph{www.mathworld.com} by Eric Weisstein et. al.
+ \item \emph{Algebra I} by van der Waerden, Springer.
+ \end{itemize}
+*}
+
+subsection {* Semigroups, Groups *}
+
+axclass semigroup_add \<subseteq> plus
+ add_assoc: "(a + b) + c = a + (b + c)"
+
+axclass ab_semigroup_add \<subseteq> semigroup_add
+ add_commute: "a + b = b + a"
+
+lemma add_left_commute: "a + (b + c) = b + (a + (c::'a::ab_semigroup_add))"
+ by (rule mk_left_commute [of "op +", OF add_assoc add_commute])
+
+theorems add_ac = add_assoc add_commute add_left_commute
+
+axclass semigroup_mult \<subseteq> times
+ mult_assoc: "(a * b) * c = a * (b * c)"
+
+axclass ab_semigroup_mult \<subseteq> semigroup_mult
+ mult_commute: "a * b = b * a"
+
+lemma mult_left_commute: "a * (b * c) = b * (a * (c::'a::ab_semigroup_mult))"
+ by (rule mk_left_commute [of "op *", OF mult_assoc mult_commute])
+
+theorems mult_ac = mult_assoc mult_commute mult_left_commute
+
+axclass comm_monoid_add \<subseteq> zero, ab_semigroup_add
+ add_0[simp]: "0 + a = a"
+
+axclass monoid_mult \<subseteq> one, semigroup_mult
+ mult_1_left[simp]: "1 * a = a"
+ mult_1_right[simp]: "a * 1 = a"
+
+axclass comm_monoid_mult \<subseteq> one, ab_semigroup_mult
+ mult_1: "1 * a = a"
+
+instance comm_monoid_mult \<subseteq> monoid_mult
+by (intro_classes, insert mult_1, simp_all add: mult_commute, auto)
+
+axclass cancel_semigroup_add \<subseteq> semigroup_add
+ add_left_imp_eq: "a + b = a + c \<Longrightarrow> b = c"
+ add_right_imp_eq: "b + a = c + a \<Longrightarrow> b = c"
+
+axclass cancel_ab_semigroup_add \<subseteq> ab_semigroup_add
+ add_imp_eq: "a + b = a + c \<Longrightarrow> b = c"
+
+instance cancel_ab_semigroup_add \<subseteq> cancel_semigroup_add
+proof
+ {
+ fix a b c :: 'a
+ assume "a + b = a + c"
+ thus "b = c" by (rule add_imp_eq)
+ }
+ note f = this
+ fix a b c :: 'a
+ assume "b + a = c + a"
+ hence "a + b = a + c" by (simp only: add_commute)
+ thus "b = c" by (rule f)
+qed
+
+axclass ab_group_add \<subseteq> minus, comm_monoid_add
+ left_minus[simp]: " - a + a = 0"
+ diff_minus: "a - b = a + (-b)"
+
+instance ab_group_add \<subseteq> cancel_ab_semigroup_add
+proof
+ fix a b c :: 'a
+ assume "a + b = a + c"
+ hence "-a + a + b = -a + a + c" by (simp only: add_assoc)
+ thus "b = c" by simp
+qed
+
+lemma add_0_right [simp]: "a + 0 = (a::'a::comm_monoid_add)"
+proof -
+ have "a + 0 = 0 + a" by (simp only: add_commute)
+ also have "... = a" by simp
+ finally show ?thesis .
+qed
+
+lemma add_left_cancel [simp]:
+ "(a + b = a + c) = (b = (c::'a::cancel_semigroup_add))"
+by (blast dest: add_left_imp_eq)
+
+lemma add_right_cancel [simp]:
+ "(b + a = c + a) = (b = (c::'a::cancel_semigroup_add))"
+ by (blast dest: add_right_imp_eq)
+
+lemma right_minus [simp]: "a + -(a::'a::ab_group_add) = 0"
+proof -
+ have "a + -a = -a + a" by (simp add: add_ac)
+ also have "... = 0" by simp
+ finally show ?thesis .
+qed
+
+lemma right_minus_eq: "(a - b = 0) = (a = (b::'a::ab_group_add))"
+proof
+ have "a = a - b + b" by (simp add: diff_minus add_ac)
+ also assume "a - b = 0"
+ finally show "a = b" by simp
+next
+ assume "a = b"
+ thus "a - b = 0" by (simp add: diff_minus)
+qed
+
+lemma minus_minus [simp]: "- (- (a::'a::ab_group_add)) = a"
+proof (rule add_left_cancel [of "-a", THEN iffD1])
+ show "(-a + -(-a) = -a + a)"
+ by simp
+qed
+
+lemma equals_zero_I: "a+b = 0 ==> -a = (b::'a::ab_group_add)"
+apply (rule right_minus_eq [THEN iffD1, symmetric])
+apply (simp add: diff_minus add_commute)
+done
+
+lemma minus_zero [simp]: "- 0 = (0::'a::ab_group_add)"
+by (simp add: equals_zero_I)
+
+lemma diff_self [simp]: "a - (a::'a::ab_group_add) = 0"
+ by (simp add: diff_minus)
+
+lemma diff_0 [simp]: "(0::'a::ab_group_add) - a = -a"
+by (simp add: diff_minus)
+
+lemma diff_0_right [simp]: "a - (0::'a::ab_group_add) = a"
+by (simp add: diff_minus)
+
+lemma diff_minus_eq_add [simp]: "a - - b = a + (b::'a::ab_group_add)"
+by (simp add: diff_minus)
+
+lemma neg_equal_iff_equal [simp]: "(-a = -b) = (a = (b::'a::ab_group_add))"
+proof
+ assume "- a = - b"
+ hence "- (- a) = - (- b)"
+ by simp
+ thus "a=b" by simp
+next
+ assume "a=b"
+ thus "-a = -b" by simp
+qed
+
+lemma neg_equal_0_iff_equal [simp]: "(-a = 0) = (a = (0::'a::ab_group_add))"
+by (subst neg_equal_iff_equal [symmetric], simp)
+
+lemma neg_0_equal_iff_equal [simp]: "(0 = -a) = (0 = (a::'a::ab_group_add))"
+by (subst neg_equal_iff_equal [symmetric], simp)
+
+text{*The next two equations can make the simplifier loop!*}
+
+lemma equation_minus_iff: "(a = - b) = (b = - (a::'a::ab_group_add))"
+proof -
+ have "(- (-a) = - b) = (- a = b)" by (rule neg_equal_iff_equal)
+ thus ?thesis by (simp add: eq_commute)
+qed
+
+lemma minus_equation_iff: "(- a = b) = (- (b::'a::ab_group_add) = a)"
+proof -
+ have "(- a = - (-b)) = (a = -b)" by (rule neg_equal_iff_equal)
+ thus ?thesis by (simp add: eq_commute)
+qed
+
+lemma minus_add_distrib [simp]: "- (a + b) = -a + -(b::'a::ab_group_add)"
+apply (rule equals_zero_I)
+apply (simp add: add_ac)
+done
+
+lemma minus_diff_eq [simp]: "- (a - b) = b - (a::'a::ab_group_add)"
+by (simp add: diff_minus add_commute)
+
+subsection {* (Partially) Ordered Groups *}
+
+axclass pordered_ab_semigroup_add \<subseteq> order, ab_semigroup_add
+ add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b"
+
+axclass pordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add, cancel_ab_semigroup_add
+
+instance pordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add ..
