Theory Lattice_Algebras

theory Lattice_Algebras
imports Complex_Main
(* Author: Steven Obua, TU Muenchen *)

header {* Various algebraic structures combined with a lattice *}

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)
apply rule
apply (rule add_le_imp_le_left[of "a"], simp only: add_assoc[symmetric], simp)+
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)
thus ?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 rule
apply (rule add_le_imp_le_left [of "a"], simp only: add_assoc[symmetric], simp)+
apply (rule le_supI)
apply (simp_all)
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)
thus ?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) ≤ 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) ≤ b"
by (rule add_le_imp_le_right [of _ "sup (uminus a) (uminus b)"])
(simp, simp add: add_sup_distrib_left)
assume "a ≤ b" "a ≤ c"
then show "a ≤ - 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 ≤ - 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 ≤ - 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 ≤ c" "b ≤ c"
then show "- inf (-a) (-b) ≤ 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 neg_sup_eq_inf: "- sup a b = inf (-a) (-b)"
by (simp add: sup_eq_neg_inf)

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)
hence "0 = sup 0 (b-a) + inf (a-b) 0"
by (simp add: inf_eq_neg_sup)
hence "0 = (-a + sup a b) + (inf a b + (-b))"
by (simp add: add_sup_distrib_left add_inf_distrib_right) (simp add: algebra_simps)
thus ?thesis by (simp add: algebra_simps)
qed


subsection {* Positive Part, Negative Part, Absolute Value *}

definition nprt :: "'a => 'a"
where "nprt x = inf x 0"

definition pprt :: "'a => '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 "… = - 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 ≤ pprt a"
by (simp add: pprt_def)

lemma nprt_le_zero[simp]: "nprt a ≤ 0"
by (simp add: nprt_def)

lemma le_eq_neg: "a ≤ - b <-> a + b ≤ 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 ≤ x ==> pprt x = x"
by (simp add: pprt_def sup_absorb1)

lemma nprt_eq_id [simp, no_atp]: "x ≤ 0 ==> nprt x = x"
by (simp add: nprt_def inf_absorb1)

lemma pprt_eq_0 [simp, no_atp]: "x ≤ 0 ==> pprt x = 0"
by (simp add: pprt_def sup_absorb2)

lemma nprt_eq_0 [simp, no_atp]: "0 ≤ x ==> nprt x = 0"
by (simp add: nprt_def inf_absorb2)

lemma sup_0_imp_0: "sup a (- a) = 0 ==> a = 0"
proof -
{
fix a::'a
assume hyp: "sup a (-a) = 0"
hence "sup a (-a) + a = a" by (simp)
hence "sup (a+a) 0 = a" by (simp add: add_sup_distrib_right)
hence "sup (a+a) 0 <= a" by (simp)
hence "0 <= a" by (blast intro: order_trans inf_sup_ord)
}
note p = this
assume hyp:"sup a (-a) = 0"
hence 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 ==> 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 <-> 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 <-> 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 ≤ a + a <-> 0 ≤ a"
proof
assume "0 <= a + a"
hence 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)
hence "?l = 0 + inf a 0" by (simp add: a, simp add: inf_commute)
hence "inf a 0 = 0" by (simp only: add_right_cancel)
then show "0 <= a" unfolding le_iff_inf by (simp add: inf_commute)
next
assume a: "0 <= a"
show "0 <= a + a" by (simp add: add_mono[OF a a, simplified])
qed

lemma double_zero [simp]: "a + a = 0 <-> 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 <-> 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 simp
done
qed

lemma double_add_le_zero_iff_single_add_le_zero [simp]:
"a + a ≤ 0 <-> a ≤ 0"
proof -
have "a + a ≤ 0 <-> 0 ≤ - (a + a)" by (subst le_minus_iff, simp)
moreover have "… <-> a ≤ 0" by simp
ultimately show ?thesis by blast
qed

lemma double_add_less_zero_iff_single_less_zero [simp]:
"a + a < 0 <-> a < 0"
proof -
have "a + a < 0 <-> 0 < - (a + a)" by (subst less_minus_iff, simp)
moreover have "… <-> a < 0" by simp
ultimately show ?thesis by blast
qed

