Git rid of the bundles (for now) use IsWhatever
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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Lattice.agda
116
Lattice.agda
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@ -16,14 +16,6 @@ record IsSemilattice {a} (A : Set a) (_⊔_ : A → A → A) : Set a where
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⊔-comm : (x y : A) → x ⊔ y ≡ y ⊔ x
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⊔-comm : (x y : A) → x ⊔ y ≡ y ⊔ x
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⊔-idemp : (x : A) → x ⊔ x ≡ x
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⊔-idemp : (x : A) → x ⊔ x ≡ x
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record Semilattice {a} (A : Set a) : Set (lsuc a) where
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field
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_⊔_ : A → A → A
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isSemilattice : IsSemilattice A _⊔_
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open IsSemilattice isSemilattice public
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record IsLattice {a} (A : Set a) (_⊔_ : A → A → A) (_⊓_ : A → A → A) : Set a where
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record IsLattice {a} (A : Set a) (_⊔_ : A → A → A) (_⊓_ : A → A → A) : Set a where
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field
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field
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joinSemilattice : IsSemilattice A _⊔_
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joinSemilattice : IsSemilattice A _⊔_
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@ -39,6 +31,14 @@ record IsLattice {a} (A : Set a) (_⊔_ : A → A → A) (_⊓_ : A → A → A)
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; ⊔-idemp to ⊓-idemp
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; ⊔-idemp to ⊓-idemp
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)
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)
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record Semilattice {a} (A : Set a) : Set (lsuc a) where
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field
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_⊔_ : A → A → A
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isSemilattice : IsSemilattice A _⊔_
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open IsSemilattice isSemilattice public
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record Lattice {a} (A : Set a) : Set (lsuc a) where
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record Lattice {a} (A : Set a) : Set (lsuc a) where
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field
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field
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@ -49,74 +49,67 @@ record Lattice {a} (A : Set a) : Set (lsuc a) where
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open IsLattice isLattice public
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open IsLattice isLattice public
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module SemilatticeInstances where
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module IsSemilatticeInstances where
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module ForNat where
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module ForNat where
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open Nat
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open Nat
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open NatProps
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open NatProps
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open Eq
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open Eq
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NatMaxSemilattice : Semilattice ℕ
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NatIsMaxSemilattice : IsSemilattice ℕ _⊔_
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NatMaxSemilattice = record
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NatIsMaxSemilattice = record
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{ _⊔_ = _⊔_
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; isSemilattice = record
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{ ⊔-assoc = ⊔-assoc
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{ ⊔-assoc = ⊔-assoc
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; ⊔-comm = ⊔-comm
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; ⊔-comm = ⊔-comm
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; ⊔-idemp = ⊔-idem
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; ⊔-idemp = ⊔-idem
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}
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}
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}
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NatMinSemilattice : Semilattice ℕ
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NatIsMinSemilattice : IsSemilattice ℕ _⊓_
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NatMinSemilattice = record
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NatIsMinSemilattice = record
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{ _⊔_ = _⊓_
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; isSemilattice = record
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{ ⊔-assoc = ⊓-assoc
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{ ⊔-assoc = ⊓-assoc
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; ⊔-comm = ⊓-comm
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; ⊔-comm = ⊓-comm
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; ⊔-idemp = ⊓-idem
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; ⊔-idemp = ⊓-idem
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}
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}
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}
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module ForProd {a} {A B : Set a} (sA : Semilattice A) (sB : Semilattice B) where
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module ForProd {a} {A B : Set a}
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(_⊔₁_ : A → A → A) (_⊔₂_ : B → B → B)
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(sA : IsSemilattice A _⊔₁_) (sB : IsSemilattice B _⊔₂_) where
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open Eq
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open Eq
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open Data.Product
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open Data.Product