+
+axclass pordered_ab_semigroup_add_imp_le \<subseteq> pordered_cancel_ab_semigroup_add
+ add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b"
+
+axclass pordered_ab_group_add \<subseteq> ab_group_add, pordered_ab_semigroup_add
+
+instance pordered_ab_group_add \<subseteq> pordered_ab_semigroup_add_imp_le
+proof
+ fix a b c :: 'a
+ assume "c + a \<le> c + b"
+ hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono)
+ hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add_assoc)
+ thus "a \<le> b" by simp
+qed
+
+axclass ordered_cancel_ab_semigroup_add \<subseteq> pordered_cancel_ab_semigroup_add, linorder
+
+instance ordered_cancel_ab_semigroup_add \<subseteq> pordered_ab_semigroup_add_imp_le
+proof
+ fix a b c :: 'a
+ assume le: "c + a <= c + b"
+ show "a <= b"
+ proof (rule ccontr)
+ assume w: "~ a \<le> b"
+ hence "b <= a" by (simp add: linorder_not_le)
+ hence le2: "c+b <= c+a" by (rule add_left_mono)
+ have "a = b"
+ apply (insert le)
+ apply (insert le2)
+ apply (drule order_antisym, simp_all)
+ done
+ with w show False
+ by (simp add: linorder_not_le [symmetric])
+ qed
+qed
+
+lemma add_right_mono: "a \<le> (b::'a::pordered_ab_semigroup_add) ==> a + c \<le> b + c"
+by (simp add: add_commute[of _ c] add_left_mono)
+
+text {* non-strict, in both arguments *}
+lemma add_mono:
+ "[|a \<le> b; c \<le> d|] ==> a + c \<le> b + (d::'a::pordered_ab_semigroup_add)"
+ apply (erule add_right_mono [THEN order_trans])
+ apply (simp add: add_commute add_left_mono)
+ done
+
+lemma add_strict_left_mono:
+ "a < b ==> c + a < c + (b::'a::pordered_cancel_ab_semigroup_add)"
+ by (simp add: order_less_le add_left_mono)
+
+lemma add_strict_right_mono:
+ "a < b ==> a + c < b + (c::'a::pordered_cancel_ab_semigroup_add)"
+ by (simp add: add_commute [of _ c] add_strict_left_mono)
+
+text{*Strict monotonicity in both arguments*}
+lemma add_strict_mono: "[|a<b; c<d|] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)"
+apply (erule add_strict_right_mono [THEN order_less_trans])
+apply (erule add_strict_left_mono)
+done
+
+lemma add_less_le_mono:
+ "[| a<b; c\<le>d |] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)"
+apply (erule add_strict_right_mono [THEN order_less_le_trans])
+apply (erule add_left_mono)
+done
+
+lemma add_le_less_mono:
+ "[| a\<le>b; c<d |] ==> a + c < b + (d::'a::pordered_cancel_ab_semigroup_add)"
+apply (erule add_right_mono [THEN order_le_less_trans])
+apply (erule add_strict_left_mono)
+done
+
+lemma add_less_imp_less_left:
+ assumes less: "c + a < c + b" shows "a < (b::'a::pordered_ab_semigroup_add_imp_le)"
+proof -
+ from less have le: "c + a <= c + b" by (simp add: order_le_less)
+ have "a <= b"
+ apply (insert le)
+ apply (drule add_le_imp_le_left)
+ by (insert le, drule add_le_imp_le_left, assumption)
+ moreover have "a \<noteq> b"
+ proof (rule ccontr)
+ assume "~(a \<noteq> b)"
+ then have "a = b" by simp
+ then have "c + a = c + b" by simp
+ with less show "False"by simp
+ qed
+ ultimately show "a < b" by (simp add: order_le_less)
+qed
+
+lemma add_less_imp_less_right:
+ "a + c < b + c ==> a < (b::'a::pordered_ab_semigroup_add_imp_le)"
+apply (rule add_less_imp_less_left [of c])
+apply (simp add: add_commute)
+done
+
+lemma add_less_cancel_left [simp]:
+ "(c+a < c+b) = (a < (b::'a::pordered_ab_semigroup_add_imp_le))"
+by (blast intro: add_less_imp_less_left add_strict_left_mono)
+
+lemma add_less_cancel_right [simp]:
+ "(a+c < b+c) = (a < (b::'a::pordered_ab_semigroup_add_imp_le))"
+by (blast intro: add_less_imp_less_right add_strict_right_mono)
+
+lemma add_le_cancel_left [simp]:
+ "(c+a \<le> c+b) = (a \<le> (b::'a::pordered_ab_semigroup_add_imp_le))"
+by (auto, drule add_le_imp_le_left, simp_all add: add_left_mono)
+
+lemma add_le_cancel_right [simp]:
+ "(a+c \<le> b+c) = (a \<le> (b::'a::pordered_ab_semigroup_add_imp_le))"
+by (simp add: add_commute[of a c] add_commute[of b c])
+
+lemma add_le_imp_le_right:
+ "a + c \<le> b + c ==> a \<le> (b::'a::pordered_ab_semigroup_add_imp_le)"
+by simp
+
+lemma add_increasing: "[|0\<le>a; b\<le>c|] ==> b \<le> a + (c::'a::{pordered_ab_semigroup_add_imp_le, comm_monoid_add})"
+by (insert add_mono [of 0 a b c], simp)
+
+subsection {* Ordering Rules for Unary Minus *}
+
+lemma le_imp_neg_le:
+ assumes "a \<le> (b::'a::{pordered_ab_semigroup_add_imp_le, ab_group_add})" shows "-b \<le> -a"
+proof -
+ have "-a+a \<le> -a+b"
+ by (rule add_left_mono)
+ hence "0 \<le> -a+b"
+ by simp
+ hence "0 + (-b) \<le> (-a + b) + (-b)"
+ by (rule add_right_mono)
+ thus ?thesis
+ by (simp add: add_assoc)
+qed
+
+lemma neg_le_iff_le [simp]: "(-b \<le> -a) = (a \<le> (b::'a::pordered_ab_group_add))"
+proof
+ assume "- b \<le> - a"
+ hence "- (- a) \<le> - (- b)"
+ by (rule le_imp_neg_le)
+ thus "a\<le>b" by simp
+next
+ assume "a\<le>b"
+ thus "-b \<le> -a" by (rule le_imp_neg_le)
+qed
+
+lemma neg_le_0_iff_le [simp]: "(-a \<le> 0) = (0 \<le> (a::'a::pordered_ab_group_add))"
+by (subst neg_le_iff_le [symmetric], simp)
+
+lemma neg_0_le_iff_le [simp]: "(0 \<le> -a) = (a \<le> (0::'a::pordered_ab_group_add))"
+by (subst neg_le_iff_le [symmetric], simp)
+
+lemma neg_less_iff_less [simp]: "(-b < -a) = (a < (b::'a::pordered_ab_group_add))"
+by (force simp add: order_less_le)
+
+lemma neg_less_0_iff_less [simp]: "(-a < 0) = (0 < (a::'a::pordered_ab_group_add))"
+by (subst neg_less_iff_less [symmetric], simp)
+
+lemma neg_0_less_iff_less [simp]: "(0 < -a) = (a < (0::'a::pordered_ab_group_add))"
+by (subst neg_less_iff_less [symmetric], simp)
+
+text{*The next several equations can make the simplifier loop!*}
+
+lemma less_minus_iff: "(a < - b) = (b < - (a::'a::pordered_ab_group_add))"
+proof -
+ have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
+ thus ?thesis by simp
+qed
+
+lemma minus_less_iff: "(- a < b) = (- b < (a::'a::pordered_ab_group_add))"
+proof -
+ have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
+ thus ?thesis by simp
+qed
+
+lemma le_minus_iff: "(a \<le> - b) = (b \<le> - (a::'a::pordered_ab_group_add))"
+proof -
+ have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff)
+ have "(- (- a) <= -b) = (b <= - a)"
+ apply (auto simp only: order_le_less)
+ apply (drule mm)
+ apply (simp_all)
+ apply (drule mm[simplified], assumption)
+ done
+ then show ?thesis by simp
+qed
+
+lemma minus_le_iff: "(- a \<le> b) = (- b \<le> (a::'a::pordered_ab_group_add))"
+by (auto simp add: order_le_less minus_less_iff)
+
+lemma add_diff_eq: "a + (b - c) = (a + b) - (c::'a::ab_group_add)"
+by (simp add: diff_minus add_ac)
+
+lemma diff_add_eq: "(a - b) + c = (a + c) - (b::'a::ab_group_add)"
+by (simp add: diff_minus add_ac)
+
+lemma diff_eq_eq: "(a-b = c) = (a = c + (b::'a::ab_group_add))"
+by (auto simp add: diff_minus add_assoc)
+
+lemma eq_diff_eq: "(a = c-b) = (a + (b::'a::ab_group_add) = c)"
+by (auto simp add: diff_minus add_assoc)
+
+lemma diff_diff_eq: "(a - b) - c = a - (b + (c::'a::ab_group_add))"
+by (simp add: diff_minus add_ac)
+
+lemma diff_diff_eq2: "a - (b - c) = (a + c) - (b::'a::ab_group_add)"
+by (simp add: diff_minus add_ac)
+
+lemma diff_add_cancel: "a - b + b = (a::'a::ab_group_add)"
+by (simp add: diff_minus add_ac)
+
+lemma add_diff_cancel: "a + b - b = (a::'a::ab_group_add)"
+by (simp add: diff_minus add_ac)
+
+text{*Further subtraction laws for ordered rings*}
+
+lemma less_iff_diff_less_0: "(a < b) = (a - b < (0::'a::pordered_ab_group_add))"
+proof -
+ have "(a < b) = (a + (- b) < b + (-b))"
+ by (simp only: add_less_cancel_right)
+ also have "... = (a - b < 0)" by (simp add: diff_minus)
+ finally show ?thesis .
