refactor(library/algebra/complete_lattice): make complete lattices more usable

I addressed two problems. First, the theorem names and notation were all in
the namespace complete_lattice. The problem was that if you opened that
namespace, names (like "sup" and "inf") and notation clashed with global notation
for lattices.

The other problem was that if you defined a lattice using Sup, the Sup you got
was not the Sup you want; it was the Sup-construction from the Inf-construction
from the Sup.

Everything seems good now.
This commit is contained in:
Jeremy Avigad 2016-02-16 17:26:11 -05:00 committed by Leonardo de Moura
parent 7fe71c972f
commit a08395b17e

View file

@ -20,6 +20,26 @@ structure complete_lattice [class] (A : Type) extends lattice A :=
(le_Sup : ∀ {a : A} {s : set A}, a ∈ s → le a (Sup s))
(Sup_le : ∀ {b : A} {s : set A} (h : ∀ (a : A), a ∈ s → le a b), le (Sup s) b)
section
variable [complete_lattice A]
definition Inf (S : set A) : A := complete_lattice.Inf S
prefix `⨅ `:70 := Inf
definition Sup (S : set A) : A := complete_lattice.Sup S
prefix `⨆ `:65 := Sup
theorem Inf_le {a : A} {s : set A} (H : a ∈ s) : (Inf s) ≤ a := complete_lattice.Inf_le H
theorem le_Inf {b : A} {s : set A} (H : ∀ (a : A), a ∈ s → b ≤ a) : b ≤ Inf s :=
complete_lattice.le_Inf H
theorem le_Sup {a : A} {s : set A} (H : a ∈ s) : a ≤ Sup s := complete_lattice.le_Sup H
theorem Sup_le {b : A} {s : set A} (H : ∀ (a : A), a ∈ s → a ≤ b) : Sup s ≤ b :=
complete_lattice.Sup_le H
end
-- Minimal complete_lattice definition based just on Inf.
-- We later show that complete_lattice_Inf is a complete_lattice.
structure complete_lattice_Inf [class] (A : Type) extends weak_order A :=
@ -93,7 +113,8 @@ definition complete_lattice_Inf_to_complete_lattice_Sup [C : complete_lattice_In
⦃ complete_lattice_Sup, C ⦄
-- Every complete_lattice_Inf is a complete_lattice
definition complete_lattice_Inf_to_complete_lattice [instance] [C : complete_lattice_Inf A] : complete_lattice A :=
definition complete_lattice_Inf_to_complete_lattice [trans_instance] [reducible] [C : complete_lattice_Inf A] :
complete_lattice A :=
⦃ complete_lattice, C ⦄
namespace complete_lattice_Sup
@ -109,26 +130,63 @@ suppose a ∈ s, Sup_le
lemma le_Inf {b : A} {s : set A} (h : ∀ (a : A), a ∈ s → b ≤ a) : b ≤ Inf s :=
le_Sup h
local prefix `⨅`:70 := Inf
local prefix `⨆`:65 := Sup
definition inf (a b : A) := ⨅ '{a, b}
definition sup (a b : A) := ⨆ '{a, b}
local infix `⊓` := inf
local infix `⊔` := sup
lemma inf_le_left (a b : A) : a ⊓ b ≤ a :=
Inf_le !mem_insert
lemma inf_le_right (a b : A) : a ⊓ b ≤ b :=
Inf_le (!mem_insert_of_mem !mem_insert)
lemma le_inf {a b c : A} : c ≤ a → c ≤ b → c ≤ a ⊓ b :=
assume h₁ h₂,
le_Inf (take x, suppose x ∈ '{a, b},
or.elim (eq_or_mem_of_mem_insert this)
(suppose x = a, begin subst x, exact h₁ end)
(suppose x ∈ '{b},
assert x = b, from !eq_of_mem_singleton this,
begin subst x, exact h₂ end))
lemma le_sup_left (a b : A) : a ≤ a ⊔ b :=
le_Sup !mem_insert
lemma le_sup_right (a b : A) : b ≤ a ⊔ b :=
le_Sup (!mem_insert_of_mem !mem_insert)
lemma sup_le {a b c : A} : a ≤ c → b ≤ c → a ⊔ b ≤ c :=
assume h₁ h₂,
Sup_le (take x, suppose x ∈ '{a, b},
or.elim (eq_or_mem_of_mem_insert this)
(assume H : x = a, by subst x; exact h₁)
(suppose x ∈ '{b},
assert x = b, from !eq_of_mem_singleton this,
by subst x; exact h₂))
end complete_lattice_Sup
-- Every complete_lattice_Sup is a complete_lattice_Inf
definition complete_lattice_Sup_to_complete_lattice_Inf [C : complete_lattice_Sup A] : complete_lattice_Inf A :=
⦃ complete_lattice_Inf, C ⦄
-- Every complete_lattice_Sup is a complete_lattice
section
local attribute complete_lattice_Sup_to_complete_lattice_Inf [instance]
definition complete_lattice_Sup_to_complete_lattice [instance] [C : complete_lattice_Sup A] : complete_lattice A :=
_
definition complete_lattice_Sup_to_complete_lattice [trans_instance] [reducible] [C : complete_lattice_Sup A] :
complete_lattice A :=
⦃ complete_lattice, C ⦄
end
namespace complete_lattice
section complete_lattice
variable [C : complete_lattice A]
include C
prefix `⨅`:70 := Inf
prefix `⨆`:65 := Sup
infix ` ⊓ ` := inf
infix ` ⊔ ` := sup
variable {f : A → A}
premise (mono : ∀ x y : A, x ≤ y → f x ≤ f y)
@ -178,6 +236,8 @@ have h₂ : ∀ b, f b = b → b ≤ a, from
le_Sup this,
exists.intro a (and.intro h₁ h₂)
/- top and bot -/
definition bot : A := ⨅ univ
definition top : A := ⨆ univ
notation `⊥` := bot
@ -199,6 +259,8 @@ assume h,
have ≤ a, from Sup_le (take b bin, h b),
le.antisymm !le_top this
/- general facts about complete lattices -/
lemma Inf_singleton {a : A} : ⨅'{a} = a :=
have ⨅'{a} ≤ a, from
Inf_le !mem_insert,
@ -269,6 +331,12 @@ have le₂ : ⨅ univ ≤ ⨆ (∅ : set A), from
Inf_le !mem_univ,
le.antisymm le₁ le₂
lemma Sup_pair (a b : A) : Sup '{a, b} = sup a b :=
by rewrite [insert_eq, Sup_union, *Sup_singleton]
lemma Inf_pair (a b : A) : Inf '{a, b} = inf a b :=
by rewrite [insert_eq, Inf_union, *Inf_singleton]
end complete_lattice
/- complete lattice instances -/