agda-spa/Lattice/Nat.agda

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module Lattice.Nat where
open import Equivalence
open import Lattice
open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; trans)
open import Data.Nat using (; _⊔_; _⊓_; _≤_)
open import Data.Nat.Properties using
( ⊔-assoc; ⊔-comm; ⊔-idem
; ⊓-assoc; ⊓-comm; ⊓-idem
; ⊓-mono-≤; ⊔-mono-≤
; m≤n⇒m≤o⊔n; m≤n⇒m⊓o≤n; ≤-refl; ≤-antisym
)
private
≡-⊔-cong : {a₁ a₂ a₃ a₄} a₁ a₂ a₃ a₄ (a₁ a₃) (a₂ a₄)
≡-⊔-cong a₁≡a₂ a₃≡a₄ rewrite a₁≡a₂ rewrite a₃≡a₄ = refl
≡-⊓-cong : {a₁ a₂ a₃ a₄} a₁ a₂ a₃ a₄ (a₁ a₃) (a₂ a₄)
≡-⊓-cong a₁≡a₂ a₃≡a₄ rewrite a₁≡a₂ rewrite a₃≡a₄ = refl
NatIsMaxSemilattice : IsSemilattice _≡_ _⊔_
NatIsMaxSemilattice = record
{ ≈-equiv = record
{ ≈-refl = refl
; ≈-sym = sym
; ≈-trans = trans
}
; ≈-⊔-cong = ≡-⊔-cong
; ⊔-assoc = ⊔-assoc
; ⊔-comm = ⊔-comm
; ⊔-idemp = ⊔-idem
}
NatIsMinSemilattice : IsSemilattice _≡_ _⊓_
NatIsMinSemilattice = record
{ ≈-equiv = record
{ ≈-refl = refl
; ≈-sym = sym
; ≈-trans = trans
}
; ≈-⊔-cong = ≡-⊓-cong
; ⊔-assoc = ⊓-assoc
; ⊔-comm = ⊓-comm
; ⊔-idemp = ⊓-idem
}
private
max-bound₁ : {x y z : } x y z x z
max-bound₁ {x} {y} {z} x⊔y≡z
rewrite sym x⊔y≡z
rewrite ⊔-comm x y = m≤n⇒m≤o⊔n y (≤-refl)
min-bound₁ : {x y z : } x y z z x
min-bound₁ {x} {y} {z} x⊓y≡z
rewrite sym x⊓y≡z = m≤n⇒m⊓o≤n y (≤-refl)
minmax-absorb : {x y : } x (x y) x
minmax-absorb {x} {y} = ≤-antisym x⊓x⊔y≤x (helper x⊓x≤x⊓x⊔y (⊓-idem x))
where
x⊓x⊔y≤x = min-bound₁ {x} {x y} {x (x y)} refl
x⊓x≤x⊓x⊔y = ⊓-mono-≤ {x} {x} ≤-refl (max-bound₁ {x} {y} {x y} refl)
-- >:(
helper : x x x (x y) x x x x x (x y)
helper x⊓x≤x⊓x⊔y x⊓x≡x rewrite x⊓x≡x = x⊓x≤x⊓x⊔y
maxmin-absorb : {x y : } x (x y) x
maxmin-absorb {x} {y} = ≤-antisym (helper x⊔x⊓y≤x⊔x (⊔-idem x)) x≤x⊔x⊓y
where
x≤x⊔x⊓y = max-bound₁ {x} {x y} {x (x y)} refl
x⊔x⊓y≤x⊔x = ⊔-mono-≤ {x} {x} ≤-refl (min-bound₁ {x} {y} {x y} refl)
-- >:(
helper : x (x y) x x x x x x (x y) x
helper x⊔x⊓y≤x⊔x x⊔x≡x rewrite x⊔x≡x = x⊔x⊓y≤x⊔x
NatIsLattice : IsLattice _≡_ _⊔_ _⊓_
NatIsLattice = record
{ joinSemilattice = NatIsMaxSemilattice
; meetSemilattice = NatIsMinSemilattice
; absorb-⊔-⊓ = λ x y maxmin-absorb {x} {y}
; absorb-⊓-⊔ = λ x y minmax-absorb {x} {y}
}