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agda-spa/Lattice/AboveBelow.agda
Danila Fedorin 040c13caba Use instances to simplify printing code 
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
2024-03-11 12:50:05 -07:00

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open import Lattice
open import Equivalence
open import Relation.Nullary using (Dec; ¬_; yes; no)
module Lattice.AboveBelow {a} (A : Set a)
(_≈₁_ : A → A → Set a)
(≈₁-equiv : IsEquivalence A _≈₁_)
(≈₁-dec : IsDecidable _≈₁_) where
open import Data.Empty using (⊥-elim)
open import Data.Product using (_,_)
open import Data.Nat using (_≤_; ℕ; z≤n; s≤s; suc)
open import Function using (_∘_)
open import Showable using (Showable; show)
open import Relation.Binary.PropositionalEquality as Eq
using (_≡_; sym; subst; refl)
import Chain
open IsEquivalence ≈₁-equiv using ()
renaming (≈-refl to ≈₁-refl; ≈-sym to ≈₁-sym; ≈-trans to ≈₁-trans)
data AboveBelow : Set a where
⊥ : AboveBelow
⊤ : AboveBelow
[_] : A → AboveBelow
instance
showable : {{ showableA : Showable A }} → Showable AboveBelow
showable = record
{ show = (λ
{ ⊥ → "⊥"
; ⊤ → "⊤"
; [ a ] → show a
})
}
data _≈_ : AboveBelow → AboveBelow → Set a where
≈-⊥-⊥ : ⊥ ≈ ⊥
≈-⊤-⊤ : ⊤ ≈ ⊤
≈-lift : ∀ {x y : A} → x ≈₁ y → [ x ] ≈ [ y ]
≈-refl : ∀ {ab : AboveBelow} → ab ≈ ab
≈-refl {⊥} = ≈-⊥-⊥
≈-refl {⊤} = ≈-⊤-⊤
≈-refl {[ x ]} = ≈-lift ≈₁-refl
≈-sym : ∀ {ab₁ ab₂ : AboveBelow} → ab₁ ≈ ab₂ → ab₂ ≈ ab₁
≈-sym ≈-⊥-⊥ = ≈-⊥-⊥
≈-sym ≈-⊤-⊤ = ≈-⊤-⊤
≈-sym (≈-lift x≈₁y) = ≈-lift (≈₁-sym x≈₁y)
≈-trans : ∀ {ab₁ ab₂ ab₃ : AboveBelow} → ab₁ ≈ ab₂ → ab₂ ≈ ab₃ → ab₁ ≈ ab₃
≈-trans ≈-⊥-⊥ ≈-⊥-⊥ = ≈-⊥-⊥
≈-trans ≈-⊤-⊤ ≈-⊤-⊤ = ≈-⊤-⊤
≈-trans (≈-lift a₁≈a₂) (≈-lift a₂≈a₃) = ≈-lift (≈₁-trans a₁≈a₂ a₂≈a₃)
≈-equiv : IsEquivalence AboveBelow _≈_
≈-equiv = record
{ ≈-refl = ≈-refl
; ≈-sym = ≈-sym
; ≈-trans = ≈-trans
}
≈-dec : IsDecidable _≈_
≈-dec ⊥ ⊥ = yes ≈-⊥-⊥
≈-dec ⊤ ⊤ = yes ≈-⊤-⊤
≈-dec [ x ] [ y ]
with ≈₁-dec x y
... | yes x≈y = yes (≈-lift x≈y)
... | no x̷≈y = no (λ { (≈-lift x≈y) → x̷≈y x≈y })
≈-dec ⊤ ⊥ = no λ ()
≈-dec ⊤ [ x ] = no λ ()
≈-dec ⊥ ⊤ = no λ ()
≈-dec ⊥ [ x ] = no λ ()
≈-dec [ x ] ⊥ = no λ ()
≈-dec [ x ] ⊤ = no λ ()
-- Any object can be wrapped in an 'above below' to make it a lattice,
-- since ⊤ and ⊥ are the largest and least elements, and the rest are left
-- unordered. That's what this module does.
