(* 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: "sup a (- a) = 0 \<Longrightarrow> a = 0"
proof -
{
fix a :: 'a
assume hyp: "sup a (- a) = 0"
then 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 have "0 \<le> a"
by (blast intro: order_trans inf_sup_ord)
}
note p = this
assume hyp:"sup a (-a) = 0"
then have hyp2:"sup (-a) (-(-a)) = 0"
by (simp add: sup_commute)
from p[OF hyp] p[OF hyp2] 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"
proof
assume "0 \<le> a + a"
then 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 "0 \<le> a"
unfolding le_iff_inf by (simp add: inf_commute)
next
assume a: "0 \<le> a"
show "0 \<le> a + a"
by (simp add: add_mono[OF a a, simplified])
qed
lemma double_zero [simp]: "a + a = 0 \<longleftrightarrow> a = 0"
proof
assume assm: "a + a = 0"
then 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 "a = 0"
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 assm
unfolding le_less
apply simp_all
done
next
assume "a = 0"
then show "a + a = 0"
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 -
from add_le_cancel_left [of "uminus a" zero "plus a a"]
have "- a \<le> a \<longleftrightarrow> 0 \<le> 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]: "\<And>a. 0 \<le> \<bar>a\<bar>"
proof -
fix a b
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: "\<And>a b. a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> 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)
{
assume "a \<le> b"
then show "- a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
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)"
proof -
note add_le_cancel_right [of a a "- a", symmetric, simplified]
moreover note add_le_cancel_right [of "-a" a a, symmetric, simplified]
then show ?thesis by (auto simp: sup_max max.absorb1 max.absorb2)
qed
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"
shows "a + b \<le> c \<Longrightarrow> a \<le> c + \<bar>b\<bar>"
proof -
assume "a + b \<le> c"
then 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)
{
fix u v :: 'a
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"
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)"
apply (subst prts[symmetric])+
apply simp
done
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
apply -
apply (rule add_mono | simp)+
done
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