DanilaFe/agda-spa
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Generalize chains to allow equivalences

Browse Source 
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
This commit is contained in:
Danila Fedorin
2023-09-03 21:05:57 -07:00
parent 67e96b27cf
commit fb86d3f84f
2 changed files with 54 additions and 23 deletions
Download Patch File Download Diff File Expand all files Collapse all files

32
Chain.agda
View File

@@ -1,21 +1,37 @@
module Chain where
open import Equivalence
module Chain {a} {A : Set a}
(_≈_ : A → A → Set a)
(≈-equiv : IsEquivalence A _≈_)
(_R_ : A → A → Set a)
(R-≈-cong : ∀ {a₁ a₁' a₂ a₂'} → a₁ ≈ a₁' → a₂ ≈ a₂' → a₁ R a₂ → a₁' R a₂') where
open import Data.Nat as Nat using (ℕ; suc; _+_; _≤_)
open import Data.Product using (_×_; Σ; _,_)
open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl)
module _ {a} {A : Set a} (_R_ : A → A → Set a) where
open IsEquivalence ≈-equiv
module _ where
data Chain : A → A → ℕ → Set a where
done : ∀ {a : A} → Chain a a 0
step : ∀ {a₁ a₂ a₃ : A} {n : ℕ} → a₁ R a₂ → Chain a₂ a₃ n → Chain a₁ a₃ (suc n)
done : ∀ {a a' : A} → a ≈ a' → Chain a a' 0
step : ∀ {a₁ a₂ a₂' a₃ : A} {n : ℕ} → a₁ R a₂ → a₂ ≈ a₂' → Chain a₂ a₃ n → Chain a₁ a₃ (suc n)
Chain-≈-cong₁ : ∀ {a₁ a₁' a₂} {n : ℕ} → a₁ ≈ a₁' → Chain a₁ a₂ n → Chain a₁' a₂ n
Chain-≈-cong₁ a₁≈a₁' (done a₁≈a₂) = done (≈-trans (≈-sym a₁≈a₁') a₁≈a₂)
Chain-≈-cong₁ a₁≈a₁' (step a₁Ra a≈a' a'a₂) = step (R-≈-cong a₁≈a₁' ≈-refl a₁Ra) a≈a' a'a₂
Chain-≈-cong₂ : ∀ {a₁ a₂ a₂'} {n : ℕ} → a₂ ≈ a₂' → Chain a₁ a₂ n → Chain a₁ a₂' n
Chain-≈-cong₂ a₂≈a₂' (done a₁≈a₂) = done (≈-trans a₁≈a₂ a₂≈a₂')
Chain-≈-cong₂ a₂≈a₂' (step a₁Ra a≈a' a'a₂) = step a₁Ra a≈a' (Chain-≈-cong₂ a₂≈a₂' a'a₂)
concat : ∀ {a₁ a₂ a₃ : A} {n₁ n₂ : ℕ} → Chain a₁ a₂ n₁ → Chain a₂ a₃ n₂ → Chain a₁ a₃ (n₁ + n₂)
concat done a₂a₃ = a₂a₃
concat (step a₁Ra aa₂) a₂a₃ = step a₁Ra (concat aa₂ a₂a₃)
concat (done a₁≈a₂) a₂a₃ = Chain-≈-cong₁ (≈-sym a₁≈a₂) a₂a₃
concat (step a₁Ra a≈a' a'a₂) a₂a₃ = step a₁Ra a≈a' (concat a'a₂ a₂a₃)
empty-≡ : ∀ {a₁ a₂ : A} → Chain a₁ a₂ 0 → a₁ ≡ a₂
empty-≡ done = refl
empty-≡ : ∀ {a₁ a₂ : A} → Chain a₁ a₂ 0 → a₁ ≈ a₂
empty-≡ (done a₁≈a₂) = a₁≈a₂
Bounded : ℕ → Set a
Bounded bound = ∀ {a₁ a₂ : A} {n : ℕ} → Chain a₁ a₂ n → n ≤ bound