--
-- For convenience, ask for the underlying type to always be inhabited, to
-- avoid requiring additional constraints in some of the proofs below.
module Plain (x : A) where
_⊔_ : AboveBelow → AboveBelow → AboveBelow
⊥ ⊔ x = x
⊤ ⊔ x = ⊤
[ x ] ⊔ [ y ] with ≈₁-dec x y
... | yes _ = [ x ]
... | no _ = ⊤
x ⊔ ⊥ = x
x ⊔ ⊤ = ⊤
⊤⊔x≡⊤ : ∀ (x : AboveBelow) → ⊤ ⊔ x ≡ ⊤
⊤⊔x≡⊤ _ = refl
x⊔⊤≡⊤ : ∀ (x : AboveBelow) → x ⊔ ⊤ ≡ ⊤
x⊔⊤≡⊤ ⊤ = refl
x⊔⊤≡⊤ ⊥ = refl
x⊔⊤≡⊤ [ x ] = refl
⊥⊔x≡x : ∀ (x : AboveBelow) → ⊥ ⊔ x ≡ x
⊥⊔x≡x _ = refl
x⊔⊥≡x : ∀ (x : AboveBelow) → x ⊔ ⊥ ≡ x
x⊔⊥≡x ⊤ = refl
x⊔⊥≡x ⊥ = refl
x⊔⊥≡x [ x ] = refl
x≈y⇒[x]⊔[y]≡[x] : ∀ {x y : A} → x ≈₁ y → [ x ] ⊔ [ y ] ≡ [ x ]
x≈y⇒[x]⊔[y]≡[x] {x} {y} x≈₁y
with ≈₁-dec x y
... | yes _ = refl
... | no x̷≈₁y = ⊥-elim (x̷≈₁y x≈₁y)
x̷≈y⇒[x]⊔[y]≡⊤ : ∀ {x y : A} → ¬ x ≈₁ y → [ x ] ⊔ [ y ] ≡ ⊤
x̷≈y⇒[x]⊔[y]≡⊤ {x} {y} x̷≈₁y
with ≈₁-dec x y
... | yes x≈₁y = ⊥-elim (x̷≈₁y x≈₁y)
... | no x̷≈₁y = refl
≈-⊔-cong : ∀ {ab₁ ab₂ ab₃ ab₄} → ab₁ ≈ ab₂ → ab₃ ≈ ab₄ →
(ab₁ ⊔ ab₃) ≈ (ab₂ ⊔ ab₄)
≈-⊔-cong ≈-⊤-⊤ ≈-⊤-⊤ = ≈-⊤-⊤
≈-⊔-cong ≈-⊤-⊤ ≈-⊥-⊥ = ≈-⊤-⊤
≈-⊔-cong ≈-⊥-⊥ ≈-⊤-⊤ = ≈-⊤-⊤
≈-⊔-cong ≈-⊥-⊥ ≈-⊥-⊥ = ≈-⊥-⊥
≈-⊔-cong ≈-⊥-⊥ (≈-lift x≈y) = ≈-lift x≈y
≈-⊔-cong (≈-lift x≈y) ≈-⊥-⊥ = ≈-lift x≈y
≈-⊔-cong ≈-⊤-⊤ (≈-lift x≈y) = ≈-⊤-⊤
≈-⊔-cong (≈-lift x≈y) ≈-⊤-⊤ = ≈-⊤-⊤
≈-⊔-cong (≈-lift {a₁} {a₂} a₁≈a₂) (≈-lift {a₃} {a₄} a₃≈a₄)
with ≈₁-dec a₁ a₃ | ≈₁-dec a₂ a₄
... | yes a₁≈a₃ | yes a₂≈a₄ = ≈-lift a₁≈a₂
... | yes a₁≈a₃ | no a₂̷≈a₄ = ⊥-elim (a₂̷≈a₄ (≈₁-trans (≈₁-sym a₁≈a₂) (≈₁-trans (a₁≈a₃) a₃≈a₄)))