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private
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private
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_⊔_ : A × B → A × B → A × B
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_⊔_ : A × B → A × B → A × B
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(a₁ , b₁) ⊔ (a₂ , b₂) = (Semilattice._⊔_ sA a₁ a₂ , Semilattice._⊔_ sB b₁ b₂)
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(a₁ , b₁) ⊔ (a₂ , b₂) = (a₁ ⊔₁ a₂ , b₁ ⊔₂ b₂)
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⊔-assoc : (p₁ p₂ p₃ : A × B) → (p₁ ⊔ p₂) ⊔ p₃ ≡ p₁ ⊔ (p₂ ⊔ p₃)
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⊔-assoc : (p₁ p₂ p₃ : A × B) → (p₁ ⊔ p₂) ⊔ p₃ ≡ p₁ ⊔ (p₂ ⊔ p₃)
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⊔-assoc (a₁ , b₁) (a₂ , b₂) (a₃ , b₃)
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⊔-assoc (a₁ , b₁) (a₂ , b₂) (a₃ , b₃)
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rewrite Semilattice.⊔-assoc sA a₁ a₂ a₃
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rewrite IsSemilattice.⊔-assoc sA a₁ a₂ a₃
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rewrite Semilattice.⊔-assoc sB b₁ b₂ b₃ = refl
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rewrite IsSemilattice.⊔-assoc sB b₁ b₂ b₃ = refl
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⊔-comm : (p₁ p₂ : A × B) → p₁ ⊔ p₂ ≡ p₂ ⊔ p₁
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⊔-comm : (p₁ p₂ : A × B) → p₁ ⊔ p₂ ≡ p₂ ⊔ p₁
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⊔-comm (a₁ , b₁) (a₂ , b₂)
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⊔-comm (a₁ , b₁) (a₂ , b₂)
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rewrite Semilattice.⊔-comm sA a₁ a₂
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rewrite IsSemilattice.⊔-comm sA a₁ a₂
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rewrite Semilattice.⊔-comm sB b₁ b₂ = refl
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rewrite IsSemilattice.⊔-comm sB b₁ b₂ = refl
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⊔-idemp : (p : A × B) → p ⊔ p ≡ p
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⊔-idemp : (p : A × B) → p ⊔ p ≡ p
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⊔-idemp (a , b)
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⊔-idemp (a , b)
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rewrite Semilattice.⊔-idemp sA a
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rewrite IsSemilattice.⊔-idemp sA a
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rewrite Semilattice.⊔-idemp sB b = refl
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rewrite IsSemilattice.⊔-idemp sB b = refl
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ProdSemilattice : Semilattice (A × B)
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ProdIsSemilattice : IsSemilattice (A × B) _⊔_
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ProdSemilattice = record
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ProdIsSemilattice = record
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{ _⊔_ = _⊔_
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; isSemilattice = record
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{ ⊔-assoc = ⊔-assoc
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{ ⊔-assoc = ⊔-assoc
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; ⊔-comm = ⊔-comm
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; ⊔-comm = ⊔-comm
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; ⊔-idemp = ⊔-idemp
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; ⊔-idemp = ⊔-idemp
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}
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}
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}
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module LatticeInstances where
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module IsLatticeInstances where
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module ForNat where
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module ForNat where
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open Nat
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open Nat
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open NatProps
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open NatProps
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open Eq
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open Eq
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open SemilatticeInstances.ForNat
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open IsSemilatticeInstances.ForNat
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open Data.Product
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open Data.Product
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private
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private
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max-bound₁ : {x y z : ℕ} → x ⊔ y ≡ z → x ≤ z
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max-bound₁ : {x y z : ℕ} → x ⊔ y ≡ z → x ≤ z
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max-bound₁ {x} {y} {z} x⊔y≡z rewrite sym x⊔y≡z rewrite ⊔-comm x y = m≤n⇒m≤o⊔n y (≤-refl)
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max-bound₁ {x} {y} {z} x⊔y≡z rewrite sym x⊔y≡z rewrite ⊔-comm x y = m≤n⇒m≤o⊔n y (≤-refl)
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@ -144,29 +137,28 @@ module LatticeInstances where
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helper : x ⊔ (x ⊓ y) ≤ x ⊔ x → x ⊔ x ≡ x → x ⊔ (x ⊓ y) ≤ x
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helper : x ⊔ (x ⊓ y) ≤ x ⊔ x → x ⊔ x ≡ x → x ⊔ (x ⊓ y) ≤ x
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helper x⊔x⊓y≤x⊔x x⊔x≡x rewrite x⊔x≡x = x⊔x⊓y≤x⊔x
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helper x⊔x⊓y≤x⊔x x⊔x≡x rewrite x⊔x≡x = x⊔x⊓y≤x⊔x
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NatLattice : Lattice ℕ
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NatIsLattice : IsLattice ℕ _⊔_ _⊓_
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NatLattice = record
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NatIsLattice = record
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{ _⊔_ = _⊔_
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{ joinSemilattice = NatIsMaxSemilattice
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; _⊓_ = _⊓_
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; meetSemilattice = NatIsMinSemilattice
