# Theory Bounds

(*  Title:      HOL/Lattice/Bounds.thy
Author:     Markus Wenzel, TU Muenchen
*)

section ‹Bounds›

theory Bounds imports Orders begin

hide_const (open) inf sup

subsection ‹Infimum and supremum›

text ‹
Given a partial order, we define infimum (greatest lower bound) and
supremum (least upper bound) wrt.\ ‹⊑› for two and for any
number of elements.
›

definition
is_inf :: "'a::partial_order ⇒ 'a ⇒ 'a ⇒ bool" where
"is_inf x y inf = (inf ⊑ x ∧ inf ⊑ y ∧ (∀z. z ⊑ x ∧ z ⊑ y ⟶ z ⊑ inf))"

definition
is_sup :: "'a::partial_order ⇒ 'a ⇒ 'a ⇒ bool" where
"is_sup x y sup = (x ⊑ sup ∧ y ⊑ sup ∧ (∀z. x ⊑ z ∧ y ⊑ z ⟶ sup ⊑ z))"

definition
is_Inf :: "'a::partial_order set ⇒ 'a ⇒ bool" where
"is_Inf A inf = ((∀x ∈ A. inf ⊑ x) ∧ (∀z. (∀x ∈ A. z ⊑ x) ⟶ z ⊑ inf))"

definition
is_Sup :: "'a::partial_order set ⇒ 'a ⇒ bool" where
"is_Sup A sup = ((∀x ∈ A. x ⊑ sup) ∧ (∀z. (∀x ∈ A. x ⊑ z) ⟶ sup ⊑ z))"

text ‹
These definitions entail the following basic properties of boundary
elements.
›

lemma is_infI [intro?]: "inf ⊑ x ⟹ inf ⊑ y ⟹
(⋀z. z ⊑ x ⟹ z ⊑ y ⟹ z ⊑ inf) ⟹ is_inf x y inf"
by (unfold is_inf_def) blast

lemma is_inf_greatest [elim?]:
"is_inf x y inf ⟹ z ⊑ x ⟹ z ⊑ y ⟹ z ⊑ inf"
by (unfold is_inf_def) blast

lemma is_inf_lower [elim?]:
"is_inf x y inf ⟹ (inf ⊑ x ⟹ inf ⊑ y ⟹ C) ⟹ C"
by (unfold is_inf_def) blast

lemma is_supI [intro?]: "x ⊑ sup ⟹ y ⊑ sup ⟹
(⋀z. x ⊑ z ⟹ y ⊑ z ⟹ sup ⊑ z) ⟹ is_sup x y sup"
by (unfold is_sup_def) blast

lemma is_sup_least [elim?]:
"is_sup x y sup ⟹ x ⊑ z ⟹ y ⊑ z ⟹ sup ⊑ z"
by (unfold is_sup_def) blast

lemma is_sup_upper [elim?]:
"is_sup x y sup ⟹ (x ⊑ sup ⟹ y ⊑ sup ⟹ C) ⟹ C"
by (unfold is_sup_def) blast

lemma is_InfI [intro?]: "(⋀x. x ∈ A ⟹ inf ⊑ x) ⟹
(⋀z. (∀x ∈ A. z ⊑ x) ⟹ z ⊑ inf) ⟹ is_Inf A inf"
by (unfold is_Inf_def) blast

lemma is_Inf_greatest [elim?]:
"is_Inf A inf ⟹ (⋀x. x ∈ A ⟹ z ⊑ x) ⟹ z ⊑ inf"
by (unfold is_Inf_def) blast

lemma is_Inf_lower [dest?]:
"is_Inf A inf ⟹ x ∈ A ⟹ inf ⊑ x"
by (unfold is_Inf_def) blast

lemma is_SupI [intro?]: "(⋀x. x ∈ A ⟹ x ⊑ sup) ⟹
(⋀z. (∀x ∈ A. x ⊑ z) ⟹ sup ⊑ z) ⟹ is_Sup A sup"
by (unfold is_Sup_def) blast

lemma is_Sup_least [elim?]:
"is_Sup A sup ⟹ (⋀x. x ∈ A ⟹ x ⊑ z) ⟹ sup ⊑ z"
by (unfold is_Sup_def) blast