... | no a₁̷≈a₃ | yes a₂≈a₄ = ⊥-elim (a₁̷≈a₃ (≈₁-trans a₁≈a₂ (≈₁-trans a₂≈a₄ (≈₁-sym a₃≈a₄))))
... | no _ | no _ = ≈-⊤-⊤
⊔-assoc : ∀ (ab₁ ab₂ ab₃ : AboveBelow) →
((ab₁ ⊔ ab₂) ⊔ ab₃) ≈ (ab₁ ⊔ (ab₂ ⊔ ab₃))
⊔-assoc ⊤ ab₂ ab₃ = ≈-⊤-⊤
⊔-assoc ⊥ ab₂ ab₃ = ≈-refl
⊔-assoc [ x₁ ] ⊤ ab₃ = ≈-⊤-⊤
⊔-assoc [ x₁ ] ⊥ ab₃ = ≈-refl
⊔-assoc [ x₁ ] [ x₂ ] ⊤ rewrite x⊔⊤≡⊤ ([ x₁ ] ⊔ [ x₂ ]) = ≈-⊤-⊤
⊔-assoc [ x₁ ] [ x₂ ] ⊥ rewrite x⊔⊥≡x ([ x₁ ] ⊔ [ x₂ ]) = ≈-refl
⊔-assoc [ x₁ ] [ x₂ ] [ x₃ ]
with ≈₁-dec x₂ x₃ | ≈₁-dec x₁ x₂
... | no x₂̷≈x₃ | no _ rewrite x̷≈y⇒[x]⊔[y]≡⊤ x₂̷≈x₃ = ≈-⊤-⊤
... | no x₂̷≈x₃ | yes x₁≈x₂ rewrite x̷≈y⇒[x]⊔[y]≡⊤ x₂̷≈x₃
rewrite x̷≈y⇒[x]⊔[y]≡⊤ (x₂̷≈x₃ ∘ (≈₁-trans (≈₁-sym x₁≈x₂))) = ≈-⊤-⊤
... | yes x₂≈x₃ | yes x₁≈x₂ rewrite x≈y⇒[x]⊔[y]≡[x] x₂≈x₃
rewrite x≈y⇒[x]⊔[y]≡[x] x₁≈x₂
rewrite x≈y⇒[x]⊔[y]≡[x] (≈₁-trans x₁≈x₂ x₂≈x₃) = ≈-refl
... | yes x₂≈x₃ | no x₁̷≈x₂ rewrite x̷≈y⇒[x]⊔[y]≡⊤ x₁̷≈x₂ = ≈-⊤-⊤
⊔-comm : ∀ (ab₁ ab₂ : AboveBelow) → (ab₁ ⊔ ab₂) ≈ (ab₂ ⊔ ab₁)
⊔-comm ⊤ x rewrite x⊔⊤≡⊤ x = ≈-refl
⊔-comm ⊥ x rewrite x⊔⊥≡x x = ≈-refl
⊔-comm x ⊤ rewrite x⊔⊤≡⊤ x = ≈-refl
⊔-comm x ⊥ rewrite x⊔⊥≡x x = ≈-refl
⊔-comm [ x₁ ] [ x₂ ] with ≈₁-dec x₁ x₂
... | yes x₁≈x₂ rewrite x≈y⇒[x]⊔[y]≡[x] (≈₁-sym x₁≈x₂) = ≈-lift x₁≈x₂
... | no x₁̷≈x₂ rewrite x̷≈y⇒[x]⊔[y]≡⊤ (x₁̷≈x₂ ∘ ≈₁-sym) = ≈-⊤-⊤
⊔-idemp : ∀ ab → (ab ⊔ ab) ≈ ab
⊔-idemp ⊤ = ≈-⊤-⊤
⊔-idemp ⊥ = ≈-⊥-⊥
⊔-idemp [ x ] rewrite x≈y⇒[x]⊔[y]≡[x] (≈₁-refl {x}) = ≈-refl
isJoinSemilattice : IsSemilattice AboveBelow _≈_ _⊔_
isJoinSemilattice = record
{ ≈-equiv = ≈-equiv
; ≈-⊔-cong = ≈-⊔-cong
; ⊔-assoc = ⊔-assoc
; ⊔-comm = ⊔-comm
; ⊔-idemp = ⊔-idemp
}
