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+{-# OPTIONS --without-K --safe #-}
+
+module Cfe.Function.Power where
+
+open import Data.Nat using (ℕ)
+open import Data.Product using (∃-syntax; _×_)
+open import Function using (id; _∘_)
+open import Level using (Level; _⊔_)
+open import Relation.Binary using (Rel; REL)
+open import Relation.Binary.PropositionalEquality using (cong; _≡_)
+
+private
+  variable
+    a : Level
+    A : Set a
+
+infix 10 _^_
+
+------------------------------------------------------------------------
+-- Definition
+
+_^_ : (A → A) → ℕ → A → A
+f ^ ℕ.zero = id
+f ^ ℕ.suc n = f ∘ f ^ n
+
+------------------------------------------------------------------------
+-- Properties
+
+f∼g⇒fⁿ∼gⁿ :
+  ∀ {a b ℓ} {A : Set a} {B : Set b} (_∼_ : REL A B ℓ) {f g 0A 0B} →
+  0A ∼ 0B → (∀ {x y} → x ∼ y → f x ∼ g y) → ∀ n → (f ^ n) 0A ∼ (g ^ n) 0B
+f∼g⇒fⁿ∼gⁿ _ refl ext    ℕ.zero = refl
+f∼g⇒fⁿ∼gⁿ ∼ refl ext (ℕ.suc n) = ext (f∼g⇒fⁿ∼gⁿ ∼ refl ext n)
+
+fⁿf≡fⁿ⁺¹ : ∀ (f : A → A) n → (f ^ n) ∘ f ≡ f ^ (ℕ.suc n)
+fⁿf≡fⁿ⁺¹ f    ℕ.zero = _≡_.refl
+fⁿf≡fⁿ⁺¹ f (ℕ.suc n) = cong (f ∘_) (fⁿf≡fⁿ⁺¹ f n)