+qed
+
+lemma diff_less_eq: "(a-b < c) = (a < c + (b::'a::pordered_ab_group_add))"
+apply (subst less_iff_diff_less_0)
+apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
+apply (simp add: diff_minus add_ac)
+done
+
+lemma less_diff_eq: "(a < c-b) = (a + (b::'a::pordered_ab_group_add) < c)"
+apply (subst less_iff_diff_less_0)
+apply (rule less_iff_diff_less_0 [of _ "c-b", THEN ssubst])
+apply (simp add: diff_minus add_ac)
+done
+
+lemma diff_le_eq: "(a-b \<le> c) = (a \<le> c + (b::'a::pordered_ab_group_add))"
+by (auto simp add: order_le_less diff_less_eq diff_add_cancel add_diff_cancel)
+
+lemma le_diff_eq: "(a \<le> c-b) = (a + (b::'a::pordered_ab_group_add) \<le> c)"
+by (auto simp add: order_le_less less_diff_eq diff_add_cancel add_diff_cancel)
+
+text{*This list of rewrites simplifies (in)equalities by bringing subtractions
+ to the top and then moving negative terms to the other side.
+ Use with @{text add_ac}*}
+lemmas compare_rls =
+ diff_minus [symmetric]
+ add_diff_eq diff_add_eq diff_diff_eq diff_diff_eq2
+ diff_less_eq less_diff_eq diff_le_eq le_diff_eq
+ diff_eq_eq eq_diff_eq
+
+
+subsection{*Lemmas for the @{text cancel_numerals} simproc*}
+
+lemma eq_iff_diff_eq_0: "(a = b) = (a-b = (0::'a::ab_group_add))"
+by (simp add: compare_rls)
+
+lemma le_iff_diff_le_0: "(a \<le> b) = (a-b \<le> (0::'a::pordered_ab_group_add))"
+by (simp add: compare_rls)
+
+subsection {* Lattice Ordered (Abelian) Groups *}
+
+axclass lordered_ab_group_meet < pordered_ab_group_add, meet_semilorder
+
+axclass lordered_ab_group_join < pordered_ab_group_add, join_semilorder
+
+lemma add_meet_distrib_left: "a + (meet b c) = meet (a + b) (a + (c::'a::{pordered_ab_group_add, meet_semilorder}))"
+apply (rule order_antisym)
+apply (rule meet_imp_le, simp_all add: meet_join_le)
+apply (rule add_le_imp_le_left [of "-a"])
+apply (simp only: add_assoc[symmetric], simp)
+apply (rule meet_imp_le)
+apply (rule add_le_imp_le_left[of "a"], simp only: add_assoc[symmetric], simp add: meet_join_le)+
+done
+
+lemma add_join_distrib_left: "a + (join b c) = join (a + b) (a+ (c::'a::{pordered_ab_group_add, join_semilorder}))"
+apply (rule order_antisym)
+apply (rule add_le_imp_le_left [of "-a"])
+apply (simp only: add_assoc[symmetric], simp)
+apply (rule join_imp_le)
+apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp add: meet_join_le)+
+apply (rule join_imp_le)
+apply (simp_all add: meet_join_le)
+done
+
+lemma is_join_neg_meet: "is_join (% (a::'a::{pordered_ab_group_add, meet_semilorder}) b. - (meet (-a) (-b)))"
+apply (auto simp add: is_join_def)
+apply (rule_tac c="meet (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_meet_distrib_left meet_join_le)
+apply (rule_tac c="meet (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_meet_distrib_left meet_join_le)
+apply (subst neg_le_iff_le[symmetric])
+apply (simp add: meet_imp_le)
+done
+
+lemma is_meet_neg_join: "is_meet (% (a::'a::{pordered_ab_group_add, join_semilorder}) b. - (join (-a) (-b)))"
+apply (auto simp add: is_meet_def)
+apply (rule_tac c="join (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_join_distrib_left meet_join_le)
+apply (rule_tac c="join (-a) (-b)" in add_le_imp_le_right, simp, simp add: add_join_distrib_left meet_join_le)
+apply (subst neg_le_iff_le[symmetric])
+apply (simp add: join_imp_le)
+done
+
+axclass lordered_ab_group \<subseteq> pordered_ab_group_add, lorder
+
+instance lordered_ab_group_meet \<subseteq> lordered_ab_group
+proof
+ show "? j. is_join (j::'a\<Rightarrow>'a\<Rightarrow>('a::lordered_ab_group_meet))" by (blast intro: is_join_neg_meet)
+qed
+
+instance lordered_ab_group_join \<subseteq> lordered_ab_group
+proof
+ show "? m. is_meet (m::'a\<Rightarrow>'a\<Rightarrow>('a::lordered_ab_group_join))" by (blast intro: is_meet_neg_join)
+qed
+
+lemma add_join_distrib_right: "(join a b) + (c::'a::lordered_ab_group) = join (a+c) (b+c)"
+proof -
+ have "c + (join a b) = join (c+a) (c+b)" by (simp add: add_join_distrib_left)
+ thus ?thesis by (simp add: add_commute)
+qed
+
+lemma add_meet_distrib_right: "(meet a b) + (c::'a::lordered_ab_group) = meet (a+c) (b+c)"
+proof -
+ have "c + (meet a b) = meet (c+a) (c+b)" by (simp add: add_meet_distrib_left)
