{-# OPTIONS --without-K #-}
open import Equality
module Fin
{reflexive} (eq : ∀ {a p} → Equality-with-J a p reflexive) where
open import Logical-equivalence hiding (id; _∘_; inverse)
open import Prelude hiding (id)
open import Bijection eq using (_↔_; module _↔_)
open Derived-definitions-and-properties eq
open import Equality.Decision-procedures eq
open import Equivalence eq as Eq using (_≃_)
open import Function-universe eq hiding (_∘_)
open import H-level eq
open import H-level.Closure eq
open import List eq
open import Nat eq using (_≤_; ≤-refl; ≤-step)
open import Univalence-axiom eq
∃-Fin-zero : ∀ {p ℓ} (P : Fin zero → Set p) → ∃ P ↔ ⊥ {ℓ = ℓ}
∃-Fin-zero P = Σ-left-zero
∃-Fin-suc : ∀ {p n} (P : Fin (suc n) → Set p) →
∃ P ↔ P fzero ⊎ ∃ (P ∘ fsuc)
∃-Fin-suc P =
∃ P ↔⟨ ∃-⊎-distrib-right ⟩
∃ (P ∘ inj₁) ⊎ ∃ (P ∘ inj₂) ↔⟨ Σ-left-identity ⊎-cong id ⟩□
P (inj₁ tt) ⊎ ∃ (P ∘ inj₂) □
abstract
Fin-set : ∀ n → Is-set (Fin n)
Fin-set zero = mono₁ 1 ⊥-propositional
Fin-set (suc n) = ⊎-closure 0 (mono 0≤2 ⊤-contractible) (Fin-set n)
where
0≤2 : 0 ≤ 2
0≤2 = ≤-step (≤-step ≤-refl)
private
well-behaved : ∀ {m n} (f : Fin (1 + m) ↔ Fin (1 + n)) →
Well-behaved (_↔_.to f)
well-behaved f {b = i} eq₁ eq₂ = ⊎.inj₁≢inj₂ (
fzero ≡⟨ sym $ to-from f eq₂ ⟩
from f fzero ≡⟨ to-from f eq₁ ⟩∎
fsuc i ∎)
where open _↔_
cancel-suc : ∀ {m n} → Fin (1 + m) ↔ Fin (1 + n) → Fin m ↔ Fin n
cancel-suc f =
⊎-left-cancellative f (well-behaved f) (well-behaved $ inverse f)
infix 4 _And_Are-related-by_
_And_Are-related-by_ :
∀ {a} {A : Set a}
(xs ys : List A) → Fin (length xs) ↔ Fin (length ys) → Set a
xs And ys Are-related-by f =
∀ i → index xs i ≡ index ys (_↔_.to f i)
abstract
cancel-suc-preserves-relatedness :
∀ {a} {A : Set a} (x : A) xs ys
(f : Fin (length (x ∷ xs)) ↔ Fin (length (x ∷ ys))) →
x ∷ xs And x ∷ ys Are-related-by f →
xs And ys Are-related-by cancel-suc f
cancel-suc-preserves-relatedness x xs ys f related = helper
where
open _↔_ f
helper : xs And ys Are-related-by cancel-suc f
helper i with inspect (to (fsuc i)) | related (fsuc i)
helper i | fsuc k , eq₁ | hyp₁ =
index xs i ≡⟨ hyp₁ ⟩
index (x ∷ ys) (to (fsuc i)) ≡⟨ cong (index (x ∷ ys)) eq₁ ⟩
index (x ∷ ys) (fsuc k) ≡⟨ refl _ ⟩∎
index ys k ∎
helper i | fzero , eq₁ | hyp₁
