25 lines
1.0 KiB
Agda
25 lines
1.0 KiB
Agda
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module Chain where
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open import Data.Nat as Nat using (ℕ; suc; _+_; _≤_)
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open import Data.Product using (_×_; Σ; _,_)
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open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl)
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module _ {a} {A : Set a} (_R_ : A → A → Set a) where
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data Chain : A → A → ℕ → Set a where
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done : ∀ {a : A} → Chain a a 0
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step : ∀ {a₁ a₂ a₃ : A} {n : ℕ} → a₁ R a₂ → Chain a₂ a₃ n → Chain a₁ a₃ (suc n)
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concat : ∀ {a₁ a₂ a₃ : A} {n₁ n₂ : ℕ} → Chain a₁ a₂ n₁ → Chain a₂ a₃ n₂ → Chain a₁ a₃ (n₁ + n₂)
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concat done a₂a₃ = a₂a₃
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concat (step a₁Ra aa₂) a₂a₃ = step a₁Ra (concat aa₂ a₂a₃)
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empty-≡ : ∀ {a₁ a₂ : A} → Chain a₁ a₂ 0 → a₁ ≡ a₂
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empty-≡ done = refl
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Bounded : ℕ → Set a
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Bounded bound = ∀ {a₁ a₂ : A} {n : ℕ} → Chain a₁ a₂ n → n ≤ bound
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Height : ℕ → Set a
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Height height = (Σ (A × A) (λ (a₁ , a₂) → Chain a₁ a₂ height) × Bounded height)
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