(*  Author:     Steven Obua, TU Muenchen *)

 section \<open>Various algebraic structures combined with a lattice\<close>

 theory Lattice_Algebras

 imports Complex_Main

 begin

 class semilattice_inf_ab_group_add = ordered_ab_group_add + semilattice_inf

 begin

 lemma add_inf_distrib_left: "a + inf b c = inf (a + b) (a + c)"

   apply (rule antisym)

   apply (simp_all add: le_infI)

   apply (rule add_le_imp_le_left [of "uminus a"])

   apply (simp only: add.assoc [symmetric], simp add: diff_le_eq add.commute)

   done

 lemma add_inf_distrib_right: "inf a b + c = inf (a + c) (b + c)"

 proof -

   have "c + inf a b = inf (c + a) (c + b)"

     by (simp add: add_inf_distrib_left)

   then show ?thesis

     by (simp add: add.commute)

 qed

 end

 class semilattice_sup_ab_group_add = ordered_ab_group_add + semilattice_sup

 begin

 lemma add_sup_distrib_left: "a + sup b c = sup (a + b) (a + c)"

   apply (rule antisym)

   apply (rule add_le_imp_le_left [of "uminus a"])

   apply (simp only: add.assoc [symmetric], simp)

   apply (simp add: le_diff_eq add.commute)

   apply (rule le_supI)

   apply (rule add_le_imp_le_left [of "a"], simp only: add.assoc[symmetric], simp)+

   done

 lemma add_sup_distrib_right: "sup a b + c = sup (a + c) (b + c)"

 proof -

   have "c + sup a b = sup (c+a) (c+b)"

     by (simp add: add_sup_distrib_left)

   then show ?thesis

     by (simp add: add.commute)

 qed

 end

 class lattice_ab_group_add = ordered_ab_group_add + lattice

 begin

 subclass semilattice_inf_ab_group_add ..

 subclass semilattice_sup_ab_group_add ..

 lemmas add_sup_inf_distribs =

   add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left

 lemma inf_eq_neg_sup: "inf a b = - sup (- a) (- b)"

 proof (rule inf_unique)

   fix a b c :: 'a

   show "- sup (- a) (- b) \<le> a"

     by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])

       (simp, simp add: add_sup_distrib_left)

   show "- sup (-a) (-b) \<le> b"

     by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])

       (simp, simp add: add_sup_distrib_left)

   assume "a \<le> b" "a \<le> c"

   then show "a \<le> - sup (-b) (-c)"

     by (subst neg_le_iff_le [symmetric]) (simp add: le_supI)

 qed

 lemma sup_eq_neg_inf: "sup a b = - inf (- a) (- b)"

 proof (rule sup_unique)

   fix a b c :: 'a

   show "a \<le> - inf (- a) (- b)"

     by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])

       (simp, simp add: add_inf_distrib_left)

   show "b \<le> - inf (- a) (- b)"

     by (rule add_le_imp_le_right [of _ "inf (uminus a) (uminus b)"])

       (simp, simp add: add_inf_distrib_left)

   assume "a \<le> c" "b \<le> c"

   then show "- inf (- a) (- b) \<le> c"

     by (subst neg_le_iff_le [symmetric]) (simp add: le_infI)

 qed

 lemma neg_inf_eq_sup: "- inf a b = sup (- a) (- b)"

   by (simp add: inf_eq_neg_sup)

 lemma diff_inf_eq_sup: "a - inf b c = a + sup (- b) (- c)"

   using neg_inf_eq_sup [of b c, symmetric] by simp

 lemma neg_sup_eq_inf: "- sup a b = inf (- a) (- b)"

   by (simp add: sup_eq_neg_inf)

 lemma diff_sup_eq_inf: "a - sup b c = a + inf (- b) (- c)"

   using neg_sup_eq_inf [of b c, symmetric] by simp

 lemma add_eq_inf_sup: "a + b = sup a b + inf a b"

 proof -

   have "0 = - inf 0 (a - b) + inf (a - b) 0"

     by (simp add: inf_commute)

   then have "0 = sup 0 (b - a) + inf (a - b) 0"

     by (simp add: inf_eq_neg_sup)

