88 lines
2.5 KiB
Text
88 lines
2.5 KiB
Text
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/-
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Copyright (c) 2015 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Leonardo de Moura
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Complete lattices
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-/
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import algebra.lattice data.set
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open set
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namespace algebra
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variable {A : Type}
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structure complete_lattice [class] (A : Type) extends lattice A :=
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(Inf : set A → A)
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(Inf_le : ∀ {a : A} {s : set A}, a ∈ s → le (Inf s) a)
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(le_Inf : ∀ {b : A} {s : set A}, (∀ (a : A), a ∈ s → le b a) → le b (Inf s))
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namespace complete_lattice
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variable [C : complete_lattice A]
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include C
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definition Sup (s : set A) : A :=
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Inf {b | ∀ a, a ∈ s → a ≤ b}
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prefix `⨅`:70 := Inf
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prefix `⨆`:65 := Sup
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lemma le_Sup {a : A} {s : set A} : a ∈ s → a ≤ ⨆ s :=
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suppose a ∈ s, le_Inf
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(show ∀ (b : A), (∀ (a : A), a ∈ s → a ≤ b) → a ≤ b, from
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take b, assume h, h a `a ∈ s`)
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lemma Sup_le {b : A} {s : set A} (h : ∀ (a : A), a ∈ s → a ≤ b) : ⨆ s ≤ b :=
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Inf_le h
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variable {f : A → A}
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premise (mono : ∀ x y : A, x ≤ y → f x ≤ f y)
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theorem knaster_tarski : ∃ a, f a = a ∧ ∀ b, f b = b → a ≤ b :=
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let a := ⨅ {u | f u ≤ u} in
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have h₁ : f a = a, from
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have ge : f a ≤ a, from
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have ∀ b, b ∈ {u | f u ≤ u} → f a ≤ b, from
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take b, suppose f b ≤ b,
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have a ≤ b, from Inf_le this,
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have f a ≤ f b, from !mono this,
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le.trans `f a ≤ f b` `f b ≤ b`,
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le_Inf this,
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have le : a ≤ f a, from
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have f (f a) ≤ f a, from !mono ge,
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have f a ∈ {u | f u ≤ u}, from this,
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Inf_le this,
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le.antisymm ge le,
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have h₂ : ∀ b, f b = b → a ≤ b, from
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take b,
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suppose f b = b,
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have b ∈ {u | f u ≤ u}, from
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show f b ≤ b, by rewrite this; apply le.refl,
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Inf_le this,
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exists.intro a (and.intro h₁ h₂)
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theorem knaster_tarski_dual : ∃ a, f a = a ∧ ∀ b, f b = b → b ≤ a :=
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let a := ⨆ {u | u ≤ f u} in
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have h₁ : f a = a, from
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have le : a ≤ f a, from
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have ∀ b, b ∈ {u | u ≤ f u} → b ≤ f a, from
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take b, suppose b ≤ f b,
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have b ≤ a, from le_Sup this,
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have f b ≤ f a, from !mono this,
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le.trans `b ≤ f b` `f b ≤ f a`,
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Sup_le this,
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have ge : f a ≤ a, from
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have f a ≤ f (f a), from !mono le,
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have f a ∈ {u | u ≤ f u}, from this,
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le_Sup this,
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le.antisymm ge le,
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have h₂ : ∀ b, f b = b → b ≤ a, from
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take b,
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suppose f b = b,
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have b ≤ f b, by rewrite this; apply le.refl,
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le_Sup this,
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exists.intro a (and.intro h₁ h₂)
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end complete_lattice
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end algebra
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