Equivalence.agda

@@ -4,18 +4,8 @@ open import Data.Product using (_×_; Σ; _,_; proj₁; proj₂)
 open import Relation.Binary.Definitions
 open import Relation.Binary.PropositionalEquality as Eq using (_≡_; sym)
 
-module _ {a} (A : Set a) (_≈_ : A → A → Set a) where
+record IsEquivalence {a} (A : Set a) (_≈_ : A → A → Set a) : Set a where
-    IsReflexive : Set a
+    field
-    IsReflexive = ∀ {a : A} → a ≈ a
+        ≈-refl : {a : A} → a ≈ a
-
+        ≈-sym : {a b : A} → a ≈ b → b ≈ a
-    IsSymmetric : Set a
+        ≈-trans : {a b c : A} → a ≈ b → b ≈ c → a ≈ c
-    IsSymmetric = ∀ {a b : A} → a ≈ b → b ≈ a
-
-    IsTransitive : Set a
-    IsTransitive = ∀ {a b c : A} → a ≈ b → b ≈ c → a ≈ c
-
-    record IsEquivalence : Set a where
-        field
-            ≈-refl : IsReflexive
-            ≈-sym : IsSymmetric
-            ≈-trans : IsTransitive

Isomorphism.agda

@@ -1,61 +0,0 @@
-module Isomorphism where
-
-open import Agda.Primitive using (Level) renaming (_⊔_ to _⊔ℓ_)
-open import Function.Definitions using (Inverseˡ; Inverseʳ; Injective)
-open import Lattice
-open import Equivalence
-open import Relation.Binary.Core using (_Preserves_⟶_ )
-open import Data.Nat using (ℕ)
-open import Data.Product using (_,_)
-open import Relation.Nullary using (¬_)
-
-module TransportFiniteHeight
-         {a b : Level} {A : Set a} {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}
-         {height : ℕ}
-         (fhlA : IsFiniteHeightLattice A height _≈₁_ _⊔₁_ _⊓₁_) (lB : IsLattice B _≈₂_ _⊔₂_ _⊓₂_)
-         {f : A → B} {g : B → A}
-         (f-preserves-≈₁ : f Preserves _≈₁_ ⟶  _≈₂_) (g-preserves-≈₂ : g Preserves _≈₂_ ⟶  _≈₁_)
-         (f-⊔-distr : ∀ (a₁ a₂ : A) → f (a₁ ⊔₁ a₂) ≈₂ ((f a₁) ⊔₂ (f a₂)))
-         (g-⊔-distr : ∀ (b₁ b₂ : B) → g (b₁ ⊔₂ b₂) ≈₁ ((g b₁) ⊔₁ (g b₂)))
-         (inverseˡ : Inverseˡ _≈₁_ _≈₂_ f g) (inverseʳ : Inverseʳ _≈₁_ _≈₂_ f g) where
-
-    open IsFiniteHeightLattice fhlA using () renaming (_≺_ to _≺₁_; ≺-cong to ≺₁-cong; ≈-equiv to ≈₁-equiv)
-    open IsLattice lB               using () renaming (_≺_ to _≺₂_; ≺-cong to ≺₂-cong; ≈-equiv to ≈₂-equiv)
-
-    open IsEquivalence ≈₁-equiv using () renaming (≈-sym to ≈₁-sym; ≈-trans to ≈₁-trans)
-    open IsEquivalence ≈₂-equiv using () renaming (≈-sym to ≈₂-sym; ≈-trans to ≈₂-trans)
-
-    open import Chain _≈₁_ ≈₁-equiv _≺₁_ ≺₁-cong using () renaming (Chain to Chain₁; done to done₁; step to step₁)
-    open import Chain _≈₂_ ≈₂-equiv _≺₂_ ≺₂-cong using () renaming (Chain to Chain₂; done to done₂; step to step₂)
-
-    f-Injective : Injective _≈₁_ _≈₂_ f
-    f-Injective {x} {y} fx≈fy = ≈₁-trans (≈₁-sym (inverseʳ x)) (≈₁-trans (g-preserves-≈₂ fx≈fy) (inverseʳ y))
-
-    g-Injective : Injective _≈₂_ _≈₁_ g
-    g-Injective {x} {y} gx≈gy = ≈₂-trans (≈₂-sym (inverseˡ x)) (≈₂-trans (f-preserves-≈₁ gx≈gy) (inverseˡ y))
-
-    f-preserves-̷≈ : f Preserves (λ x y → ¬ x ≈₁ y) ⟶  (λ x y → ¬ x ≈₂ y)
-    f-preserves-̷≈ x̷≈y = λ fx≈fy → x̷≈y (f-Injective fx≈fy)
-
-    g-preserves-̷≈ : g Preserves (λ x y → ¬ x ≈₂ y) ⟶  (λ x y → ¬ x ≈₁ y)
-    g-preserves-̷≈ x̷≈y = λ gx≈gy → x̷≈y (g-Injective gx≈gy)
-
-    portChain₁ : ∀ {a₁ a₂ : A} {h : ℕ} → Chain₁ a₁ a₂ h → Chain₂ (f a₁) (f a₂) h
-    portChain₁ (done₁ a₁≈a₂) = done₂ (f-preserves-≈₁ a₁≈a₂)
-    portChain₁ (step₁ {a₁} {a₂} (a₁≼a₂ , a₁̷≈a₂) a₂≈a₂' c) = step₂ (≈₂-trans (≈₂-sym (f-⊔-distr a₁ a₂)) (f-preserves-≈₁ a₁≼a₂) , f-preserves-̷≈ a₁̷≈a₂) (f-preserves-≈₁ a₂≈a₂') (portChain₁ c)
-
-    portChain₂ : ∀ {b₁ b₂ : B} {h : ℕ} → Chain₂ b₁ b₂ h → Chain₁ (g b₁) (g b₂) h
-    portChain₂ (done₂ a₂≈a₁) = done₁ (g-preserves-≈₂ a₂≈a₁)
-    portChain₂ (step₂ {b₁} {b₂} (b₁≼b₂ , b₁̷≈b₂) b₂≈b₂' c) = step₁ (≈₁-trans (≈₁-sym (g-⊔-distr b₁ b₂)) (g-preserves-≈₂ b₁≼b₂) , g-preserves-̷≈ b₁̷≈b₂) (g-preserves-≈₂ b₂≈b₂') (portChain₂ c)
-
-    isFiniteHeightLattice : IsFiniteHeightLattice B height _≈₂_ _⊔₂_ _⊓₂_
-    isFiniteHeightLattice =
-        let
-            (((a₁ , a₂) , c) , bounded₁) = IsFiniteHeightLattice.fixedHeight fhlA
-        in record
-            { isLattice = lB
-            ; fixedHeight = (((f a₁ , f a₂), portChain₁ c) , λ c' → bounded₁ (portChain₂ c'))
-            }

Lattice/IterProd.agda

@@ -20,7 +20,7 @@ IterProd k = iterate k (λ t → A × t) B
 -- To make iteration more convenient, package the definitions in Lattice
 -- records, perform the recursion, and unpackage.
 
-module _ where
+private module _ where
     lattice : ∀ {k : ℕ} → Lattice (IterProd k)
     lattice {0} = record
         { _≈_ = _≈₂_