declare neg_inf_eq_sup [simp] neg_sup_eq_inf [simp]

lemma le_minus_self_iff: "a ≤ - a <-> a ≤ 0"
proof -
from add_le_cancel_left [of "uminus a" "plus a a" zero]
have "(a <= -a) = (a+a <= 0)"
by (simp add: add_assoc[symmetric])
thus ?thesis by simp
qed

lemma minus_le_self_iff: "- a ≤ a <-> 0 ≤ a"
proof -
from add_le_cancel_left [of "uminus a" zero "plus a a"]
have "(-a <= a) = (0 <= a+a)"
by (simp add: add_assoc[symmetric])
thus ?thesis by simp
qed

lemma zero_le_iff_zero_nprt: "0 ≤ a <-> nprt a = 0"
unfolding le_iff_inf by (simp add: nprt_def inf_commute)

lemma le_zero_iff_zero_pprt: "a ≤ 0 <-> pprt a = 0"
unfolding le_iff_sup by (simp add: pprt_def sup_commute)

lemma le_zero_iff_pprt_id: "0 ≤ a <-> pprt a = a"
unfolding le_iff_sup by (simp add: pprt_def sup_commute)

lemma zero_le_iff_nprt_id: "a ≤ 0 <-> nprt a = a"
unfolding le_iff_inf by (simp add: nprt_def inf_commute)

lemma pprt_mono [simp, no_atp]: "a ≤ b ==> pprt a ≤ 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 ≤ b ==> nprt a ≤ 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: "¦a¦ = sup a (- a)"
begin

lemma abs_prts: "¦a¦ = pprt a - nprt a"
proof -
have "0 ≤ ¦a¦"
proof -
have a: "a ≤ ¦a¦" and b: "- a ≤ ¦a¦" by (auto simp add: abs_lattice)
show ?thesis by (rule add_mono [OF a b, simplified])
qed
then have "0 ≤ 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 sup_aci pprt_def nprt_def diff_minus abs_lattice)
qed

subclass ordered_ab_group_add_abs
proof
have abs_ge_zero [simp]: "!!a. 0 ≤ ¦a¦"
proof -
fix a b
have a: "a ≤ ¦a¦" and b: "- a ≤ ¦a¦"
by (auto simp add: abs_lattice)
show "0 ≤ ¦a¦"
by (rule add_mono [OF a b, simplified])
qed
have abs_leI: "!!a b. a ≤ b ==> - a ≤ b ==> ¦a¦ ≤ b"
by (simp add: abs_lattice le_supI)
fix a b
show "0 ≤ ¦a¦" by simp
show "a ≤ ¦a¦"
by (auto simp add: abs_lattice)
show "¦-a¦ = ¦a¦"
by (simp add: abs_lattice sup_commute)
{
assume "a ≤ b"
then show "- a ≤ b ==> ¦a¦ ≤ b"
by (rule abs_leI)
}
show "¦a + b¦ ≤ ¦a¦ + ¦b¦"
proof -
have g:"abs a + abs b = sup (a+b) (sup (-a-b) (sup (-a+b) (a + (-b))))" (is "_=sup ?m ?n")
by (simp add: abs_lattice add_sup_inf_distribs sup_aci diff_minus)
have a: "a + b <= sup ?m ?n" by simp
have b: "- a - b <= ?n" by simp
have c: "?n <= sup ?m ?n" by simp
from b c have d: "-a-b <= sup ?m ?n" by (rule order_trans)
have e:"-a-b = -(a+b)" by (simp add: diff_minus)
from a d e have "abs(a+b) <= sup ?m ?n"
apply -
apply (drule abs_leI)
apply auto
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 min_max.sup_absorb1 min_max.sup_absorb2)
qed

lemma abs_if_lattice:
fixes a :: "'a::{lattice_ab_group_add_abs, linorder}"
shows "¦a¦ = (if a < 0 then - a else a)"
by auto