   then have "0 = (- a + sup a b) + (inf a b + (- b))"

     by (simp only: add_sup_distrib_left add_inf_distrib_right) simp

   then show ?thesis

     by (simp add: algebra_simps)

 qed

 subsection \<open>Positive Part, Negative Part, Absolute Value\<close>

 definition nprt :: "'a \<Rightarrow> 'a"

   where "nprt x = inf x 0"

 definition pprt :: "'a \<Rightarrow> 'a"

   where "pprt x = sup x 0"

 lemma pprt_neg: "pprt (- x) = - nprt x"

 proof -

   have "sup (- x) 0 = sup (- x) (- 0)"

     unfolding minus_zero ..

   also have "\<dots> = - inf x 0"

     unfolding neg_inf_eq_sup ..

   finally have "sup (- x) 0 = - inf x 0" .

   then show ?thesis

     unfolding pprt_def nprt_def .

 qed

 lemma nprt_neg: "nprt (- x) = - pprt x"

 proof -

   from pprt_neg have "pprt (- (- x)) = - nprt (- x)" .

   then have "pprt x = - nprt (- x)" by simp

   then show ?thesis by simp

 qed

 lemma prts: "a = pprt a + nprt a"

   by (simp add: pprt_def nprt_def add_eq_inf_sup[symmetric])

 lemma zero_le_pprt[simp]: "0 \<le> pprt a"

   by (simp add: pprt_def)

 lemma nprt_le_zero[simp]: "nprt a \<le> 0"

   by (simp add: nprt_def)

 lemma le_eq_neg: "a \<le> - b \<longleftrightarrow> a + b \<le> 0"

   (is "?l = ?r")

 proof

   assume ?l

   then show ?r

     apply -

     apply (rule add_le_imp_le_right[of _ "uminus b" _])

     apply (simp add: add.assoc)

     done

 next

   assume ?r

   then show ?l

     apply -

     apply (rule add_le_imp_le_right[of _ "b" _])

     apply simp

     done

 qed

 lemma pprt_0[simp]: "pprt 0 = 0" by (simp add: pprt_def)

 lemma nprt_0[simp]: "nprt 0 = 0" by (simp add: nprt_def)

 lemma pprt_eq_id [simp, no_atp]: "0 \<le> x \<Longrightarrow> pprt x = x"

   by (simp add: pprt_def sup_absorb1)

 lemma nprt_eq_id [simp, no_atp]: "x \<le> 0 \<Longrightarrow> nprt x = x"

   by (simp add: nprt_def inf_absorb1)

 lemma pprt_eq_0 [simp, no_atp]: "x \<le> 0 \<Longrightarrow> pprt x = 0"

   by (simp add: pprt_def sup_absorb2)

 lemma nprt_eq_0 [simp, no_atp]: "0 \<le> x \<Longrightarrow> nprt x = 0"

   by (simp add: nprt_def inf_absorb2)

 lemma sup_0_imp_0:

   assumes "sup a (- a) = 0"

   shows "a = 0"

 proof -

   have p: "0 \<le> a" if "sup a (- a) = 0" for a :: 'a

   proof -

     from that have "sup a (- a) + a = a"

       by simp

     then have "sup (a + a) 0 = a"

       by (simp add: add_sup_distrib_right)

     then have "sup (a + a) 0 \<le> a"

       by simp

     then show ?thesis

       by (blast intro: order_trans inf_sup_ord)

   qed

   from assms have **: "sup (-a) (-(-a)) = 0"

     by (simp add: sup_commute)

   from p[OF assms] p[OF **] show "a = 0"

     by simp

 qed

 lemma inf_0_imp_0: "inf a (- a) = 0 \<Longrightarrow> a = 0"

   apply (simp add: inf_eq_neg_sup)

   apply (simp add: sup_commute)

   apply (erule sup_0_imp_0)

   done

 lemma inf_0_eq_0 [simp, no_atp]: "inf a (- a) = 0 \<longleftrightarrow> a = 0"

   apply rule

   apply (erule inf_0_imp_0)

   apply simp

   done

 lemma sup_0_eq_0 [simp, no_atp]: "sup a (- a) = 0 \<longleftrightarrow> a = 0"

   apply rule

   apply (erule sup_0_imp_0)

   apply simp

   done

 lemma zero_le_double_add_iff_zero_le_single_add [simp]: "0 \<le> a + a \<longleftrightarrow> 0 \<le> a"