lemma is_Sup_upper [dest?]:
"is_Sup A sup ⟹ x ∈ A ⟹ x ⊑ sup"
by (unfold is_Sup_def) blast

subsection ‹Duality›

text ‹
Infimum and supremum are dual to each other.
›

theorem dual_inf [iff?]:
"is_inf (dual x) (dual y) (dual sup) = is_sup x y sup"
by (simp add: is_inf_def is_sup_def dual_all [symmetric] dual_leq)

theorem dual_sup [iff?]:
"is_sup (dual x) (dual y) (dual inf) = is_inf x y inf"
by (simp add: is_inf_def is_sup_def dual_all [symmetric] dual_leq)

theorem dual_Inf [iff?]:
"is_Inf (dual  A) (dual sup) = is_Sup A sup"
by (simp add: is_Inf_def is_Sup_def dual_all [symmetric] dual_leq)

theorem dual_Sup [iff?]:
"is_Sup (dual  A) (dual inf) = is_Inf A inf"
by (simp add: is_Inf_def is_Sup_def dual_all [symmetric] dual_leq)

subsection ‹Uniqueness›

text ‹
Infima and suprema on partial orders are unique; this is mainly due
to anti-symmetry of the underlying relation.
›

theorem is_inf_uniq: "is_inf x y inf ⟹ is_inf x y inf' ⟹ inf = inf'"
proof -
assume inf: "is_inf x y inf"
assume inf': "is_inf x y inf'"
show ?thesis
proof (rule leq_antisym)
from inf' show "inf ⊑ inf'"
proof (rule is_inf_greatest)
from inf show "inf ⊑ x" ..
from inf show "inf ⊑ y" ..
qed
from inf show "inf' ⊑ inf"
proof (rule is_inf_greatest)
from inf' show "inf' ⊑ x" ..
from inf' show "inf' ⊑ y" ..
qed
qed
qed

theorem is_sup_uniq: "is_sup x y sup ⟹ is_sup x y sup' ⟹ sup = sup'"
proof -
assume sup: "is_sup x y sup" and sup': "is_sup x y sup'"
have "dual sup = dual sup'"
proof (rule is_inf_uniq)
from sup show "is_inf (dual x) (dual y) (dual sup)" ..
from sup' show "is_inf (dual x) (dual y) (dual sup')" ..
qed
then show "sup = sup'" ..
qed

theorem is_Inf_uniq: "is_Inf A inf ⟹ is_Inf A inf' ⟹ inf = inf'"
proof -
assume inf: "is_Inf A inf"
assume inf': "is_Inf A inf'"
show ?thesis
proof (rule leq_antisym)
from inf' show "inf ⊑ inf'"
proof (rule is_Inf_greatest)
fix x assume "x ∈ A"
with inf show "inf ⊑ x" ..
qed
from inf show "inf' ⊑ inf"
proof (rule is_Inf_greatest)
fix x assume "x ∈ A"
with inf' show "inf' ⊑ x" ..
qed
qed
qed

theorem is_Sup_uniq: "is_Sup A sup ⟹ is_Sup A sup' ⟹ sup = sup'"
proof -
assume sup: "is_Sup A sup" and sup': "is_Sup A sup'"
have "dual sup = dual sup'"
proof (rule is_Inf_uniq)
from sup show "is_Inf (dual  A) (dual sup)" ..
from sup' show "is_Inf (dual  A) (dual sup')" ..
qed
then show "sup = sup'" ..
qed

subsection ‹Related elements›

text ‹
The binary bound of related elements is either one of the argument.
›

theorem is_inf_related [elim?]: "x ⊑ y ⟹ is_inf x y x"
proof -
assume "x ⊑ y"
show ?thesis
proof
show "x ⊑ x" ..
show "x ⊑ y" by fact
fix z assume "z ⊑ x" and "z ⊑ y" show "z ⊑ x" by fact
qed
qed

theorem is_sup_related [elim?]: "x ⊑ y ⟹ is_sup x y y"
proof -
assume "x ⊑ y"
show ?thesis
proof
show "x ⊑ y" by fact
show "y ⊑ y" ..
fix z assume "x ⊑ z" and "y ⊑ z"
show "y ⊑ z" by fact
qed
qed