+ thus ?thesis by (simp add: add_commute)
+qed
+
+lemmas add_meet_join_distribs = add_meet_distrib_right add_meet_distrib_left add_join_distrib_right add_join_distrib_left
+
+lemma join_eq_neg_meet: "join a (b::'a::lordered_ab_group) = - meet (-a) (-b)"
+by (simp add: is_join_unique[OF is_join_join is_join_neg_meet])
+
+lemma meet_eq_neg_join: "meet a (b::'a::lordered_ab_group) = - join (-a) (-b)"
+by (simp add: is_meet_unique[OF is_meet_meet is_meet_neg_join])
+
+lemma add_eq_meet_join: "a + b = (join a b) + (meet a (b::'a::lordered_ab_group))"
+proof -
+ have "0 = - meet 0 (a-b) + meet (a-b) 0" by (simp add: meet_comm)
+ hence "0 = join 0 (b-a) + meet (a-b) 0" by (simp add: meet_eq_neg_join)
+ hence "0 = (-a + join a b) + (meet a b + (-b))"
+ apply (simp add: add_join_distrib_left add_meet_distrib_right)
+ by (simp add: diff_minus add_commute)
+ thus ?thesis
+ apply (simp add: compare_rls)
+ apply (subst add_left_cancel[symmetric, of "a+b" "join a b + meet a b" "-a"])
+ apply (simp only: add_assoc, simp add: add_assoc[symmetric])
+ done
+qed
+
+subsection {* Positive Part, Negative Part, Absolute Value *}
+
+constdefs
+ pprt :: "'a \<Rightarrow> ('a::lordered_ab_group)"
+ "pprt x == join x 0"
+ nprt :: "'a \<Rightarrow> ('a::lordered_ab_group)"
+ "nprt x == meet x 0"
+
+lemma prts: "a = pprt a + nprt a"
+by (simp add: pprt_def nprt_def add_eq_meet_join[symmetric])
+
+lemma zero_le_pprt[simp]: "0 \<le> pprt a"
+by (simp add: pprt_def meet_join_le)
+
+lemma nprt_le_zero[simp]: "nprt a \<le> 0"
+by (simp add: nprt_def meet_join_le)
+
+lemma le_eq_neg: "(a \<le> -b) = (a + b \<le> (0::_::lordered_ab_group))" (is "?l = ?r")
+proof -
+ have a: "?l \<longrightarrow> ?r"
+ apply (auto)
+ apply (rule add_le_imp_le_right[of _ "-b" _])
+ apply (simp add: add_assoc)
+ done
+ have b: "?r \<longrightarrow> ?l"
+ apply (auto)
+ apply (rule add_le_imp_le_right[of _ "b" _])
+ apply (simp)
+ done
+ from a b show ?thesis by blast
+qed
+
+lemma join_0_imp_0: "join a (-a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)"
+proof -
+ {
+ fix a::'a
+ assume hyp: "join a (-a) = 0"
+ hence "join a (-a) + a = a" by (simp)
+ hence "join (a+a) 0 = a" by (simp add: add_join_distrib_right)
+ hence "join (a+a) 0 <= a" by (simp)
+ hence "0 <= a" by (blast intro: order_trans meet_join_le)
+ }
+ note p = this
+ thm p
+ assume hyp:"join a (-a) = 0"
+ hence hyp2:"join (-a) (-(-a)) = 0" by (simp add: join_comm)
+ from p[OF hyp] p[OF hyp2] show "a = 0" by simp
+qed
+
+lemma meet_0_imp_0: "meet a (-a) = 0 \<Longrightarrow> a = (0::'a::lordered_ab_group)"
+apply (simp add: meet_eq_neg_join)
+apply (simp add: join_comm)
+apply (subst join_0_imp_0)
+by auto
+
+lemma join_0_eq_0[simp]: "(join a (-a) = 0) = (a = (0::'a::lordered_ab_group))"
+by (auto, erule join_0_imp_0)
+
+lemma meet_0_eq_0[simp]: "(meet a (-a) = 0) = (a = (0::'a::lordered_ab_group))"
+by (auto, erule meet_0_imp_0)
+
+lemma zero_le_double_add_iff_zero_le_single_add[simp]: "(0 \<le> a + a) = (0 \<le> (a::'a::lordered_ab_group))"
+proof
+ assume "0 <= a + a"
+ hence a:"meet (a+a) 0 = 0" by (simp add: le_def_meet meet_comm)
+ have "(meet a 0)+(meet a 0) = meet (meet (a+a) 0) a" (is "?l=_") by (simp add: add_meet_join_distribs meet_aci)
+ hence "?l = 0 + meet a 0" by (simp add: a, simp add: meet_comm)
+ hence "meet a 0 = 0" by (simp only: add_right_cancel)
+ then show "0 <= a" by (simp add: le_def_meet meet_comm)
+next
+ assume a: "0 <= a"
+ show "0 <= a + a" by (simp add: add_mono[OF a a, simplified])
+qed
+
+lemma double_add_le_zero_iff_single_add_le_zero[simp]: "(a + a <= 0) = ((a::'a::lordered_ab_group) <= 0)"
+proof -
+ have "(a + a <= 0) = (0 <= -(a+a))" by (subst le_minus_iff, simp)
+ moreover have "\<dots> = (a <= 0)" by (simp add: zero_le_double_add_iff_zero_le_single_add)
+ ultimately show ?thesis by blast
+qed
+
+lemma double_add_less_zero_iff_single_less_zero[simp]: "(a+a<0) = ((a::'a::{pordered_ab_group_add,linorder}) < 0)" (is ?s)
+proof cases
+ assume a: "a < 0"
+ thus ?s by (simp add: add_strict_mono[OF a a, simplified])
+next
+ assume "~(a < 0)"
+ hence a:"0 <= a" by (simp)
+ hence "0 <= a+a" by (simp add: add_mono[OF a a, simplified])