   (is "?lhs \<longleftrightarrow> ?rhs")

 proof

   show ?rhs if ?lhs

   proof -

     from that have a: "inf (a + a) 0 = 0"

       by (simp add: inf_commute inf_absorb1)

     have "inf a 0 + inf a 0 = inf (inf (a + a) 0) a"  (is "?l = _")

       by (simp add: add_sup_inf_distribs inf_aci)

     then have "?l = 0 + inf a 0"

       by (simp add: a, simp add: inf_commute)

     then have "inf a 0 = 0"

       by (simp only: add_right_cancel)

     then show ?thesis

       unfolding le_iff_inf by (simp add: inf_commute)

   qed

   show ?lhs if ?rhs

     by (simp add: add_mono[OF that that, simplified])

 qed

 lemma double_zero [simp]: "a + a = 0 \<longleftrightarrow> a = 0"

   (is "?lhs \<longleftrightarrow> ?rhs")

 proof

   show ?rhs if ?lhs

   proof -

     from that have "a + a + - a = - a"

       by simp

     then have "a + (a + - a) = - a"

       by (simp only: add.assoc)

     then have a: "- a = a"

       by simp

     show ?thesis

       apply (rule antisym)

       apply (unfold neg_le_iff_le [symmetric, of a])

       unfolding a

       apply simp

       unfolding zero_le_double_add_iff_zero_le_single_add [symmetric, of a]

       unfolding that

       unfolding le_less

       apply simp_all

       done

   qed

   show ?lhs if ?rhs

     using that by simp

 qed

 lemma zero_less_double_add_iff_zero_less_single_add [simp]: "0 < a + a \<longleftrightarrow> 0 < a"

 proof (cases "a = 0")

   case True

   then show ?thesis by auto

 next

   case False

   then show ?thesis

     unfolding less_le

     apply simp

     apply rule

     apply clarify

     apply rule

     apply assumption

     apply (rule notI)

     unfolding double_zero [symmetric, of a]

     apply blast

     done

 qed

 lemma double_add_le_zero_iff_single_add_le_zero [simp]: "a + a \<le> 0 \<longleftrightarrow> a \<le> 0"

 proof -

   have "a + a \<le> 0 \<longleftrightarrow> 0 \<le> - (a + a)"

     by (subst le_minus_iff) simp

   moreover have "\<dots> \<longleftrightarrow> a \<le> 0"

     by (simp only: minus_add_distrib zero_le_double_add_iff_zero_le_single_add) simp

   ultimately show ?thesis

     by blast

 qed

 lemma double_add_less_zero_iff_single_less_zero [simp]: "a + a < 0 \<longleftrightarrow> a < 0"

 proof -

   have "a + a < 0 \<longleftrightarrow> 0 < - (a + a)"

     by (subst less_minus_iff) simp

   moreover have "\<dots> \<longleftrightarrow> a < 0"

     by (simp only: minus_add_distrib zero_less_double_add_iff_zero_less_single_add) simp

   ultimately show ?thesis

     by blast

 qed

 declare neg_inf_eq_sup [simp] neg_sup_eq_inf [simp] diff_inf_eq_sup [simp] diff_sup_eq_inf [simp]

 lemma le_minus_self_iff: "a \<le> - a \<longleftrightarrow> a \<le> 0"

 proof -

   from add_le_cancel_left [of "uminus a" "plus a a" zero]

   have "a \<le> - a \<longleftrightarrow> a + a \<le> 0"

     by (simp add: add.assoc[symmetric])

   then show ?thesis

     by simp

 qed

 lemma minus_le_self_iff: "- a \<le> a \<longleftrightarrow> 0 \<le> a"

 proof -

   have "- a \<le> a \<longleftrightarrow> 0 \<le> a + a"

     using add_le_cancel_left [of "uminus a" zero "plus a a"]

     by (simp add: add.assoc[symmetric])

   then show ?thesis

     by simp

 qed

 lemma zero_le_iff_zero_nprt: "0 \<le> a \<longleftrightarrow> nprt a = 0"

