Factor the Semilattice instances for Nat into their own module
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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Lattice.agda
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Lattice.agda
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@ -106,20 +106,17 @@ module PreorderInstances where
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_≼_ : A × B → A × B → Set a
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(a₁ , b₁) ≼ (a₂ , b₂) = Preorder._≼_ pA a₁ a₂ × Preorder._≼_ pB b₁ b₂
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ispA = Preorder.isPreorder pA
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ispB = Preorder.isPreorder pB
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≼-refl : {p : A × B} → p ≼ p
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≼-refl {(a , b)} = (IsPreorder.≼-refl ispA {a}, IsPreorder.≼-refl ispB {b})
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≼-refl {(a , b)} = (Preorder.≼-refl pA {a}, Preorder.≼-refl pB {b})
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≼-trans : {p₁ p₂ p₃ : A × B} → p₁ ≼ p₂ → p₂ ≼ p₃ → p₁ ≼ p₃
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≼-trans (a₁≼a₂ , b₁≼b₂) (a₂≼a₃ , b₂≼b₃) =
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( IsPreorder.≼-trans ispA a₁≼a₂ a₂≼a₃
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, IsPreorder.≼-trans ispB b₁≼b₂ b₂≼b₃
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( Preorder.≼-trans pA a₁≼a₂ a₂≼a₃
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, Preorder.≼-trans pB b₁≼b₂ b₂≼b₃
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)
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≼-antisym : {p₁ p₂ : A × B} → p₁ ≼ p₂ → p₂ ≼ p₁ → p₁ ≡ p₂
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≼-antisym (a₁≼a₂ , b₁≼b₂) (a₂≼a₁ , b₂≼b₁) = cong₂ (_,_) (IsPreorder.≼-antisym ispA a₁≼a₂ a₂≼a₁) (IsPreorder.≼-antisym ispB b₁≼b₂ b₂≼b₁)
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≼-antisym (a₁≼a₂ , b₁≼b₂) (a₂≼a₁ , b₂≼b₁) = cong₂ (_,_) (Preorder.≼-antisym pA a₁≼a₂ a₂≼a₁) (Preorder.≼-antisym pB b₁≼b₂ b₂≼b₁)
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ProdPreorder : Preorder (A × B)
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ProdPreorder = record
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@ -131,8 +128,8 @@ module PreorderInstances where
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}
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}
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private module NatInstances where
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module SemilatticeInstances where
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module ForNat where
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open Nat
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open NatProps
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open Eq
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@ -191,12 +188,20 @@ private module NatInstances where
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}
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}
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private module NatInstances where
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open Nat
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open NatProps
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open Eq
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open SemilatticeInstances.ForNat
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open Data.Product
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private
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minmax-absorb : {x y : ℕ} → x ⊓ (x ⊔ y) ≡ x
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minmax-absorb {x} {y} = ≤-antisym x⊓x⊔y≤x (helper x⊓x≤x⊓x⊔y (⊓-idem x))
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where
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x⊓x⊔y≤x = min-bound₁ {x} {x ⊔ y} {x ⊓ (x ⊔ y)} refl
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x⊓x≤x⊓x⊔y = ⊓-mono-≤ {x} {x} ≤-refl (max-bound₁ {x} {y} {x ⊔ y} refl)
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x⊓x⊔y≤x = proj₁ (Semilattice.⊔-bound NatMinSemilattice x (x ⊔ y) (x ⊓ (x ⊔ y)) refl)
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x⊓x≤x⊓x⊔y = ⊓-mono-≤ {x} {x} ≤-refl (proj₁ (Semilattice.⊔-bound NatMaxSemilattice x y (x ⊔ y) refl))
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-- >:(
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helper : x ⊓ x ≤ x ⊓ (x ⊔ y) → x ⊓ x ≡ x → x ≤ x ⊓ (x ⊔ y)
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@ -205,8 +210,8 @@ private module NatInstances where
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maxmin-absorb : {x y : ℕ} → x ⊔ (x ⊓ y) ≡ x
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maxmin-absorb {x} {y} = ≤-antisym (helper x⊔x⊓y≤x⊔x (⊔-idem x)) x≤x⊔x⊓y
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where
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x≤x⊔x⊓y = max-bound₁ {x} {x ⊓ y} {x ⊔ (x ⊓ y)} refl
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x⊔x⊓y≤x⊔x = ⊔-mono-≤ {x} {x} ≤-refl (min-bound₁ {x} {y} {x ⊓ y} refl)
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x≤x⊔x⊓y = proj₁ (Semilattice.⊔-bound NatMaxSemilattice x (x ⊓ y) (x ⊔ (x ⊓ y)) refl)
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x⊔x⊓y≤x⊔x = ⊔-mono-≤ {x} {x} ≤-refl (proj₁ (Semilattice.⊔-bound NatMinSemilattice x y (x ⊓ y) refl))
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-- >:(
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helper : x ⊔ (x ⊓ y) ≤ x ⊔ x → x ⊔ x ≡ x → x ⊔ (x ⊓ y) ≤ x
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