(*  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