+ hence "~(a+a < 0)" by simp
+ with a show ?thesis by simp
+qed
+
+axclass lordered_ab_group_abs \<subseteq> lordered_ab_group
+ abs_lattice: "abs x = join x (-x)"
+
+lemma abs_zero[simp]: "abs 0 = (0::'a::lordered_ab_group_abs)"
+by (simp add: abs_lattice)
+
+lemma abs_eq_0[simp]: "(abs a = 0) = (a = (0::'a::lordered_ab_group_abs))"
+by (simp add: abs_lattice)
+
+lemma abs_0_eq[simp]: "(0 = abs a) = (a = (0::'a::lordered_ab_group_abs))"
+proof -
+ have "(0 = abs a) = (abs a = 0)" by (simp only: eq_ac)
+ thus ?thesis by simp
+qed
+
+lemma neg_meet_eq_join[simp]: "- meet a (b::_::lordered_ab_group) = join (-a) (-b)"
+by (simp add: meet_eq_neg_join)
+
+lemma neg_join_eq_meet[simp]: "- join a (b::_::lordered_ab_group) = meet (-a) (-b)"
+by (simp del: neg_meet_eq_join add: join_eq_neg_meet)
+
+lemma join_eq_if: "join a (-a) = (if a < 0 then -a else (a::'a::{lordered_ab_group, linorder}))"
+proof -
+ note b = add_le_cancel_right[of a a "-a",symmetric,simplified]
+ have c: "a + a = 0 \<Longrightarrow> -a = a" by (rule add_right_imp_eq[of _ a], simp)
+ show ?thesis
+ apply (auto simp add: join_max max_def b linorder_not_less)
+ apply (drule order_antisym, auto)
+ done
+qed
+
+lemma abs_if_lattice: "\<bar>a\<bar> = (if a < 0 then -a else (a::'a::{lordered_ab_group_abs, linorder}))"
+proof -
+ show ?thesis by (simp add: abs_lattice join_eq_if)
+qed
+
+lemma abs_ge_zero[simp]: "0 \<le> abs (a::'a::lordered_ab_group_abs)"
+proof -
+ have a:"a <= abs a" and b:"-a <= abs a" by (auto simp add: abs_lattice meet_join_le)
+ show ?thesis by (rule add_mono[OF a b, simplified])
+qed
+
+lemma abs_le_zero_iff [simp]: "(abs a \<le> (0::'a::lordered_ab_group_abs)) = (a = 0)"
+proof
+ assume "abs a <= 0"
+ hence "abs a = 0" by (auto dest: order_antisym)
+ thus "a = 0" by simp
+next
+ assume "a = 0"
+ thus "abs a <= 0" by simp
+qed
+
+lemma zero_less_abs_iff [simp]: "(0 < abs a) = (a \<noteq> (0::'a::lordered_ab_group_abs))"
+by (simp add: order_less_le)
+
+lemma abs_not_less_zero [simp]: "~ abs a < (0::'a::lordered_ab_group_abs)"
+proof -
+ have a:"!! x (y::_::order). x <= y \<Longrightarrow> ~(y < x)" by auto
+ show ?thesis by (simp add: a)
+qed
+
+lemma abs_ge_self: "a \<le> abs (a::'a::lordered_ab_group_abs)"
+by (simp add: abs_lattice meet_join_le)
+
+lemma abs_ge_minus_self: "-a \<le> abs (a::'a::lordered_ab_group_abs)"
+by (simp add: abs_lattice meet_join_le)
+
+lemma le_imp_join_eq: "a \<le> b \<Longrightarrow> join a b = b"
+by (simp add: le_def_join)
+
+lemma ge_imp_join_eq: "b \<le> a \<Longrightarrow> join a b = a"
+by (simp add: le_def_join join_aci)
+
+lemma le_imp_meet_eq: "a \<le> b \<Longrightarrow> meet a b = a"
+by (simp add: le_def_meet)
+
+lemma ge_imp_meet_eq: "b \<le> a \<Longrightarrow> meet a b = b"
+by (simp add: le_def_meet meet_aci)
+
+lemma abs_prts: "abs (a::_::lordered_ab_group_abs) = pprt a - nprt a"
+apply (simp add: pprt_def nprt_def diff_minus)
+apply (simp add: add_meet_join_distribs join_aci abs_lattice[symmetric])
+apply (subst le_imp_join_eq, auto)
+done
+
+lemma abs_minus_cancel [simp]: "abs (-a) = abs(a::'a::lordered_ab_group_abs)"
+by (simp add: abs_lattice join_comm)
+
+lemma abs_idempotent [simp]: "abs (abs a) = abs (a::'a::lordered_ab_group_abs)"
+apply (simp add: abs_lattice[of "abs a"])
+apply (subst ge_imp_join_eq)
+apply (rule order_trans[of _ 0])
+by auto
+
+lemma zero_le_iff_zero_nprt: "(0 \<le> a) = (nprt a = 0)"
+by (simp add: le_def_meet nprt_def meet_comm)
+
+lemma le_zero_iff_zero_pprt: "(a \<le> 0) = (pprt a = 0)"
+by (simp add: le_def_join pprt_def join_comm)
+
+lemma le_zero_iff_pprt_id: "(0 \<le> a) = (pprt a = a)"
+by (simp add: le_def_join pprt_def join_comm)
+
+lemma zero_le_iff_nprt_id: "(a \<le> 0) = (nprt a = a)"
+by (simp add: le_def_meet nprt_def meet_comm)
+
+lemma iff2imp: "(A=B) \<Longrightarrow> (A \<Longrightarrow> B)"
+by (simp)
+
+lemma imp_abs_id: "0 \<le> a \<Longrightarrow> abs a = (a::'a::lordered_ab_group_abs)"
+by (simp add: iff2imp[OF zero_le_iff_zero_nprt] iff2imp[OF le_zero_iff_pprt_id] abs_prts)
+
+lemma imp_abs_neg_id: "a \<le> 0 \<Longrightarrow> abs a = -(a::'a::lordered_ab_group_abs)"
+by (simp add: iff2imp[OF le_zero_iff_zero_pprt] iff2imp[OF zero_le_iff_nprt_id] abs_prts)
