module FreeTheorems where

open import Level using () renaming (zero to ℓ₀)
open import Data.Nat using (ℕ)
open import Data.List using (List ; map)
open import Data.Vec using (Vec) renaming (map to mapV)
open import Function using (_∘_)
open import Function.Equality using (_⟶_ ; _⟨$⟩_)
open import Function.Injection using (module Injection) renaming (Injection to _↪_)
open import Relation.Binary.PropositionalEquality using (_≗_ ; cong) renaming (setoid to EqSetoid)
open import Relation.Binary using (Setoid)
open Injection using (to)

open import Generic using (≡-to-Π)

module ListList where
  get-type : Set₁
  get-type = {A : Set} → List A → List A

  postulate
    free-theorem : (get : get-type) → {α β : Set} → (f : α → β) → get ∘ map f ≗ map f ∘ get

module VecVec where
  get-type : (ℕ → ℕ) → Set₁
  get-type getlen = {A : Set} {n : ℕ} → Vec A n → Vec A (getlen n)

  free-theorem-type : Set₁
  free-theorem-type = {getlen : ℕ → ℕ} → (get : get-type getlen) → {α β : Set} → (f : α → β) → {n : ℕ} → get {_} {n} ∘ mapV f ≗ mapV f ∘ get

  postulate
    free-theorem : free-theorem-type

module PartialVecVec where
  get-type : {I : Setoid ℓ₀ ℓ₀} → (I ↪ (EqSetoid ℕ)) → (I ⟶ (EqSetoid ℕ)) → Set₁
  get-type {I} gl₁ gl₂ = {A : Set} {i : Setoid.Carrier I} → Vec A (to gl₁ ⟨$⟩ i) → Vec A (gl₂ ⟨$⟩ i)

  postulate
    free-theorem : {I : Setoid ℓ₀ ℓ₀} → (gl₁ : I ↪ (EqSetoid ℕ)) → (gl₂ : I ⟶ (EqSetoid ℕ)) (get : get-type gl₁ gl₂)  → {α β : Set} → (f : α → β) → {i : Setoid.Carrier I} → get {_} {i} ∘ mapV f ≗ mapV f ∘ get

  open VecVec using () renaming (free-theorem-type to VecVec-free-theorem-type)

  VecVec-free-theorem : VecVec-free-theorem-type
  VecVec-free-theorem {getlen} get = free-theorem Function.Injection.id (≡-to-Π getlen) get