Tom de Jong, 1-18 March 2021

We show that the induction principle for 𝕊¹ with propositional computation rules
follows from the universal property of 𝕊¹.

This is claimed at the end of Section 6.2 in the HoTT Book and follows from a
general result by Sojakova in her PhD Thesis "Higher Inductive Types as
Homotopy-Initial Algebras" (CMU-CS-16-125). The proof of the general result is
quite complicated (see for instance Lemma 105 in the PhD thesis) and the
development below offers an alternative proof for 𝕊¹.

\begin{code}

{-# OPTIONS --safe --without-K #-}

open import MLTT.Spartan
open import UF.Base
open import UF.Equiv
open import UF.FunExt
open import UF.Sets
open import UF.Sets-Properties
open import UF.Subsingletons

module Circle.Induction where

\end{code}

First some preliminaries on the free loop space.

\begin{code}

𝓛 : (X : 𝓤 ̇ ) → 𝓤 ̇
𝓛 X = Σ x ꞉ X , x ＝ x

𝓛-functor : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (f : X → Y) → 𝓛 X → 𝓛 Y
𝓛-functor f (x , p) = f x , ap f p

𝓛-functor-id : {X : 𝓤 ̇ } → 𝓛-functor id ∼ id {𝓤} {𝓛 X}
𝓛-functor-id {𝓤} {X} (x , p) = to-Σ-＝ (refl , γ p)
 where
  γ : {y z : X} (q : y ＝ z) → transport (λ - → y ＝ -) q refl ＝ q
  γ refl = refl

𝓛-functor-comp : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } {Z : 𝓦 ̇ } (f : X → Y) (g : Y → Z)
               → 𝓛-functor g ∘ 𝓛-functor f ∼ 𝓛-functor (g ∘ f)
𝓛-functor-comp f g (x , p) = to-Σ-＝ (refl , (ap-ap f g p))

ap-𝓛-functor-lemma : {A : 𝓤 ̇ } {B : 𝓥 ̇ } (f g : A → B)
                     {a : A} (p : a ＝ a) {b : B} (q : b ＝ b)
                     (u : 𝓛-functor f (a , p) ＝ (b , q))
                     (v : 𝓛-functor g (a , p) ＝ (b , q))
                     (w : (f , u) ＝ (g , v))
                   → ap (λ - → 𝓛-functor - (a , p)) (ap pr₁ w) ＝ u ∙ v ⁻¹
ap-𝓛-functor-lemma f g p q refl v refl = refl

happly-𝓛-functor-lemma : {A : 𝓤 ̇ } {B : 𝓥 ̇ } (f g : A → B)
                         {a : A} (p : a ＝ a) {b : B} (q : b ＝ b)
                         (u : 𝓛-functor f (a , p) ＝ (b , q))
                         (v : 𝓛-functor g (a , p) ＝ (b , q))
                         (w : (f , u) ＝ (g , v))
                       → happly (ap pr₁ w) a ＝ (ap pr₁ u) ∙ (ap pr₁ v) ⁻¹
happly-𝓛-functor-lemma f g p q refl v refl = refl

\end{code}

Next we introduce the circle 𝕊¹ with its point base, its loop at base and its
universal property.

\begin{code}

module _
        (𝕊¹ : 𝓤 ̇ )
        (base : 𝕊¹)
        (loop : base ＝ base)
       where

 𝕊¹-universal-map : (A : 𝓥 ̇ )
                  → (𝕊¹ → A) → 𝓛 A
 𝕊¹-universal-map A f = (f base , ap f loop)

 module _
         (𝕊¹-universal-property : {𝓥 : Universe} (A : 𝓥 ̇ )
                                → is-equiv (𝕊¹-universal-map A))
        where

  𝕊¹-uniqueness-principle : {A : 𝓥 ̇ } (a : A) (p : a ＝ a)
                          → ∃! r ꞉ (𝕊¹ → A) , (r base , ap r loop) ＝ (a , p)
  𝕊¹-uniqueness-principle {𝓥} {A} a p =
    equivs-are-vv-equivs (𝕊¹-universal-map A)
                         (𝕊¹-universal-property A) (a , p)

  𝕊¹-at-most-one-function : {A : 𝓥 ̇ } (a : A) (p : a ＝ a)
                          → is-prop (Σ r ꞉ (𝕊¹ → A) , (r base , ap r loop) ＝ (a , p))
  𝕊¹-at-most-one-function a p = singletons-are-props (𝕊¹-uniqueness-principle a p)