+
+lemma abs_leI: "[|a \<le> b; -a \<le> b|] ==> abs a \<le> (b::'a::lordered_ab_group_abs)"
+by (simp add: abs_lattice join_imp_le)
+
+lemma le_minus_self_iff: "(a \<le> -a) = (a \<le> (0::'a::lordered_ab_group))"
+proof -
+ from add_le_cancel_left[of "-a" "a+a" "0"] have "(a <= -a) = (a+a <= 0)"
+ by (simp add: add_assoc[symmetric])
+ thus ?thesis by simp
+qed
+
+lemma minus_le_self_iff: "(-a \<le> a) = (0 \<le> (a::'a::lordered_ab_group))"
+proof -
+ from add_le_cancel_left[of "-a" "0" "a+a"] have "(-a <= a) = (0 <= a+a)"
+ by (simp add: add_assoc[symmetric])
+ thus ?thesis by simp
+qed
+
+lemma abs_le_D1: "abs a \<le> b ==> a \<le> (b::'a::lordered_ab_group_abs)"
+by (insert abs_ge_self, blast intro: order_trans)
+
+lemma abs_le_D2: "abs a \<le> b ==> -a \<le> (b::'a::lordered_ab_group_abs)"
+by (insert abs_le_D1 [of "-a"], simp)
+
+lemma abs_le_iff: "(abs a \<le> b) = (a \<le> b & -a \<le> (b::'a::lordered_ab_group_abs))"
+by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)
+
+lemma abs_triangle_ineq: "abs (a+b) \<le> abs a + abs (b::'a::lordered_ab_group_abs)"
+proof -
+ have g:"abs a + abs b = join (a+b) (join (-a-b) (join (-a+b) (a + (-b))))" (is "_=join ?m ?n")
+ apply (simp add: abs_lattice add_meet_join_distribs join_aci)
+ by (simp only: diff_minus)
+ have a:"a+b <= join ?m ?n" by (simp add: meet_join_le)
+ have b:"-a-b <= ?n" by (simp add: meet_join_le)
+ have c:"?n <= join ?m ?n" by (simp add: meet_join_le)
+ from b c have d: "-a-b <= join ?m ?n" by simp
+ have e:"-a-b = -(a+b)" by (simp add: diff_minus)
+ from a d e have "abs(a+b) <= join ?m ?n"
+ by (drule_tac abs_leI, auto)
+ with g[symmetric] show ?thesis by simp
+qed
+
+lemma abs_diff_triangle_ineq:
+ "\<bar>(a::'a::lordered_ab_group_abs) + b - (c+d)\<bar> \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>"
+proof -
+ have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: diff_minus add_ac)
+ also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq)
+ finally show ?thesis .
+qed
+
+ML {*
+val add_zero_left = thm"add_0";
+val add_zero_right = thm"add_0_right";
+*}
+
+ML {*
+val add_assoc = thm "add_assoc";
+val add_commute = thm "add_commute";
+val add_left_commute = thm "add_left_commute";
+val add_ac = thms "add_ac";
+val mult_assoc = thm "mult_assoc";
+val mult_commute = thm "mult_commute";
+val mult_left_commute = thm "mult_left_commute";
+val mult_ac = thms "mult_ac";
+val add_0 = thm "add_0";
+val mult_1_left = thm "mult_1_left";
+val mult_1_right = thm "mult_1_right";
+val mult_1 = thm "mult_1";
+val add_left_imp_eq = thm "add_left_imp_eq";
+val add_right_imp_eq = thm "add_right_imp_eq";
+val add_imp_eq = thm "add_imp_eq";
+val left_minus = thm "left_minus";
+val diff_minus = thm "diff_minus";
+val add_0_right = thm "add_0_right";
+val add_left_cancel = thm "add_left_cancel";
+val add_right_cancel = thm "add_right_cancel";
+val right_minus = thm "right_minus";
+val right_minus_eq = thm "right_minus_eq";
+val minus_minus = thm "minus_minus";
+val equals_zero_I = thm "equals_zero_I";
+val minus_zero = thm "minus_zero";
+val diff_self = thm "diff_self";
+val diff_0 = thm "diff_0";
+val diff_0_right = thm "diff_0_right";
+val diff_minus_eq_add = thm "diff_minus_eq_add";
+val neg_equal_iff_equal = thm "neg_equal_iff_equal";
+val neg_equal_0_iff_equal = thm "neg_equal_0_iff_equal";
+val neg_0_equal_iff_equal = thm "neg_0_equal_iff_equal";
+val equation_minus_iff = thm "equation_minus_iff";
+val minus_equation_iff = thm "minus_equation_iff";
+val minus_add_distrib = thm "minus_add_distrib";
+val minus_diff_eq = thm "minus_diff_eq";
+val add_left_mono = thm "add_left_mono";
+val add_le_imp_le_left = thm "add_le_imp_le_left";
+val add_right_mono = thm "add_right_mono";
+val add_mono = thm "add_mono";
+val add_strict_left_mono = thm "add_strict_left_mono";
+val add_strict_right_mono = thm "add_strict_right_mono";
+val add_strict_mono = thm "add_strict_mono";
+val add_less_le_mono = thm "add_less_le_mono";
+val add_le_less_mono = thm "add_le_less_mono";
+val add_less_imp_less_left = thm "add_less_imp_less_left";
+val add_less_imp_less_right = thm "add_less_imp_less_right";
