(*  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