\end{code}

The recursion principle for 𝕊¹ with its computation rule follows immediately
from the universal property of 𝕊¹.

\begin{code}

  𝕊¹-rec : {A : 𝓥 ̇ } (a : A) (p : a ＝ a)
         → 𝕊¹ → A
  𝕊¹-rec {𝓥} {A} a p = ∃!-witness (𝕊¹-uniqueness-principle a p)

  𝕊¹-rec-comp : {A : 𝓥 ̇ } (a : A) (p : a ＝ a)
              → 𝓛-functor (𝕊¹-rec a p) (base , loop) ＝[ 𝓛 A ] (a , p)
  𝕊¹-rec-comp {𝓥} {A} a p = ∃!-is-witness (𝕊¹-uniqueness-principle a p)

  𝕊¹-rec-on-base : {A : 𝓥 ̇ } (a : A) (p : a ＝ a)
                  → 𝕊¹-rec a p base ＝ a
  𝕊¹-rec-on-base a p = ap pr₁ (𝕊¹-rec-comp a p)

  𝕊¹-rec-on-loop : {A : 𝓥 ̇ } (a : A) (p : a ＝ a)
                 → transport (λ - → - ＝ -) (𝕊¹-rec-on-base a p)
                     (ap (𝕊¹-rec a p) loop)
                 ＝ p
  𝕊¹-rec-on-loop a p = from-Σ-＝' (𝕊¹-rec-comp a p)

\end{code}

The induction principle for 𝕊¹ also follows quite directly. The idea is to turn
a type family A over 𝕊¹ to the type Σ A and consider a nondependent map 𝕊¹ → Σ A
as a substitute for the dependent function (x : 𝕊¹) → A x.

What is significantly harder is showing that it obeys the computation rules.

\begin{code}

  module 𝕊¹-induction
          (A : 𝕊¹ → 𝓥 ̇ )
          (a : A base)
          (l : transport A loop a ＝ a)
         where

   l⁺ : (base , a) ＝[ Σ A ] (base , a)
   l⁺ = to-Σ-＝ (loop , l)

   r : 𝕊¹ → Σ A
   r = 𝕊¹-rec (base , a) l⁺

\end{code}

Next we show that r is a retraction of pr₁ : Σ A → 𝕊¹. This tells us that
r (x) = (x , pr₂ (r x)), so that we can define 𝕊¹-induction by transport.

\begin{code}

   r-retraction-lemma : 𝓛-functor (pr₁ ∘ r) (base , loop) ＝[ 𝓛 𝕊¹ ] (base , loop)
   r-retraction-lemma =
    ((pr₁ ∘ r) base , ap (pr₁ ∘ r) loop) ＝⟨ I   ⟩
    𝓛-functor pr₁ (r base , ap r loop)   ＝⟨ II  ⟩
    (base , ap pr₁ (to-Σ-＝ (loop , l)))  ＝⟨ III ⟩
    (base , loop)                        ∎
     where
      I   = to-Σ-＝ (refl , ((ap-ap r pr₁ loop) ⁻¹))
      II  = ap (𝓛-functor pr₁) (𝕊¹-rec-comp (base , a) l⁺)
      III = to-Σ-＝ (refl , (ap-pr₁-to-Σ-＝ (loop , l)))

   r-is-retraction-of-pr₁ : pr₁ ∘ r ＝ id
   r-is-retraction-of-pr₁ = ap pr₁ (𝕊¹-at-most-one-function base loop
                                     (pr₁ ∘ r , r-retraction-lemma)
                                     (id , to-Σ-＝ (refl , ap-id-is-id loop)))

   𝕊¹-induction : (x : 𝕊¹) → A x
   𝕊¹-induction x = transport A (happly r-is-retraction-of-pr₁ x) (pr₂ (r x))

\end{code}

Next we set out to prove the computation rules for 𝕊¹-induction.

\begin{code}

   ρ : 𝕊¹ → Σ A
   ρ x = (x , 𝕊¹-induction x)

   r-comp : (r base , ap r loop) ＝[ 𝓛 (Σ A) ] ((base , a) , l⁺)
   r-comp = 𝕊¹-rec-comp (base , a) l⁺