subsection ‹General versus binary bounds \label{sec:gen-bin-bounds}›

text ‹
General bounds of two-element sets coincide with binary bounds.
›

theorem is_Inf_binary: "is_Inf {x, y} inf = is_inf x y inf"
proof -
let ?A = "{x, y}"
show ?thesis
proof
assume is_Inf: "is_Inf ?A inf"
show "is_inf x y inf"
proof
have "x ∈ ?A" by simp
with is_Inf show "inf ⊑ x" ..
have "y ∈ ?A" by simp
with is_Inf show "inf ⊑ y" ..
fix z assume zx: "z ⊑ x" and zy: "z ⊑ y"
from is_Inf show "z ⊑ inf"
proof (rule is_Inf_greatest)
fix a assume "a ∈ ?A"
then have "a = x ∨ a = y" by blast
then show "z ⊑ a"
proof
assume "a = x"
with zx show ?thesis by simp
next
assume "a = y"
with zy show ?thesis by simp
qed
qed
qed
next
assume is_inf: "is_inf x y inf"
show "is_Inf {x, y} inf"
proof
fix a assume "a ∈ ?A"
then have "a = x ∨ a = y" by blast
then show "inf ⊑ a"
proof
assume "a = x"
also from is_inf have "inf ⊑ x" ..
finally show ?thesis .
next
assume "a = y"
also from is_inf have "inf ⊑ y" ..
finally show ?thesis .
qed
next
fix z assume z: "∀a ∈ ?A. z ⊑ a"
from is_inf show "z ⊑ inf"
proof (rule is_inf_greatest)
from z show "z ⊑ x" by blast
from z show "z ⊑ y" by blast
qed
qed
qed
qed

theorem is_Sup_binary: "is_Sup {x, y} sup = is_sup x y sup"
proof -
have "is_Sup {x, y} sup = is_Inf (dual  {x, y}) (dual sup)"
by (simp only: dual_Inf)
also have "dual  {x, y} = {dual x, dual y}"
by simp
also have "is_Inf … (dual sup) = is_inf (dual x) (dual y) (dual sup)"
by (rule is_Inf_binary)
also have "… = is_sup x y sup"
by (simp only: dual_inf)
finally show ?thesis .
qed

subsection ‹Connecting general bounds \label{sec:connect-bounds}›

text ‹
Either kind of general bounds is sufficient to express the other.
The least upper bound (supremum) is the same as the the greatest
lower bound of the set of all upper bounds; the dual statements
holds as well; the dual statement holds as well.
›

theorem Inf_Sup: "is_Inf {b. ∀a ∈ A. a ⊑ b} sup ⟹ is_Sup A sup"
proof -
let ?B = "{b. ∀a ∈ A. a ⊑ b}"
assume is_Inf: "is_Inf ?B sup"
show "is_Sup A sup"
proof
fix x assume x: "x ∈ A"
from is_Inf show "x ⊑ sup"
proof (rule is_Inf_greatest)
fix y assume "y ∈ ?B"
then have "∀a ∈ A. a ⊑ y" ..
from this x show "x ⊑ y" ..
qed
next
fix z assume "∀x ∈ A. x ⊑ z"
then have "z ∈ ?B" ..
with is_Inf show "sup ⊑ z" ..
qed
qed

theorem Sup_Inf: "is_Sup {b. ∀a ∈ A. b ⊑ a} inf ⟹ is_Inf A inf"
proof -
assume "is_Sup {b. ∀a ∈ A. b ⊑ a} inf"
then have "is_Inf (dual  {b. ∀a ∈ A. dual a ⊑ dual b}) (dual inf)"
by (simp only: dual_Inf dual_leq)
also have "dual  {b. ∀a ∈ A. dual a ⊑ dual b} = {b'. ∀a' ∈ dual  A. a' ⊑ b'}"
by (auto iff: dual_ball dual_Collect simp add: image_Collect)  (* FIXME !? *)
finally have "is_Inf … (dual inf)" .
then have "is_Sup (dual  A) (dual inf)"
by (rule Inf_Sup)
then show ?thesis ..
qed

end