_⊓_ : AboveBelow → AboveBelow → AboveBelow
⊥ ⊓ x = ⊥
⊤ ⊓ x = x
[ x ] ⊓ [ y ] with ≈₁-dec x y
... | yes _ = [ x ]
... | no _ = ⊥
x ⊓ ⊥ = ⊥
x ⊓ ⊤ = x
⊥⊓x≡⊥ : ∀ (x : AboveBelow) → ⊥ ⊓ x ≡ ⊥
⊥⊓x≡⊥ _ = refl
x⊓⊥≡⊥ : ∀ (x : AboveBelow) → x ⊓ ⊥ ≡ ⊥
x⊓⊥≡⊥ ⊤ = refl
x⊓⊥≡⊥ ⊥ = refl
x⊓⊥≡⊥ [ x ] = refl
⊤⊓x≡x : ∀ (x : AboveBelow) → ⊤ ⊓ x ≡ x
⊤⊓x≡x _ = refl
x⊓⊤≡x : ∀ (x : AboveBelow) → x ⊓ ⊤ ≡ x
x⊓⊤≡x ⊤ = refl
x⊓⊤≡x ⊥ = refl
x⊓⊤≡x [ x ] = refl
x≈y⇒[x]⊓[y]≡[x] : ∀ {x y : A} → x ≈₁ y → [ x ] ⊓ [ y ] ≡ [ x ]
x≈y⇒[x]⊓[y]≡[x] {x} {y} x≈₁y
with ≈₁-dec x y
... | yes _ = refl
... | no x̷≈₁y = ⊥-elim (x̷≈₁y x≈₁y)
x̷≈y⇒[x]⊓[y]≡⊥ : ∀ {x y : A} → ¬ x ≈₁ y → [ x ] ⊓ [ y ] ≡ ⊥
x̷≈y⇒[x]⊓[y]≡⊥ {x} {y} x̷≈₁y
with ≈₁-dec x y
... | yes x≈₁y = ⊥-elim (x̷≈₁y x≈₁y)
... | no x̷≈₁y = refl
≈-⊓-cong : ∀ {ab₁ ab₂ ab₃ ab₄} → ab₁ ≈ ab₂ → ab₃ ≈ ab₄ → (ab₁ ⊓ ab₃) ≈ (ab₂ ⊓ ab₄)
≈-⊓-cong ≈-⊤-⊤ ≈-⊤-⊤ = ≈-⊤-⊤
≈-⊓-cong ≈-⊤-⊤ ≈-⊥-⊥ = ≈-⊥-⊥
≈-⊓-cong ≈-⊥-⊥ ≈-⊤-⊤ = ≈-⊥-⊥
≈-⊓-cong ≈-⊥-⊥ ≈-⊥-⊥ = ≈-⊥-⊥
≈-⊓-cong ≈-⊤-⊤ (≈-lift x≈y) = ≈-lift x≈y
≈-⊓-cong (≈-lift x≈y) ≈-⊤-⊤ = ≈-lift x≈y
≈-⊓-cong ≈-⊥-⊥ (≈-lift x≈y) = ≈-⊥-⊥
≈-⊓-cong (≈-lift x≈y) ≈-⊥-⊥ = ≈-⊥-⊥
≈-⊓-cong (≈-lift {a₁} {a₂} a₁≈a₂) (≈-lift {a₃} {a₄} a₃≈a₄)
with ≈₁-dec a₁ a₃ | ≈₁-dec a₂ a₄
... | yes a₁≈a₃ | yes a₂≈a₄ = ≈-lift a₁≈a₂
... | yes a₁≈a₃ | no a₂̷≈a₄ = ⊥-elim (a₂̷≈a₄ (≈₁-trans (≈₁-sym a₁≈a₂) (≈₁-trans (a₁≈a₃) a₃≈a₄)))
... | no a₁̷≈a₃ | yes a₂≈a₄ = ⊥-elim (a₁̷≈a₃ (≈₁-trans a₁≈a₂ (≈₁-trans a₂≈a₄ (≈₁-sym a₃≈a₄))))
... | no _ | no _ = ≈-⊥-⊥
⊓-assoc : ∀ (ab₁ ab₂ ab₃ : AboveBelow) → ((ab₁ ⊓ ab₂) ⊓ ab₃) ≈ (ab₁ ⊓ (ab₂ ⊓ ab₃))
⊓-assoc ⊥ ab₂ ab₃ = ≈-⊥-⊥
