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+{-# OPTIONS --without-K --safe #-}
+
+open import Relation.Binary as B using (Setoid)
+
+module Cfe.Language.Indexed.Construct.Iterate
+  {c ℓ} (over : Setoid c ℓ)
+  where
+
+open Setoid over using () renaming (Carrier to C)
+
+open import Cfe.Language over
+open import Cfe.Language.Indexed.Homogeneous over
+open import Data.List
+open import Data.Nat as ℕ hiding (_⊔_)
+open import Data.Product as Product
+open import Function
+open import Level hiding (Lift) renaming (suc to lsuc)
+open import Relation.Binary.Indexed.Heterogeneous
+open import Relation.Binary.PropositionalEquality as ≡
+
+open IndexedLanguage
+
+iter : ∀ {a} {A : Set a} → (A → A) → ℕ → A → A
+iter f ℕ.zero = id
+iter f (ℕ.suc n) = f ∘ iter f n
+
+Iterate : ∀ {a aℓ} → (Language a aℓ → Language a aℓ) → IndexedLanguage 0ℓ 0ℓ a aℓ
+Iterate {a} {aℓ} f = record
+  { Carrierᵢ = ℕ
+  ; _≈ᵢ_ = ≡._≡_
+  ; isEquivalenceᵢ = ≡.isEquivalence
+  ; F = λ n → iter f n (Lift a aℓ ∅)
+  ; cong = λ {≡.refl → ≈-refl}
+  }
+
+≈ᵗ-refl : ∀ {a aℓ} →
+          (g : Language a aℓ → Language a aℓ) →
+          Reflexive (Tagged (Iterate g)) (_≈ᵗ_ (Iterate g))
+≈ᵗ-refl g {_} {n , _} = refl , (≈ᴸ-refl (Iter.F n))
+  where
+  module Iter = IndexedLanguage (Iterate g)
+
+≈ᵗ-sym : ∀ {a aℓ} →
+         (g : Language a aℓ → Language a aℓ) →
+         Symmetric (Tagged (Iterate g)) (_≈ᵗ_ (Iterate g))
+≈ᵗ-sym g {_} {_} {n , _} (refl , x∈Fn≈y∈Fn) =
+  refl , (≈ᴸ-sym (Iter.F n) x∈Fn≈y∈Fn)
+  where
+  module Iter = IndexedLanguage (Iterate g)
+
+≈ᵗ-trans : ∀ {a aℓ} →
+          (g : Language a aℓ → Language a aℓ) →
+          Transitive (Tagged (Iterate g)) (_≈ᵗ_ (Iterate g))
+≈ᵗ-trans g {_} {_} {_} {n , _} (refl , x∈Fn≈y∈Fn) (refl , y∈Fn≈z∈Fn) =
+  refl , (≈ᴸ-trans (Iter.F n) x∈Fn≈y∈Fn y∈Fn≈z∈Fn)
+  where
+  module Iter = IndexedLanguage (Iterate g)
+
+⋃ : ∀ {a aℓ} → (Language a aℓ → Language a aℓ) → Language a aℓ
+⋃ f = record
+  { Carrier = Iter.Tagged
+  ; _≈_ = Iter._≈ᵗ_
+  ; isEquivalence = record
+    { refl = ≈ᵗ-refl f
+    ; sym = ≈ᵗ-sym f
+    ; trans = ≈ᵗ-trans f
+    }
+  }
+  where
+  module Iter = IndexedLanguage (Iterate f)