   ρ-r-homotopy : ρ ∼ r
   ρ-r-homotopy x = to-Σ-＝ ((γ₁ ⁻¹) , γ₂)
    where
     γ₁ : pr₁ (r x) ＝ pr₁ (ρ x)
     γ₁ = happly r-is-retraction-of-pr₁ x
     γ₂ = transport A (γ₁ ⁻¹) (pr₂ (ρ x))                  ＝⟨ refl ⟩
          transport A (γ₁ ⁻¹) (transport A γ₁ (pr₂ (r x))) ＝⟨ I    ⟩
          transport A (γ₁ ∙ γ₁ ⁻¹) (pr₂ (r x))             ＝⟨ II   ⟩
          transport A refl (pr₂ (r x))                     ＝⟨ refl ⟩
          pr₂ (r x)                                        ∎
      where
       I  = (transport-∙ A γ₁ (γ₁ ⁻¹)) ⁻¹
       II = ap (λ - → transport A - (pr₂ (r x))) ((right-inverse γ₁) ⁻¹)

   ρ-and-r-on-base-and-loop : (ρ base , ap ρ loop) ＝[ 𝓛 (Σ A) ] (r base , ap r loop)
   ρ-and-r-on-base-and-loop = to-Σ-＝ (ρ-r-homotopy base , γ)
    where
     γ = transport (λ - → - ＝ -) (ρ-r-homotopy base) (ap ρ loop) ＝⟨ I  ⟩
         ρ-r-homotopy base ⁻¹ ∙ ap ρ loop ∙ ρ-r-homotopy base    ＝⟨ II ⟩
         ap r loop                                               ∎
      where
       I  = transport-along-＝ (ρ-r-homotopy base) (ap ρ loop)
       II = homotopies-are-natural'' ρ r ρ-r-homotopy {base} {base} {loop}

   ρ-comp : (ρ base , ap ρ loop) ＝[ 𝓛 (Σ A) ] ((base , a) , l⁺)
   ρ-comp = ρ-and-r-on-base-and-loop ∙ r-comp

\end{code}

Looking at ρ-comp, we see that ρ base = (base , 𝕊¹-induction base) ＝ (base , a),
which looks promising, for if we can show that the equality in the first
component is refl, then 𝕊¹-induction base ＝ a would follow. So that's exactly
what we do next.