⊓-assoc ⊤ ab₂ ab₃ = ≈-refl
⊓-assoc [ x₁ ] ⊥ ab₃ = ≈-⊥-⊥
⊓-assoc [ x₁ ] ⊤ ab₃ = ≈-refl
⊓-assoc [ x₁ ] [ x₂ ] ⊥ rewrite x⊓⊥≡⊥ ([ x₁ ] ⊓ [ x₂ ]) = ≈-⊥-⊥
⊓-assoc [ x₁ ] [ x₂ ] ⊤ rewrite x⊓⊤≡x ([ x₁ ] ⊓ [ x₂ ]) = ≈-refl
⊓-assoc [ x₁ ] [ x₂ ] [ x₃ ]
with ≈₁-dec x₂ x₃ | ≈₁-dec x₁ x₂
... | no x₂̷≈x₃ | no _ rewrite x̷≈y⇒[x]⊓[y]≡⊥ x₂̷≈x₃ = ≈-⊥-⊥
... | no x₂̷≈x₃ | yes x₁≈x₂ rewrite x̷≈y⇒[x]⊓[y]≡⊥ x₂̷≈x₃
rewrite x̷≈y⇒[x]⊓[y]≡⊥ λ x₁≈x₃ → x₂̷≈x₃ (≈₁-trans (≈₁-sym x₁≈x₂) x₁≈x₃) = ≈-⊥-⊥
... | yes x₂≈x₃ | yes x₁≈x₂ rewrite x≈y⇒[x]⊓[y]≡[x] x₂≈x₃
rewrite x≈y⇒[x]⊓[y]≡[x] x₁≈x₂
rewrite x≈y⇒[x]⊓[y]≡[x] (≈₁-trans x₁≈x₂ x₂≈x₃) = ≈-refl
... | yes x₂≈x₃ | no x₁̷≈x₂ rewrite x̷≈y⇒[x]⊓[y]≡⊥ x₁̷≈x₂ = ≈-⊥-⊥
⊓-comm : ∀ (ab₁ ab₂ : AboveBelow) → (ab₁ ⊓ ab₂) ≈ (ab₂ ⊓ ab₁)
⊓-comm ⊥ x rewrite x⊓⊥≡⊥ x = ≈-refl
⊓-comm ⊤ x rewrite x⊓⊤≡x x = ≈-refl
⊓-comm x ⊥ rewrite x⊓⊥≡⊥ x = ≈-refl
⊓-comm x ⊤ rewrite x⊓⊤≡x x = ≈-refl
⊓-comm [ x₁ ] [ x₂ ] with ≈₁-dec x₁ x₂
... | yes x₁≈x₂ rewrite x≈y⇒[x]⊓[y]≡[x] (≈₁-sym x₁≈x₂) = ≈-lift x₁≈x₂
... | no x₁̷≈x₂ rewrite x̷≈y⇒[x]⊓[y]≡⊥ λ x₂≈x₁ → (x₁̷≈x₂ (≈₁-sym x₂≈x₁)) = ≈-⊥-⊥
⊓-idemp : ∀ ab → (ab ⊓ ab) ≈ ab
⊓-idemp ⊥ = ≈-⊥-⊥
⊓-idemp ⊤ = ≈-⊤-⊤
⊓-idemp [ x ] rewrite x≈y⇒[x]⊓[y]≡[x] (≈₁-refl {x}) = ≈-refl
isMeetSemilattice : IsSemilattice AboveBelow _≈_ _⊓_
isMeetSemilattice = record
{ ≈-equiv = ≈-equiv
; ≈-⊔-cong = ≈-⊓-cong
; ⊔-assoc = ⊓-assoc
; ⊔-comm = ⊓-comm
; ⊔-idemp = ⊓-idemp
}
absorb-⊔-⊓ : ∀ (ab₁ ab₂ : AboveBelow) → (ab₁ ⊔ (ab₁ ⊓ ab₂)) ≈ ab₁
absorb-⊔-⊓ ⊥ ab₂ rewrite ⊥⊓x≡⊥ ab₂ = ≈-⊥-⊥
absorb-⊔-⊓ ⊤ _ = ≈-⊤-⊤
absorb-⊔-⊓ [ x ] ⊥ rewrite x⊓⊥≡⊥ [ x ]