   unfolding le_iff_inf by (simp add: nprt_def inf_commute)

 lemma le_zero_iff_zero_pprt: "a \<le> 0 \<longleftrightarrow> pprt a = 0"

   unfolding le_iff_sup by (simp add: pprt_def sup_commute)

 lemma le_zero_iff_pprt_id: "0 \<le> a \<longleftrightarrow> pprt a = a"

   unfolding le_iff_sup by (simp add: pprt_def sup_commute)

 lemma zero_le_iff_nprt_id: "a \<le> 0 \<longleftrightarrow> nprt a = a"

   unfolding le_iff_inf by (simp add: nprt_def inf_commute)

 lemma pprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> pprt a \<le> pprt b"

   unfolding le_iff_sup by (simp add: pprt_def sup_aci sup_assoc [symmetric, of a])

 lemma nprt_mono [simp, no_atp]: "a \<le> b \<Longrightarrow> nprt a \<le> nprt b"

   unfolding le_iff_inf by (simp add: nprt_def inf_aci inf_assoc [symmetric, of a])

 end

 lemmas add_sup_inf_distribs =

   add_inf_distrib_right add_inf_distrib_left add_sup_distrib_right add_sup_distrib_left

 class lattice_ab_group_add_abs = lattice_ab_group_add + abs +

   assumes abs_lattice: "\<bar>a\<bar> = sup a (- a)"

 begin

 lemma abs_prts: "\<bar>a\<bar> = pprt a - nprt a"

 proof -

   have "0 \<le> \<bar>a\<bar>"

   proof -

     have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>"

       by (auto simp add: abs_lattice)

     show ?thesis

       by (rule add_mono [OF a b, simplified])

   qed

   then have "0 \<le> sup a (- a)"

     unfolding abs_lattice .

   then have "sup (sup a (- a)) 0 = sup a (- a)"

     by (rule sup_absorb1)

   then show ?thesis

     by (simp add: add_sup_inf_distribs ac_simps pprt_def nprt_def abs_lattice)

 qed

 subclass ordered_ab_group_add_abs

 proof

   have abs_ge_zero [simp]: "0 \<le> \<bar>a\<bar>" for a

   proof -

     have a: "a \<le> \<bar>a\<bar>" and b: "- a \<le> \<bar>a\<bar>"

       by (auto simp add: abs_lattice)

     show "0 \<le> \<bar>a\<bar>"

       by (rule add_mono [OF a b, simplified])

   qed

   have abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" for a b

     by (simp add: abs_lattice le_supI)

   fix a b

   show "0 \<le> \<bar>a\<bar>"

     by simp

   show "a \<le> \<bar>a\<bar>"

     by (auto simp add: abs_lattice)

   show "\<bar>-a\<bar> = \<bar>a\<bar>"

     by (simp add: abs_lattice sup_commute)

   show "- a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b" if "a \<le> b"

     using that by (rule abs_leI)

   show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"

   proof -

     have g: "\<bar>a\<bar> + \<bar>b\<bar> = sup (a + b) (sup (- a - b) (sup (- a + b) (a + (- b))))"

       (is "_ = sup ?m ?n")

       by (simp add: abs_lattice add_sup_inf_distribs ac_simps)

     have a: "a + b \<le> sup ?m ?n"

       by simp

     have b: "- a - b \<le> ?n"

       by simp

     have c: "?n \<le> sup ?m ?n"

       by simp

     from b c have d: "- a - b \<le> sup ?m ?n"

       by (rule order_trans)

     have e: "- a - b = - (a + b)"

       by simp

     from a d e have "\<bar>a + b\<bar> \<le> sup ?m ?n"

       apply -

       apply (drule abs_leI)

       apply (simp_all only: algebra_simps minus_add)

       apply (metis add_uminus_conv_diff d sup_commute uminus_add_conv_diff)

       done

     with g[symmetric] show ?thesis by simp

   qed

 qed

 end

 lemma sup_eq_if:

   fixes a :: "'a::{lattice_ab_group_add,linorder}"

   shows "sup a (- a) = (if a < 0 then - a else a)"