\begin{code}

   ρ-comp-lemma : ap pr₁ (ap pr₁ ρ-comp) ＝ refl
   ρ-comp-lemma =
    ap pr₁ (ap pr₁ ρ-comp)                                          ＝⟨ I   ⟩
    ap (pr₁ ∘ pr₁) ρ-comp                                           ＝⟨ II  ⟩
    ap (pr₁ ∘ pr₁) ρ-and-r-on-base-and-loop ∙ ap (pr₁ ∘ pr₁) r-comp ＝⟨ III ⟩
    p ⁻¹ ∙ p                                                        ＝⟨ IV  ⟩
    refl                                                            ∎
    where
     p = happly r-is-retraction-of-pr₁ base
     I   = ap-ap pr₁ pr₁ ρ-comp
     II  = ap-∙ (pr₁ ∘ pr₁) ρ-and-r-on-base-and-loop r-comp
     IV  = left-inverse p
     III = ap₂ _∙_ γ₁ γ₂
      where
       γ₁ : ap (pr₁ ∘ pr₁) ρ-and-r-on-base-and-loop  ＝ p ⁻¹
       γ₁ = ap (pr₁ ∘ pr₁) ρ-and-r-on-base-and-loop  ＝⟨ I₁   ⟩
            ap pr₁ (ap pr₁ ρ-and-r-on-base-and-loop) ＝⟨ II₁  ⟩
            ap pr₁ (ρ-r-homotopy base)               ＝⟨ III₁ ⟩
            p ⁻¹                                     ∎
        where
         I₁   = (ap-ap pr₁ pr₁ ρ-and-r-on-base-and-loop) ⁻¹
         II₁  = ap (ap pr₁) (ap-pr₁-to-Σ-＝ (ρ-r-homotopy base , _))
         III₁ = ap-pr₁-to-Σ-＝ ((p ⁻¹) , _)
       γ₂ : ap (pr₁ ∘ pr₁) r-comp ＝ p
       γ₂ = ϕ ⁻¹
        where
         κ = r-retraction-lemma
         ϕ = p                                                     ＝⟨ I₂    ⟩
             ap pr₁ κ ∙ ap π (to-Σ-＝ (refl , ap-id-is-id loop)) ⁻¹ ＝⟨ II₂   ⟩
             ap pr₁ κ ∙ refl ⁻¹                                    ＝⟨ refl  ⟩
             ap pr₁ κ                                              ＝⟨ III₂  ⟩
             ap pr₁ (ap pr₁ r-comp)                                ＝⟨ IV₂   ⟩
             ap (pr₁ ∘ pr₁) r-comp                                 ∎
          where
           π : 𝓛 (𝕊¹) → 𝕊¹
           π = pr₁
           I₂   = happly-𝓛-functor-lemma (pr₁ ∘ r) id loop loop
                   κ (to-Σ-＝ (refl , ap-id-is-id loop))
                   (𝕊¹-at-most-one-function base loop
                     (pr₁ ∘ r , r-retraction-lemma)
                     (id , to-Σ-＝ (refl , ap-id-is-id loop)))
           II₂  = ap (λ - → ap pr₁ κ ∙ - ⁻¹)
                   (ap-pr₁-to-Σ-＝ {𝓤} {𝓤} {𝕊¹} {λ - → (- ＝ -)} {_} {_}
                    (refl , ap-id-is-id loop))
           IV₂  = ap-ap pr₁ pr₁ r-comp
           III₂ = ap pr₁ κ                        ＝⟨ refl ⟩
                  ap pr₁ (κ₁ ∙ (κ₂ ∙ κ₃))         ＝⟨ I'   ⟩
                  ap pr₁ κ₁ ∙ ap pr₁ (κ₂ ∙ κ₃)    ＝⟨ II'  ⟩
                  refl ∙ ap pr₁ (κ₂ ∙ κ₃)         ＝⟨ III' ⟩
                  ap pr₁ (κ₂ ∙ κ₃)                ＝⟨ IV'  ⟩
                  ap pr₁ κ₂ ∙ ap pr₁ κ₃           ＝⟨ V'   ⟩
                  ap pr₁ κ₂ ∙ refl                ＝⟨ refl ⟩
                  ap pr₁ κ₂                       ＝⟨ VI'  ⟩
                  ap (pr₁ ∘ 𝓛-functor pr₁) r-comp ＝⟨ refl ⟩
                  ap (pr₁ ∘ pr₁) r-comp           ＝⟨ VII' ⟩
                  ap pr₁ (ap pr₁ r-comp)          ∎
                  where
                   κ₁ = to-Σ-＝ (refl , ((ap-ap r pr₁ loop) ⁻¹))
                   κ₂ = ap (𝓛-functor pr₁) r-comp
                   κ₃ = to-Σ-＝ (refl , (ap-pr₁-to-Σ-＝ (loop , l)))
                   I'   = ap-∙ pr₁ κ₁ (κ₂ ∙ κ₃)
                   II'  = ap (_∙ (ap pr₁ (κ₂ ∙ κ₃)))
                           (ap-pr₁-to-Σ-＝ {𝓤} {𝓤} {𝕊¹} {λ - → (- ＝ -)} {_} {_}
                            (refl , ((ap-ap r pr₁ loop) ⁻¹)))
                   III' = refl-left-neutral
                   IV'  = ap-∙ pr₁ κ₂ κ₃
                   V'   = ap ((ap pr₁ κ₂) ∙_)
                           (ap-pr₁-to-Σ-＝ {𝓤} {𝓤} {𝕊¹} {λ - → (- ＝ -)} {_} {_}
                            (refl , ap-pr₁-to-Σ-＝ (loop , l)))
                   VI'  = ap-ap (𝓛-functor pr₁) pr₁ r-comp
                   VII' = (ap-ap pr₁ pr₁ r-comp) ⁻¹

   𝕊¹-induction-on-base : 𝕊¹-induction base ＝ a
   𝕊¹-induction-on-base =
    transport (λ - → transport A - (𝕊¹-induction base) ＝ a) ρ-comp-lemma γ
     where
      γ : transport A (ap pr₁ (ap pr₁ ρ-comp)) (𝕊¹-induction base) ＝ a
      γ = from-Σ-＝' (ap pr₁ ρ-comp)

\end{code}

This takes care of the first computation rule for 𝕊¹-induction. We can
get a fairly direct proof of the second computation rule (the one for
loop) by assuming that base ＝ base is a set, because this tells us
that every element of loop ＝ loop must be refl.

We can satisfy this assumption for our intended application (see
CircleConstruction.lagda), because for the construction involving ℤ-torsors it's
is quite easy to prove that base ＝ base is a set.