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; isLattice = record
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{ joinSemilattice = Semilattice.isSemilattice NatMaxSemilattice
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; meetSemilattice = Semilattice.isSemilattice NatMinSemilattice
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; absorb-⊔-⊓ = λ x y → maxmin-absorb {x} {y}
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; absorb-⊔-⊓ = λ x y → maxmin-absorb {x} {y}
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; absorb-⊓-⊔ = λ x y → minmax-absorb {x} {y}
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; absorb-⊓-⊔ = λ x y → minmax-absorb {x} {y}
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}
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}
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}
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module ForProd {a} {A B : Set a} (lA : Lattice A) (lB : Lattice B) where
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module ForProd {a} {A B : Set a}
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(_⊔₁_ : A → A → A) (_⊓₁_ : A → A → A)
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(_⊔₂_ : B → B → B) (_⊓₂_ : B → B → B)
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(lA : IsLattice A _⊔₁_ _⊓₁_) (lB : IsLattice B _⊔₂_ _⊓₂_) where
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private
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private
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module ProdJoin = SemilatticeInstances.ForProd
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module ProdJoin = IsSemilatticeInstances.ForProd _⊔₁_ _⊔₂_ (IsLattice.joinSemilattice lA) (IsLattice.joinSemilattice lB)
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record { _⊔_ = Lattice._⊔_ lA; isSemilattice = Lattice.joinSemilattice lA }
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module ProdMeet = IsSemilatticeInstances.ForProd _⊓₁_ _⊓₂_ (IsLattice.meetSemilattice lA) (IsLattice.meetSemilattice lB)
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record { _⊔_ = Lattice._⊔_ lB; isSemilattice = Lattice.joinSemilattice lB }
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module ProdMeet = SemilatticeInstances.ForProd
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record { _⊔_ = Lattice._⊓_ lA; isSemilattice = Lattice.meetSemilattice lA }
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record { _⊔_ = Lattice._⊓_ lB; isSemilattice = Lattice.meetSemilattice lB }
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_⊔_ = Semilattice._⊔_ ProdJoin.ProdSemilattice
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_⊔_ : (A × B) → (A × B) → (A × B)
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_⊓_ = Semilattice._⊔_ ProdMeet.ProdSemilattice
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(a₁ , b₁) ⊔ (a₂ , b₂) = (a₁ ⊔₁ a₂ , b₁ ⊔₂ b₂)
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_⊓_ : (A × B) → (A × B) → (A × B)
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(a₁ , b₁) ⊓ (a₂ , b₂) = (a₁ ⊓₁ a₂ , b₁ ⊓₂ b₂)
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open Eq
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open Eq
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open Data.Product
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open Data.Product
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@ -174,22 +166,18 @@ module LatticeInstances where
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private
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private
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absorb-⊔-⊓ : (p₁ p₂ : A × B) → p₁ ⊔ (p₁ ⊓ p₂) ≡ p₁
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absorb-⊔-⊓ : (p₁ p₂ : A × B) → p₁ ⊔ (p₁ ⊓ p₂) ≡ p₁
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absorb-⊔-⊓ (a₁ , b₁) (a₂ , b₂)
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absorb-⊔-⊓ (a₁ , b₁) (a₂ , b₂)
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rewrite Lattice.absorb-⊔-⊓ lA a₁ a₂
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rewrite IsLattice.absorb-⊔-⊓ lA a₁ a₂
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rewrite Lattice.absorb-⊔-⊓ lB b₁ b₂ = refl
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rewrite IsLattice.absorb-⊔-⊓ lB b₁ b₂ = refl
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absorb-⊓-⊔ : (p₁ p₂ : A × B) → p₁ ⊓ (p₁ ⊔ p₂) ≡ p₁
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absorb-⊓-⊔ : (p₁ p₂ : A × B) → p₁ ⊓ (p₁ ⊔ p₂) ≡ p₁
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absorb-⊓-⊔ (a₁ , b₁) (a₂ , b₂)
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absorb-⊓-⊔ (a₁ , b₁) (a₂ , b₂)
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rewrite Lattice.absorb-⊓-⊔ lA a₁ a₂
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rewrite IsLattice.absorb-⊓-⊔ lA a₁ a₂
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rewrite Lattice.absorb-⊓-⊔ lB b₁ b₂ = refl
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rewrite IsLattice.absorb-⊓-⊔ lB b₁ b₂ = refl
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ProdLattice : Lattice (A × B)
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ProdIsLattice : IsLattice (A × B) _⊔_ _⊓_
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ProdLattice = record
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ProdIsLattice = record
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{ _⊔_ = _⊔_
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{ joinSemilattice = ProdJoin.ProdIsSemilattice
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; _⊓_ = _⊓_
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; meetSemilattice = ProdMeet.ProdIsSemilattice
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; isLattice = record
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{ joinSemilattice = Semilattice.isSemilattice ProdJoin.ProdSemilattice
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; meetSemilattice = Semilattice.isSemilattice ProdMeet.ProdSemilattice
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; absorb-⊔-⊓ = absorb-⊔-⊓
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; absorb-⊔-⊓ = absorb-⊔-⊓
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; absorb-⊓-⊔ = absorb-⊓-⊔
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; absorb-⊓-⊔ = absorb-⊓-⊔
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}
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}
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}
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