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Add a preorder instance for product

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
Danila Fedorin 2023-07-14 19:42:29 -07:00
parent 2b3d429631
commit c9b514e9af
1 changed files with 48 additions and 11 deletions
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45
Lattice.agda
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@ -85,11 +85,9 @@ record Lattice {a} (A : Set a) : Set (lsuc a) where
open IsLattice isLattice public
private module NatInstances where
open Nat
module PreorderInstances where
module ForNat where
open NatProps
open Eq
open Data.Sum
NatPreorder : Preorder ℕ
NatPreorder = record
@ -101,6 +99,45 @@ private module NatInstances where
}
}
module ForProd {a} {A B : Set a} {{ pA : Preorder A }} {{ pB : Preorder B }} where
open Eq
private
_≼_ : A × B → A × B → Set a
(a₁ , b₁) ≼ (a₂ , b₂) = Preorder._≼_ pA a₁ a₂ × Preorder._≼_ pB b₁ b₂
ispA = Preorder.isPreorder pA
ispB = Preorder.isPreorder pB
≼-refl : {p : A × B} → p ≼ p
≼-refl {(a , b)} = (IsPreorder.≼-refl ispA {a}, IsPreorder.≼-refl ispB {b})
≼-trans : {p₁ p₂ p₃ : A × B} → p₁ ≼ p₂ → p₂ ≼ p₃ → p₁ ≼ p₃
≼-trans (a₁≼a₂ , b₁≼b₂) (a₂≼a₃ , b₂≼b₃) =
( IsPreorder.≼-trans ispA a₁≼a₂ a₂≼a₃
, IsPreorder.≼-trans ispB b₁≼b₂ b₂≼b₃
)
≼-antisym : {p₁ p₂ : A × B} → p₁ ≼ p₂ → p₂ ≼ p₁ → p₁ ≡ p₂
≼-antisym (a₁≼a₂ , b₁≼b₂) (a₂≼a₁ , b₂≼b₁) = cong₂ (_,_) (IsPreorder.≼-antisym ispA a₁≼a₂ a₂≼a₁) (IsPreorder.≼-antisym ispB b₁≼b₂ b₂≼b₁)
ProdPreorder : Preorder (A × B)
ProdPreorder = record
{ _≼_ = _≼_
; isPreorder = record
{ ≼-refl = ≼-refl
; ≼-trans = ≼-trans
; ≼-antisym = ≼-antisym
}
}
private module NatInstances where
open Nat
open NatProps
open Eq
open PreorderInstances.ForNat
private
max-bound₁ : {x y z : ℕ} → 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)