rewrite x⊔⊥≡x [ x ] = ≈-refl
absorb-⊔-⊓ [ x ] ⊤ rewrite x⊓⊤≡x [ x ] = ⊔-idemp _
absorb-⊔-⊓ [ x ] [ y ]
with ≈₁-dec x y
... | yes x≈y rewrite x≈y⇒[x]⊓[y]≡[x] x≈y = ⊔-idemp _
... | no x̷≈y rewrite x̷≈y⇒[x]⊓[y]≡⊥ x̷≈y rewrite x⊔⊥≡x [ x ] = ≈-refl
absorb-⊓-⊔ : ∀ (ab₁ ab₂ : AboveBelow) → (ab₁ ⊓ (ab₁ ⊔ ab₂)) ≈ ab₁
absorb-⊓-⊔ ⊤ ab₂ rewrite ⊤⊔x≡⊤ ab₂ = ≈-⊤-⊤
absorb-⊓-⊔ ⊥ _ = ≈-⊥-⊥
absorb-⊓-⊔ [ x ] ⊤ rewrite x⊔⊤≡⊤ [ x ]
rewrite x⊓⊤≡x [ x ] = ≈-refl
absorb-⊓-⊔ [ x ] ⊥ rewrite x⊔⊥≡x [ x ] = ⊓-idemp _
absorb-⊓-⊔ [ x ] [ y ]
with ≈₁-dec x y
... | yes x≈y rewrite x≈y⇒[x]⊔[y]≡[x] x≈y = ⊓-idemp _
... | no x̷≈y rewrite x̷≈y⇒[x]⊔[y]≡⊤ x̷≈y rewrite x⊓⊤≡x [ x ] = ≈-refl
isLattice : IsLattice AboveBelow _≈_ _⊔_ _⊓_
isLattice = record
{ joinSemilattice = isJoinSemilattice
; meetSemilattice = isMeetSemilattice
; absorb-⊔-⊓ = absorb-⊔-⊓
; absorb-⊓-⊔ = absorb-⊓-⊔
}
lattice : Lattice AboveBelow
lattice = record
{ _≈_ = _≈_
; _⊔_ = _⊔_
; _⊓_ = _⊓_
; isLattice = isLattice
}
open IsLattice isLattice using (_≼_; _≺_; ⊔-Monotonicˡ; ⊔-Monotonicʳ) public
⊥≺[x] : ∀ (x : A) → ⊥ ≺ [ x ]
⊥≺[x] x = (≈-refl , λ ())
x≺[y]⇒x≡⊥ : ∀ (x : AboveBelow) (y : A) → x ≺ [ y ] → x ≡ ⊥
x≺[y]⇒x≡⊥ x y ((x⊔[y]≈[y]) , x̷≈[y]) with x
... | ⊥ = refl
... | ⊤ with () ← x⊔[y]≈[y]
... | [ b ] with ≈₁-dec b y
... | yes b≈y = ⊥-elim (x̷≈[y] (≈-lift b≈y))
... | no _ with () ← x⊔[y]≈[y]
[x]≺⊤ : ∀ (x : A) → [ x ] ≺ ⊤
[x]≺⊤ x rewrite x⊔⊤≡⊤ [ x ] = (≈-⊤-⊤ , λ ())
[x]≺y⇒y≡⊤ : ∀ (x : A) (y : AboveBelow) → [ x ] ≺ y → y ≡ ⊤
[x]≺y⇒y≡⊤ x y ([x]⊔y≈y , [x]̷≈y) with y
... | ⊥ with () ← [x]⊔y≈y
... | ⊤ = refl