However, for completeness sake, below we also show that assuming function
extensionality and univalence, it is possible to prove that base ＝ base is a
set, by using both computation rules for 𝕊¹-rec and the first computation rule
for 𝕊¹-induction.

\begin{code}

   𝕊¹-induction-on-loop-lemma : (loop , transport (λ - → transport A loop - ＝ -)
                                         𝕊¹-induction-on-base
                                         (apd 𝕊¹-induction loop))
                              ＝ (loop , l)
   𝕊¹-induction-on-loop-lemma =
      (fromto-Σ-＝ (loop , transport (λ - → transport A loop - ＝ -) σ τ)) ⁻¹
    ∙ (ap from-Σ-＝ γ) ∙ (fromto-Σ-＝ (loop , l))
     where
      σ = 𝕊¹-induction-on-base
      τ = apd 𝕊¹-induction loop
      γ : to-Σ-＝ (loop , transport (λ - → transport A loop - ＝ -) σ τ)
        ＝ to-Σ-＝ (loop , l)
      γ = to-Σ-＝ (loop , transport (λ - → transport A loop - ＝ -) σ τ)    ＝⟨ I   ⟩
          transport (λ - → - ＝ -) (to-Σ-＝ (refl , σ)) (to-Σ-＝ (loop , τ)) ＝⟨ II  ⟩
          transport (λ - → - ＝ -) (ap pr₁ ρ-comp) (to-Σ-＝ (loop , τ))     ＝⟨ III ⟩
          transport (λ - → - ＝ -) (ap pr₁ ρ-comp) (ap ρ loop)             ＝⟨ IV  ⟩
          to-Σ-＝ (loop , l)                                               ∎
       where
        I   = h loop σ τ
         where
          h : {X : 𝓦 ̇ } {Y : X → 𝓣 ̇ } {x : X} (p : x ＝ x) {y y' : Y x}
              (q : y ＝ y') (q' : transport Y p y ＝ y)
            → to-Σ-＝ (p , transport (λ - → transport Y p - ＝ -) q q')
            ＝ transport (λ - → - ＝ -) (to-Σ-＝ (refl , q)) (to-Σ-＝ (p , q'))
          h p refl q' = refl
        II  = ap (λ - → transport (λ - → - ＝ -) - (to-Σ-＝ (loop , τ))) h
         where
          h = to-Σ-＝ (refl , σ)                 ＝⟨ I'  ⟩
              to-Σ-＝ (from-Σ-＝ (ap pr₁ ρ-comp)) ＝⟨ II' ⟩
              ap pr₁ ρ-comp                     ∎
           where
            I'  = (ap to-Σ-＝ (to-Σ-＝ (ρ-comp-lemma , refl))) ⁻¹
            II' = tofrom-Σ-＝ (ap pr₁ ρ-comp)
        III = ap (transport (λ - → - ＝ -) (ap pr₁ ρ-comp)) (h 𝕊¹-induction loop)
         where
          h : {X : 𝓦 ̇ } {Y : X → 𝓣 ̇ } (f : (x : X) → Y x)
              {x x' : X} (p : x ＝ x')
            → to-Σ-＝ (p , apd f p) ＝ ap (λ x → (x , f x)) p
          h f refl = refl
        IV  = from-Σ-＝' ρ-comp

   module _
           (base-sethood : is-set (base ＝ base))
          where

    𝕊¹-induction-on-loop : transport (λ - → transport A loop - ＝ -)
                            𝕊¹-induction-on-base (apd 𝕊¹-induction loop)
                         ＝ l
    𝕊¹-induction-on-loop =
     ap-pr₁-refl-lemma (λ - → transport A - a ＝ a) 𝕊¹-induction-on-loop-lemma γ
     where
      γ : ap pr₁ 𝕊¹-induction-on-loop-lemma ＝ refl
      γ = base-sethood (ap pr₁ 𝕊¹-induction-on-loop-lemma) refl

    𝕊¹-induction-comp : (𝕊¹-induction base , apd 𝕊¹-induction loop)
                      ＝[ Σ y ꞉ A base , transport A loop y ＝ y ] (a , l)
    𝕊¹-induction-comp = to-Σ-＝ (𝕊¹-induction-on-base , 𝕊¹-induction-on-loop)

\end{code}

As promised above, here follows a proof, assuming function
extensionality and univalence, that base ＝ base is a set, using both
computation rules for 𝕊¹-rec and the first computation rule for
𝕊¹-induction.

