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Prove monotonicity of lub in one argument

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Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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Danila Fedorin 2024-03-01 23:26:25 -08:00
parent ae3e2c28b0
commit 65d1590358
1 changed files with 26 additions and 6 deletions
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Lattice.agda
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@ -14,6 +14,12 @@ open import Function.Definitions using (Injective)
IsDecidable : ∀ {a} {A : Set a} (R : A → A → Set a) → Set a IsDecidable : ∀ {a} {A : Set a} (R : A → A → Set a) → Set a
IsDecidable {a} {A} R = ∀ (a₁ a₂ : A) → Dec (R a₁ a₂) IsDecidable {a} {A} R = ∀ (a₁ a₂ : A) → Dec (R a₁ a₂)
module _ {a b} {A : Set a} {B : Set b}
(_≼₁_ : A → A → Set a) (_≼₂_ : B → B → Set b) where
Monotonic : (A → B) → Set (a ⊔ℓ b)
Monotonic f = ∀ {a₁ a₂ : A} → a₁ ≼₁ a₂ → f a₁ ≼₂ f a₂
record IsSemilattice {a} (A : Set a) record IsSemilattice {a} (A : Set a)
(_≈_ : A → A → Set a) (_≈_ : A → A → Set a)
(_⊔_ : A → A → A) : Set a where (_⊔_ : A → A → A) : Set a where
@ -36,6 +42,26 @@ record IsSemilattice {a} (A : Set a)
open import Relation.Binary.Reasoning.Base.Single _≈_ ≈-refl ≈-trans open import Relation.Binary.Reasoning.Base.Single _≈_ ≈-refl ≈-trans
⊔-Monotonicˡ : ∀ (a₁ : A) → Monotonic _≼_ _≼_ (λ a₂ → a₁ ⊔ a₂)
⊔-Monotonicˡ a {a₁} {a₂} a₁≼a₂ = ≈-trans (≈-sym lhs) (≈-⊔-cong (≈-refl {a}) a₁≼a₂)
where
lhs =
begin
a ⊔ (a₁ ⊔ a₂)
∼⟨ ≈-⊔-cong (≈-sym (⊔-idemp _)) ≈-refl ⟩
(a ⊔ a) ⊔ (a₁ ⊔ a₂)
∼⟨ ⊔-assoc _ _ _ ⟩
a ⊔ (a ⊔ (a₁ ⊔ a₂))
∼⟨ ≈-⊔-cong ≈-refl (≈-sym (⊔-assoc _ _ _)) ⟩
a ⊔ ((a ⊔ a₁) ⊔ a₂)
∼⟨ ≈-⊔-cong ≈-refl (≈-⊔-cong (⊔-comm _ _) ≈-refl) ⟩
a ⊔ ((a₁ ⊔ a) ⊔ a₂)
∼⟨ ≈-⊔-cong ≈-refl (⊔-assoc _ _ _) ⟩
a ⊔ (a₁ ⊔ (a ⊔ a₂))
∼⟨ ≈-sym (⊔-assoc _ _ _) ⟩
(a ⊔ a₁) ⊔ (a ⊔ a₂)
∎
≼-refl : ∀ (a : A) → a ≼ a ≼-refl : ∀ (a : A) → a ≼ a
≼-refl a = ⊔-idemp a ≼-refl a = ⊔-idemp a
@ -97,12 +123,6 @@ record IsFiniteHeightLattice {a} (A : Set a)
field field
fixedHeight : FixedHeight h fixedHeight : FixedHeight h
module _ {a b} {A : Set a} {B : Set b}
(_≼₁_ : A → A → Set a) (_≼₂_ : B → B → Set b) where
Monotonic : (A → B) → Set (a ⊔ℓ b)
Monotonic f = ∀ {a₁ a₂ : A} → a₁ ≼₁ a₂ → f a₁ ≼₂ f a₂
module ChainMapping {a b} {A : Set a} {B : Set b} module ChainMapping {a b} {A : Set a} {B : Set b}
{_≈₁_ : A → A → Set a} {_≈₂_ : B → B → Set b} {_≈₁_ : A → A → Set a} {_≈₂_ : B → B → Set b}
{_⊔₁_ : A → A → A} {_⊔₂_ : B → B → B} {_⊔₁_ : A → A → A} {_⊔₂_ : B → B → B}