... | [ a ] with ≈₁-dec x a
... | yes x≈a = ⊥-elim ([x]̷≈y (≈-lift x≈a))
... | no _ with () ← [x]⊔y≈y
open Chain _≈_ ≈-equiv (IsLattice._≺_ isLattice) (IsLattice.≺-cong isLattice)
longestChain : Chain ⊥ ⊤ 2
longestChain = step (⊥≺[x] x) ≈-refl (step ([x]≺⊤ x) ≈-⊤-⊤ (done ≈-⊤-⊤))
¬-Chain-⊤ : ∀ {ab : AboveBelow} {n : ℕ} → ¬ Chain ⊤ ab (suc n)
¬-Chain-⊤ {x} (step (⊤⊔x≈x , ⊤̷≈x) _ _) rewrite ⊤⊔x≡⊤ x = ⊥-elim (⊤̷≈x ⊤⊔x≈x)
isLongest : ∀ {ab₁ ab₂ : AboveBelow} {n : ℕ} → Chain ab₁ ab₂ n → n ≤ 2
isLongest (done _) = z≤n
isLongest (step _ _ (done _)) = s≤s z≤n
isLongest (step _ _ (step _ _ (done _))) = s≤s (s≤s z≤n)
isLongest {⊤} c@(step _ _ _) = ⊥-elim (¬-Chain-⊤ c)
isLongest {[ x ]} (step {_} {y} [x]≺y y≈y' c@(step _ _ _))
rewrite [x]≺y⇒y≡⊤ x y [x]≺y with ≈-⊤-⊤ ← y≈y' = ⊥-elim (¬-Chain-⊤ c)
isLongest {⊥} (step {_} {⊥} (_ , ⊥̷≈⊥) _ _) = ⊥-elim (⊥̷≈⊥ ≈-⊥-⊥)
isLongest {⊥} (step {_} {⊤} _ ≈-⊤-⊤ c@(step _ _ _)) = ⊥-elim (¬-Chain-⊤ c)
isLongest {⊥} (step {_} {[ x ]} _ (≈-lift _) (step [x]≺y y≈z c@(step _ _ _)))
rewrite [x]≺y⇒y≡⊤ _ _ [x]≺y with ≈-⊤-⊤ ← y≈z = ⊥-elim (¬-Chain-⊤ c)
fixedHeight : IsLattice.FixedHeight isLattice 2
fixedHeight = (((⊥ , ⊤) , longestChain) , isLongest)
isFiniteHeightLattice : IsFiniteHeightLattice AboveBelow 2 _≈_ _⊔_ _⊓_
isFiniteHeightLattice = record
{ isLattice = isLattice
; fixedHeight = fixedHeight
}
finiteHeightLattice : FiniteHeightLattice AboveBelow
finiteHeightLattice = record
{ height = 2
; _≈_ = _≈_
; _⊔_ = _⊔_
; _⊓_ = _⊓_
; isFiniteHeightLattice = isFiniteHeightLattice
}
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