The proof uses the encode-decode method (Section 8.1.4 of the HoTT
Book) to show that base ＝ base is a retract of ℤ. Since sets are
closed under retracts, the claim follows.

\begin{code}

  open import Circle.Integers
  open import Circle.Integers-Properties

  open import UF.Univalence

  module _
          (ua : is-univalent 𝓤₀)
         where

   succ-ℤ-＝ : ℤ ＝ ℤ
   succ-ℤ-＝ = eqtoid ua ℤ ℤ succ-ℤ-≃

   code : 𝕊¹ → 𝓤₀ ̇
   code = 𝕊¹-rec ℤ succ-ℤ-＝

\end{code}

   Using the first computation rule for 𝕊¹-rec:

\begin{code}

   code-on-base : code base ＝ ℤ
   code-on-base = 𝕊¹-rec-on-base ℤ succ-ℤ-＝

   ℤ-to-code-base : ℤ → code base
   ℤ-to-code-base = Idtofun (code-on-base ⁻¹)

   code-base-to-ℤ : code base → ℤ
   code-base-to-ℤ = Idtofun code-on-base

   transport-code-loop-is-succ-ℤ : code-base-to-ℤ
                                 ∘ transport code loop
                                 ∘ ℤ-to-code-base
                                 ＝ succ-ℤ
   transport-code-loop-is-succ-ℤ =
    δ ∘ transport code loop ∘ ε                  ＝⟨ I    ⟩
    δ ∘ transport id acl ∘ ε                     ＝⟨ refl ⟩
    Idtofun cob ∘ Idtofun acl ∘ Idtofun (cob ⁻¹) ＝⟨ II   ⟩
    Idtofun cob ∘ Idtofun (cob ⁻¹ ∙ acl)         ＝⟨ III  ⟩
    Idtofun (cob ⁻¹ ∙ acl ∙ cob)                 ＝⟨ IV   ⟩
    Idtofun succ-ℤ-＝                             ＝⟨ V    ⟩
    succ-ℤ                                       ∎
     where
      cob = code-on-base
      acl = ap code loop
      ε = ℤ-to-code-base
      δ = code-base-to-ℤ
      I   = ap (λ - → δ ∘ - ∘ ε) (transport-ap' id code loop)
      II  = ap (_∘_ (Idtofun cob)) ((Idtofun-∙ ua (cob ⁻¹) acl) ⁻¹)
      III = (Idtofun-∙ ua (cob ⁻¹ ∙ acl) cob) ⁻¹

\end{code}

      Using the second computation rule for 𝕊¹-rec

\begin{code}

      IV  = ap Idtofun ((transport-along-＝ cob acl) ⁻¹
                       ∙ (𝕊¹-rec-on-loop ℤ succ-ℤ-＝))
      V   = Idtofun-eqtoid ua succ-ℤ-≃

   transport-code-loop⁻¹-is-pred-ℤ : code-base-to-ℤ
                                   ∘ transport code (loop ⁻¹)
                                   ∘ ℤ-to-code-base
                                   ∼ pred-ℤ
   transport-code-loop⁻¹-is-pred-ℤ x = equivs-are-lc succ-ℤ succ-ℤ-is-equiv γ
    where
     γ : (succ-ℤ ∘ code-base-to-ℤ ∘ transport code (loop ⁻¹) ∘ ℤ-to-code-base) x
       ＝ (succ-ℤ ∘ pred-ℤ) x
     γ = (succ-ℤ ∘ δ ∘ t⁻¹ ∘ ε) x    ＝⟨ I   ⟩
         (δ ∘ t ∘ ε ∘ δ ∘ t⁻¹ ∘ ε) x ＝⟨ II  ⟩
         (δ ∘ t ∘ t⁻¹ ∘ ε) x         ＝⟨ III ⟩
         (δ ∘ ε) x                   ＝⟨ IV  ⟩
         x                           ＝⟨ V   ⟩
         (succ-ℤ ∘ pred-ℤ) x         ∎
      where
       ε = ℤ-to-code-base
       δ = code-base-to-ℤ
       t⁻¹ = transport code (loop ⁻¹)
       t   = transport code loop
       I   = happly (transport-code-loop-is-succ-ℤ ⁻¹) ((δ ∘ t⁻¹ ∘ ε) x)
       II  = ap (δ ∘ t) (Idtofun-section code-on-base (t⁻¹ (ε x)))
       III = ap δ (back-and-forth-transport loop)
       IV  = Idtofun-retraction code-on-base x
       V   = (succ-ℤ-is-retraction x) ⁻¹