   using add_le_cancel_right [of a a "- a", symmetric, simplified]

     and add_le_cancel_right [of "-a" a a, symmetric, simplified]

   by (auto simp: sup_max max.absorb1 max.absorb2)

 lemma abs_if_lattice:

   fixes a :: "'a::{lattice_ab_group_add_abs,linorder}"

   shows "\<bar>a\<bar> = (if a < 0 then - a else a)"

   by auto

 lemma estimate_by_abs:

   fixes a b c :: "'a::lattice_ab_group_add_abs"

   assumes "a + b \<le> c"

   shows "a \<le> c + \<bar>b\<bar>"

 proof -

   from assms have "a \<le> c + (- b)"

     by (simp add: algebra_simps)

   have "- b \<le> \<bar>b\<bar>"

     by (rule abs_ge_minus_self)

   then have "c + (- b) \<le> c + \<bar>b\<bar>"

     by (rule add_left_mono)

   with \<open>a \<le> c + (- b)\<close> show ?thesis

     by (rule order_trans)

 qed

 class lattice_ring = ordered_ring + lattice_ab_group_add_abs

 begin

 subclass semilattice_inf_ab_group_add ..

 subclass semilattice_sup_ab_group_add ..

 end

 lemma abs_le_mult:

   fixes a b :: "'a::lattice_ring"

   shows "\<bar>a * b\<bar> \<le> \<bar>a\<bar> * \<bar>b\<bar>"

 proof -

   let ?x = "pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b"

   let ?y = "pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"

   have a: "\<bar>a\<bar> * \<bar>b\<bar> = ?x"

     by (simp only: abs_prts[of a] abs_prts[of b] algebra_simps)

   have bh: "u = a \<Longrightarrow> v = b \<Longrightarrow>

             u * v = pprt a * pprt b + pprt a * nprt b +

                     nprt a * pprt b + nprt a * nprt b" for u v :: 'a

     apply (subst prts[of u], subst prts[of v])

     apply (simp add: algebra_simps)

     done

   note b = this[OF refl[of a] refl[of b]]

   have xy: "- ?x \<le> ?y"

     apply simp

     apply (metis (full_types) add_increasing add_uminus_conv_diff

       lattice_ab_group_add_class.minus_le_self_iff minus_add_distrib mult_nonneg_nonneg

       mult_nonpos_nonpos nprt_le_zero zero_le_pprt)

     done

   have yx: "?y \<le> ?x"

     apply simp

     apply (metis (full_types) add_nonpos_nonpos add_uminus_conv_diff

       lattice_ab_group_add_class.le_minus_self_iff minus_add_distrib mult_nonneg_nonpos

       mult_nonpos_nonneg nprt_le_zero zero_le_pprt)

     done

   have i1: "a * b \<le> \<bar>a\<bar> * \<bar>b\<bar>"

     by (simp only: a b yx)

   have i2: "- (\<bar>a\<bar> * \<bar>b\<bar>) \<le> a * b"

     by (simp only: a b xy)

   show ?thesis

     apply (rule abs_leI)

     apply (simp add: i1)

     apply (simp add: i2[simplified minus_le_iff])

     done

 qed

 instance lattice_ring \<subseteq> ordered_ring_abs

 proof

   fix a b :: "'a::lattice_ring"

   assume a: "(0 \<le> a \<or> a \<le> 0) \<and> (0 \<le> b \<or> b \<le> 0)"

   show "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"

   proof -

     have s: "(0 \<le> a * b) \<or> (a * b \<le> 0)"

       apply auto

       apply (rule_tac split_mult_pos_le)

       apply (rule_tac contrapos_np[of "a * b \<le> 0"])

       apply simp

       apply (rule_tac split_mult_neg_le)

       using a

       apply blast

       done

     have mulprts: "a * b = (pprt a + nprt a) * (pprt b + nprt b)"

       by (simp add: prts[symmetric])

     show ?thesis

     proof (cases "0 \<le> a * b")

       case True

       then show ?thesis

         apply (simp_all add: mulprts abs_prts)

         using a

         apply (auto simp add:

           algebra_simps

           iffD1[OF zero_le_iff_zero_nprt] iffD1[OF le_zero_iff_zero_pprt]