+val add_less_cancel_left = thm "add_less_cancel_left";
+val add_less_cancel_right = thm "add_less_cancel_right";
+val add_le_cancel_left = thm "add_le_cancel_left";
+val add_le_cancel_right = thm "add_le_cancel_right";
+val add_le_imp_le_right = thm "add_le_imp_le_right";
+val add_increasing = thm "add_increasing";
+val le_imp_neg_le = thm "le_imp_neg_le";
+val neg_le_iff_le = thm "neg_le_iff_le";
+val neg_le_0_iff_le = thm "neg_le_0_iff_le";
+val neg_0_le_iff_le = thm "neg_0_le_iff_le";
+val neg_less_iff_less = thm "neg_less_iff_less";
+val neg_less_0_iff_less = thm "neg_less_0_iff_less";
+val neg_0_less_iff_less = thm "neg_0_less_iff_less";
+val less_minus_iff = thm "less_minus_iff";
+val minus_less_iff = thm "minus_less_iff";
+val le_minus_iff = thm "le_minus_iff";
+val minus_le_iff = thm "minus_le_iff";
+val add_diff_eq = thm "add_diff_eq";
+val diff_add_eq = thm "diff_add_eq";
+val diff_eq_eq = thm "diff_eq_eq";
+val eq_diff_eq = thm "eq_diff_eq";
+val diff_diff_eq = thm "diff_diff_eq";
+val diff_diff_eq2 = thm "diff_diff_eq2";
+val diff_add_cancel = thm "diff_add_cancel";
+val add_diff_cancel = thm "add_diff_cancel";
+val less_iff_diff_less_0 = thm "less_iff_diff_less_0";
+val diff_less_eq = thm "diff_less_eq";
+val less_diff_eq = thm "less_diff_eq";
+val diff_le_eq = thm "diff_le_eq";
+val le_diff_eq = thm "le_diff_eq";
+val compare_rls = thms "compare_rls";
+val eq_iff_diff_eq_0 = thm "eq_iff_diff_eq_0";
+val le_iff_diff_le_0 = thm "le_iff_diff_le_0";
+val add_meet_distrib_left = thm "add_meet_distrib_left";
+val add_join_distrib_left = thm "add_join_distrib_left";
+val is_join_neg_meet = thm "is_join_neg_meet";
+val is_meet_neg_join = thm "is_meet_neg_join";
+val add_join_distrib_right = thm "add_join_distrib_right";
+val add_meet_distrib_right = thm "add_meet_distrib_right";
+val add_meet_join_distribs = thms "add_meet_join_distribs";
+val join_eq_neg_meet = thm "join_eq_neg_meet";
+val meet_eq_neg_join = thm "meet_eq_neg_join";
+val add_eq_meet_join = thm "add_eq_meet_join";
+val prts = thm "prts";
+val zero_le_pprt = thm "zero_le_pprt";
+val nprt_le_zero = thm "nprt_le_zero";
+val le_eq_neg = thm "le_eq_neg";
+val join_0_imp_0 = thm "join_0_imp_0";
+val meet_0_imp_0 = thm "meet_0_imp_0";
+val join_0_eq_0 = thm "join_0_eq_0";
+val meet_0_eq_0 = thm "meet_0_eq_0";
+val zero_le_double_add_iff_zero_le_single_add = thm "zero_le_double_add_iff_zero_le_single_add";
+val double_add_le_zero_iff_single_add_le_zero = thm "double_add_le_zero_iff_single_add_le_zero";
+val double_add_less_zero_iff_single_less_zero = thm "double_add_less_zero_iff_single_less_zero";
+val abs_lattice = thm "abs_lattice";
+val abs_zero = thm "abs_zero";
+val abs_eq_0 = thm "abs_eq_0";
+val abs_0_eq = thm "abs_0_eq";
+val neg_meet_eq_join = thm "neg_meet_eq_join";
+val neg_join_eq_meet = thm "neg_join_eq_meet";
+val join_eq_if = thm "join_eq_if";
+val abs_if_lattice = thm "abs_if_lattice";
+val abs_ge_zero = thm "abs_ge_zero";
+val abs_le_zero_iff = thm "abs_le_zero_iff";
+val zero_less_abs_iff = thm "zero_less_abs_iff";
+val abs_not_less_zero = thm "abs_not_less_zero";
+val abs_ge_self = thm "abs_ge_self";
+val abs_ge_minus_self = thm "abs_ge_minus_self";
+val le_imp_join_eq = thm "le_imp_join_eq";
+val ge_imp_join_eq = thm "ge_imp_join_eq";
+val le_imp_meet_eq = thm "le_imp_meet_eq";
+val ge_imp_meet_eq = thm "ge_imp_meet_eq";
+val abs_prts = thm "abs_prts";
+val abs_minus_cancel = thm "abs_minus_cancel";
+val abs_idempotent = thm "abs_idempotent";
+val zero_le_iff_zero_nprt = thm "zero_le_iff_zero_nprt";
+val le_zero_iff_zero_pprt = thm "le_zero_iff_zero_pprt";
+val le_zero_iff_pprt_id = thm "le_zero_iff_pprt_id";
+val zero_le_iff_nprt_id = thm "zero_le_iff_nprt_id";
+val iff2imp = thm "iff2imp";
+val imp_abs_id = thm "imp_abs_id";
+val imp_abs_neg_id = thm "imp_abs_neg_id";
+val abs_leI = thm "abs_leI";
+val le_minus_self_iff = thm "le_minus_self_iff";
+val minus_le_self_iff = thm "minus_le_self_iff";
+val abs_le_D1 = thm "abs_le_D1";
+val abs_le_D2 = thm "abs_le_D2";
+val abs_le_iff = thm "abs_le_iff";
+val abs_triangle_ineq = thm "abs_triangle_ineq";
+val abs_diff_triangle_ineq = thm "abs_diff_triangle_ineq";
+*}
+
+end