   transport-code-loop⁻¹-is-pred-ℤ' : transport code (loop ⁻¹)
                                    ∼ ℤ-to-code-base ∘ pred-ℤ ∘ code-base-to-ℤ
   transport-code-loop⁻¹-is-pred-ℤ' x =
    transport code (loop ⁻¹) x                   ＝⟨ I   ⟩
    (ε ∘ δ ∘ transport code (loop ⁻¹)) x         ＝⟨ II  ⟩
    (ε ∘ δ ∘ transport code (loop ⁻¹) ∘ ε ∘ δ) x ＝⟨ III ⟩
    (ε ∘ pred-ℤ ∘ δ) x                           ∎
     where
      ε = ℤ-to-code-base
      δ = code-base-to-ℤ
      I   = (Idtofun-section code-on-base (transport code (loop ⁻¹) x)) ⁻¹
      II  = ap (ε ∘ δ ∘ transport code (loop ⁻¹))
             ((Idtofun-section code-on-base x) ⁻¹)
      III = ap ε (transport-code-loop⁻¹-is-pred-ℤ (δ x))

   encode : (x : 𝕊¹) → (base ＝ x) → code x
   encode x p = transport code p (ℤ-to-code-base 𝟎)

   iterated-path : {X : 𝓦 ̇ } {x : X} → x ＝ x → ℕ → x ＝ x
   iterated-path p zero     = refl
   iterated-path p (succ n) = p ∙ iterated-path p n

   iterated-path-comm : {X : 𝓦 ̇ } {x : X} (p : x ＝ x) (n : ℕ)
                      → iterated-path p n ∙ p ＝ p ∙ iterated-path p n
   iterated-path-comm p zero = refl ∙ p ＝⟨ refl-left-neutral ⟩
                               p        ＝⟨ refl              ⟩
                               p ∙ refl ∎
   iterated-path-comm p (succ n) = p ∙ iterated-path p n ∙ p   ＝⟨ I  ⟩
                                   p ∙ (iterated-path p n ∙ p) ＝⟨ II ⟩
                                   p ∙ (p ∙ iterated-path p n) ∎
    where
     I  =  ∙assoc p (iterated-path p n) p
     II = ap (p ∙_) (iterated-path-comm p n)

   loops : ℤ → base ＝ base
   loops 𝟎       = refl
   loops (pos n) = iterated-path loop (succ n)
   loops (neg n) = iterated-path (loop ⁻¹) (succ n)

   module _
           (fe : funext 𝓤₀ 𝓤)
          where

    open import UF.Lower-FunExt

    loops-lemma : (_∙ loop) ∘ loops ∘ pred-ℤ ＝ loops
    loops-lemma = dfunext fe h
     where
      h : (k : ℤ) → loops (pred-ℤ k) ∙ loop ＝ loops k
      h 𝟎 = loop ⁻¹ ∙ refl ∙ loop ＝⟨ refl              ⟩
            loop ⁻¹ ∙ loop        ＝⟨ left-inverse loop ⟩
            refl                  ∎
      h (pos n) = g n
       where
        g : (n : ℕ) → loops (pred-ℤ (pos n)) ∙ loop ＝ loops (pos n)
        g zero     = iterated-path-comm loop zero
        g (succ n) = iterated-path-comm loop (succ n)
      h (neg n) =
       loop ⁻¹ ∙ (loop ⁻¹ ∙ iterated-path (loop ⁻¹) n) ∙ loop ＝⟨ I'   ⟩
       loop ⁻¹ ∙ (iterated-path (loop ⁻¹) n ∙ loop ⁻¹) ∙ loop ＝⟨ II'  ⟩
       loop ⁻¹ ∙ iterated-path (loop ⁻¹) n ∙ (loop ⁻¹ ∙ loop) ＝⟨ III' ⟩
       loop ⁻¹ ∙ iterated-path (loop ⁻¹) n                    ∎
        where
         I'   = ap (λ - → loop ⁻¹ ∙ - ∙ loop)
                 ((iterated-path-comm (loop ⁻¹) n) ⁻¹)
         II'  = ∙assoc (loop ⁻¹) (iterated-path (loop ⁻¹) n ∙ loop ⁻¹) loop
              ∙ ap (loop ⁻¹ ∙_)
                 (∙assoc (iterated-path (loop ⁻¹) n) (loop ⁻¹) loop)
              ∙ (∙assoc (loop ⁻¹) (iterated-path (loop ⁻¹) n)
                  (loop ⁻¹ ∙ loop)) ⁻¹
         III' = ap ((loop ⁻¹ ∙ iterated-path (loop ⁻¹) n) ∙_)
                 (left-inverse loop)