           iffD1[OF le_zero_iff_pprt_id] iffD1[OF zero_le_iff_nprt_id])

         apply(drule (1) mult_nonneg_nonpos[of a b], simp)

         apply(drule (1) mult_nonneg_nonpos2[of b a], simp)

         done

     next

       case False

       with s have "a * b \<le> 0"

         by simp

       then show ?thesis

         apply (simp_all add: mulprts abs_prts)

         apply (insert a)

         apply (auto simp add: algebra_simps)

         apply(drule (1) mult_nonneg_nonneg[of a b],simp)

         apply(drule (1) mult_nonpos_nonpos[of a b],simp)

         done

     qed

   qed

 qed

 lemma mult_le_prts:

   fixes a b :: "'a::lattice_ring"

   assumes "a1 \<le> a"

     and "a \<le> a2"

     and "b1 \<le> b"

     and "b \<le> b2"

   shows "a * b \<le>

     pprt a2 * pprt b2 + pprt a1 * nprt b2 + nprt a2 * pprt b1 + nprt a1 * nprt b1"

 proof -

   have "a * b = (pprt a + nprt a) * (pprt b + nprt b)"

     by (subst prts[symmetric])+ simp

   then have "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"

     by (simp add: algebra_simps)

   moreover have "pprt a * pprt b \<le> pprt a2 * pprt b2"

     by (simp_all add: assms mult_mono)

   moreover have "pprt a * nprt b \<le> pprt a1 * nprt b2"

   proof -

     have "pprt a * nprt b \<le> pprt a * nprt b2"

       by (simp add: mult_left_mono assms)

     moreover have "pprt a * nprt b2 \<le> pprt a1 * nprt b2"

       by (simp add: mult_right_mono_neg assms)

     ultimately show ?thesis

       by simp

   qed

   moreover have "nprt a * pprt b \<le> nprt a2 * pprt b1"

   proof -

     have "nprt a * pprt b \<le> nprt a2 * pprt b"

       by (simp add: mult_right_mono assms)

     moreover have "nprt a2 * pprt b \<le> nprt a2 * pprt b1"

       by (simp add: mult_left_mono_neg assms)

     ultimately show ?thesis

       by simp

   qed

   moreover have "nprt a * nprt b \<le> nprt a1 * nprt b1"

   proof -

     have "nprt a * nprt b \<le> nprt a * nprt b1"

       by (simp add: mult_left_mono_neg assms)

     moreover have "nprt a * nprt b1 \<le> nprt a1 * nprt b1"

       by (simp add: mult_right_mono_neg assms)

     ultimately show ?thesis

       by simp

   qed

   ultimately show ?thesis

     by - (rule add_mono | simp)+

 qed

 lemma mult_ge_prts:

   fixes a b :: "'a::lattice_ring"

   assumes "a1 \<le> a"

     and "a \<le> a2"

     and "b1 \<le> b"

     and "b \<le> b2"

   shows "a * b \<ge>

     nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"

 proof -

   from assms have a1: "- a2 \<le> -a"

     by auto

   from assms have a2: "- a \<le> -a1"

     by auto

   from mult_le_prts[of "- a2" "- a" "- a1" "b1" b "b2",

     OF a1 a2 assms(3) assms(4), simplified nprt_neg pprt_neg]

   have le: "- (a * b) \<le>

     - nprt a1 * pprt b2 + - nprt a2 * nprt b2 +

     - pprt a1 * pprt b1 + - pprt a2 * nprt b1"

     by simp

   then have "- (- nprt a1 * pprt b2 + - nprt a2 * nprt b2 +

       - pprt a1 * pprt b1 + - pprt a2 * nprt b1) \<le> a * b"

     by (simp only: minus_le_iff)

   then show ?thesis

     by (simp add: algebra_simps)

 qed

 instance int :: lattice_ring

 proof

   fix k :: int

   show "\<bar>k\<bar> = sup k (- k)"

     by (auto simp add: sup_int_def)

 qed

 instance real :: lattice_ring

 proof

   fix a :: real

   show "\<bar>a\<bar> = sup a (- a)"

     by (auto simp add: sup_real_def)

 qed

 end