lemma estimate_by_abs:
"a + b <= (c::'a::lattice_ab_group_add_abs) ==> a <= c + abs b"
proof -
assume "a+b <= c"
then have "a <= c+(-b)" by (simp add: algebra_simps)
have "(-b) <= abs b" by (rule abs_ge_minus_self)
then have "c + (- b) ≤ c + ¦b¦" by (rule add_left_mono)
with `a ≤ c + (- b)` 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: "abs (a * b) ≤ (abs a) * (abs (b::'a::lattice_ring))"
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: "(abs a) * (abs b) = ?x"
by (simp only: abs_prts[of a] abs_prts[of b] algebra_simps)
{
fix u v :: 'a
have bh: "[|u = a; v = b|] ==>
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 <= ?y"
apply (simp)
apply (rule order_trans [OF add_nonpos_nonpos add_nonneg_nonneg])
apply (simp_all add: mult_nonneg_nonneg mult_nonpos_nonpos)
done
have yx: "?y <= ?x"
apply (simp add:diff_minus)
apply (rule order_trans [OF add_nonpos_nonpos add_nonneg_nonneg])
apply (simp_all add: mult_nonneg_nonpos mult_nonpos_nonneg)
done
have i1: "a*b <= abs a * abs b" by (simp only: a b yx)
have i2: "- (abs a * abs b) <= 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 ordered_ring_abs
proof
fix a b :: "'a:: lattice_ring"
assume a: "(0 ≤ a ∨ a ≤ 0) ∧ (0 ≤ b ∨ b ≤ 0)"
show "abs (a*b) = abs a * abs b"
proof -
have s: "(0 <= a*b) | (a*b <= 0)"
apply (auto)
apply (rule_tac split_mult_pos_le)
apply (rule_tac contrapos_np[of "a*b <= 0"])
apply (simp)
apply (rule_tac split_mult_neg_le)
apply (insert 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
assume "0 <= a * b"
then show ?thesis
apply (simp_all add: mulprts abs_prts)
apply (insert 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
assume "~(0 <= a*b)"
with s have "a*b <= 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:
assumes "a1 <= (a::'a::lattice_ring)"
and "a <= a2"
and "b1 <= b"
and "b <= b2"
shows "a * b <=
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 <= pprt a2 * pprt b2"
by (simp_all add: assms mult_mono)
moreover have "pprt a * nprt b <= pprt a1 * nprt b2"
proof -
have "pprt a * nprt b <= pprt a * nprt b2"
by (simp add: mult_left_mono assms)
moreover have "pprt a * nprt b2 <= pprt a1 * nprt b2"
by (simp add: mult_right_mono_neg assms)
ultimately show ?thesis
by simp
qed
moreover have "nprt a * pprt b <= nprt a2 * pprt b1"
proof -
have "nprt a * pprt b <= nprt a2 * pprt b"
by (simp add: mult_right_mono assms)
moreover have "nprt a2 * pprt b <= nprt a2 * pprt b1"
by (simp add: mult_left_mono_neg assms)
ultimately show ?thesis
by simp
qed
moreover have "nprt a * nprt b <= nprt a1 * nprt b1"
proof -
have "nprt a * nprt b <= nprt a * nprt b1"
by (simp add: mult_left_mono_neg assms)
moreover have "nprt a * nprt b1 <= 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:
assumes "a1 <= (a::'a::lattice_ring)"
and "a <= a2"
and "b1 <= b"
and "b <= b2"
shows "a * b >=
nprt a1 * pprt b2 + nprt a2 * nprt b2 + pprt a1 * pprt b1 + pprt a2 * nprt b1"

proof -
from assms have a1:"- a2 <= -a"
by auto
from assms have a2: "-a <= -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) <= - 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) <= a * b"
by (simp only: minus_le_iff)
then show ?thesis by simp
qed

instance int :: lattice_ring
proof
fix k :: int
show "abs k = sup k (- k)"
by (auto simp add: sup_int_def)
qed

instance real :: lattice_ring
proof
fix a :: real
show "abs a = sup a (- a)"
by (auto simp add: sup_real_def)
qed

end