    transport-loops-lemma : transport (λ - → code - → base ＝ -) loop
                             (loops ∘ code-base-to-ℤ)
                          ＝ (loops ∘ code-base-to-ℤ)
    transport-loops-lemma =
     transport (λ - → code - → base ＝ -) loop f                     ＝⟨ I   ⟩
     transport (λ - → base ＝ -) loop ∘ f ∘ transport code (loop ⁻¹) ＝⟨ II  ⟩
     (_∙ loop) ∘ f ∘ transport code (loop ⁻¹)                       ＝⟨ III ⟩
     (_∙ loop) ∘ loops ∘ δ ∘ ε ∘ pred-ℤ ∘ δ                         ＝⟨ IV  ⟩
     (_∙ loop) ∘ loops ∘ pred-ℤ ∘ δ                                 ＝⟨ V   ⟩
     loops ∘ δ                                                      ∎
      where
       ε : ℤ → code base
       ε = ℤ-to-code-base
       δ : code base → ℤ
       δ = code-base-to-ℤ
       f : code base → base ＝ base
       f = loops ∘ δ
       I   = transport-along-→ code (_＝_ base) loop f
       II  = refl
       III = ap ((_∙ loop) ∘ f ∘_)
              (dfunext (lower-funext 𝓤₀ 𝓤 fe) transport-code-loop⁻¹-is-pred-ℤ')
       IV  = ap (λ - → (_∙ loop) ∘ loops ∘ - ∘ pred-ℤ ∘ δ)
              (dfunext (lower-funext 𝓤₀ 𝓤 fe) (Idtofun-retraction code-on-base))
       V   = ap (_∘ δ) loops-lemma


    open 𝕊¹-induction (λ - → code - → base ＝ -)
                      (loops ∘ code-base-to-ℤ)
                      transport-loops-lemma

    decode : (x : 𝕊¹) → code x → base ＝ x
    decode = 𝕊¹-induction

    decode-encode : (x : 𝕊¹) (p : base ＝ x) → decode x (encode x p) ＝ p
    decode-encode base refl =
     decode base (encode base refl)                       ＝⟨ refl ⟩
     decode base (transport code refl (ℤ-to-code-base 𝟎)) ＝⟨ refl ⟩
     decode base (ℤ-to-code-base 𝟎)                       ＝⟨ I    ⟩
     loops (code-base-to-ℤ (ℤ-to-code-base 𝟎))            ＝⟨ II   ⟩
     loops 𝟎                                              ＝⟨ refl ⟩
     refl                                                 ∎
      where

\end{code}

       Using the first computation rule for 𝕊¹-induction

\begin{code}

       I  = happly 𝕊¹-induction-on-base (ℤ-to-code-base 𝟎)
       II = ap loops (Idtofun-retraction code-on-base 𝟎)

    open import UF.Retracts

    Ω𝕊¹-is-set : is-set (base ＝ base)
    Ω𝕊¹-is-set = subtypes-of-sets-are-sets' (encode base)
                  (sections-are-lc (encode base)
                   ((decode base) , (decode-encode base)))
                   (transport is-set (code-on-base ⁻¹) ℤ-is-set)

  module 𝕊¹-induction'
          {𝓥 : Universe}
          (A : 𝕊¹ → 𝓥 ̇ )
          (a : A base)
          (l : transport A loop a ＝ a)
          (fe : funext 𝓤₀ 𝓤)
          (ua : is-univalent 𝓤₀)
         where

   open 𝕊¹-induction A a l

   𝕊¹-induction-on-loop' : transport (λ - → transport A loop - ＝ -)
                            𝕊¹-induction-on-base (apd 𝕊¹-induction loop)
                         ＝ l
   𝕊¹-induction-on-loop' = 𝕊¹-induction-on-loop (Ω𝕊¹-is-set ua fe)

   𝕊¹-induction-comp' : (𝕊¹-induction base , apd 𝕊¹-induction loop)
                      ＝[ Σ y ꞉ A base , transport A loop y ＝ y ] (a , l)
   𝕊¹-induction-comp' = 𝕊¹-induction-comp (Ω𝕊¹-is-set ua fe)

\end{code}