with inspect (to (fzero)) | related (fzero)
helper i | fzero , eq₁ | hyp₁ | fsuc j , eq₂ | hyp₂ =
index xs i ≡⟨ hyp₁ ⟩
index (x ∷ ys) (to (fsuc i)) ≡⟨ cong (index (x ∷ ys)) eq₁ ⟩
index (x ∷ ys) (fzero) ≡⟨ refl _ ⟩
x ≡⟨ hyp₂ ⟩
index (x ∷ ys) (to (fzero)) ≡⟨ cong (index (x ∷ ys)) eq₂ ⟩
index (x ∷ ys) (fsuc j) ≡⟨ refl _ ⟩∎
index ys j ∎
helper i | fzero , eq₁ | hyp₁ | fzero , eq₂ | hyp₂ =
⊥-elim $ well-behaved f eq₁ eq₂
isomorphic-same-size : ∀ {m n} → (Fin m ↔ Fin n) ⇔ m ≡ n
isomorphic-same-size {m} {n} = record
{ to = to m n
; from = λ m≡n → subst (λ n → Fin m ↔ Fin n) m≡n id
}
where
abstract
to : ∀ m n → (Fin m ↔ Fin n) → m ≡ n
to zero zero _ = refl _
to (suc m) (suc n) 1+m↔1+n = cong suc $ to m n $ cancel-suc 1+m↔1+n
to zero (suc n) 0↔1+n = ⊥-elim $ _↔_.from 0↔1+n fzero
to (suc m) zero 1+m↔0 = ⊥-elim $ _↔_.to 1+m↔0 fzero
Fin⊎Fin↔Fin+ : ∀ m n → Fin m ⊎ Fin n ↔ Fin (m + n)
Fin⊎Fin↔Fin+ zero n =
Fin 0 ⊎ Fin n ↝⟨ id ⟩
⊥ ⊎ Fin n ↝⟨ ⊎-left-identity ⟩
Fin n ↝⟨ id ⟩□
Fin (0 + n) □
Fin⊎Fin↔Fin+ (suc m) n =
Fin (suc m) ⊎ Fin n ↝⟨ id ⟩
(⊤ ⊎ Fin m) ⊎ Fin n ↝⟨ inverse ⊎-assoc ⟩
⊤ ⊎ (Fin m ⊎ Fin n) ↝⟨ id ⊎-cong Fin⊎Fin↔Fin+ m n ⟩
⊤ ⊎ Fin (m + n) ↝⟨ id ⟩
Fin (suc (m + n)) ↝⟨ id ⟩□
Fin (suc m + n) □
Fin×Fin↔Fin* : ∀ m n → Fin m × Fin n ↔ Fin (m * n)
Fin×Fin↔Fin* zero n =
Fin 0 × Fin n ↝⟨ id ⟩
⊥ × Fin n ↝⟨ ×-left-zero ⟩
⊥ ↝⟨ id ⟩□
Fin 0 □
Fin×Fin↔Fin* (suc m) n =
Fin (suc m) × Fin n ↝⟨ id ⟩
(⊤ ⊎ Fin m) × Fin n ↝⟨ ∃-⊎-distrib-right ⟩
⊤ × Fin n ⊎ Fin m × Fin n ↝⟨ ×-left-identity ⊎-cong Fin×Fin↔Fin* m n ⟩
Fin n ⊎ Fin (m * n) ↝⟨ Fin⊎Fin↔Fin+ _ _ ⟩
Fin (n + m * n) ↝⟨ id ⟩□
Fin (suc m * n) □
[Fin→Fin]↔Fin^ :
Extensionality (# 0) (# 0) →
∀ m n → (Fin m → Fin n) ↔ Fin (n ^ m)
[Fin→Fin]↔Fin^ ext zero n =
(Fin 0 → Fin n) ↝⟨ id ⟩
(⊥ → Fin n) ↝⟨ Π⊥↔⊤ ext ⟩
⊤ ↝⟨ inverse ⊎-right-identity ⟩□
Fin 1 □
[Fin→Fin]↔Fin^ ext (suc m) n =
(Fin (suc m) → Fin n) ↝⟨ id ⟩
(⊤ ⊎ Fin m → Fin n) ↝⟨ Π⊎↔Π×Π ext ⟩
(⊤ → Fin n) × (Fin m → Fin n) ↝⟨ Π-left-identity ×-cong [Fin→Fin]↔Fin^ ext m n ⟩
Fin n × Fin (n ^ m) ↝⟨ Fin×Fin↔Fin* _ _ ⟩
Fin (n * n ^ m) ↝⟨ id ⟩□
Fin (n ^ suc m) □
[Fin↔Fin]↔Fin! :
Extensionality (# 0) (# 0) →
∀ n → (Fin n ↔ Fin n) ↔ Fin (n !)
[Fin↔Fin]↔Fin! ext zero =
Fin 0 ↔ Fin 0 ↝⟨ Eq.↔↔≃ ext (Fin-set 0) ⟩
Fin 0 ≃ Fin 0 ↝⟨ id ⟩
⊥ ≃ ⊥ ↔⟨ ≃⊥≃¬ ext ⟩
¬ ⊥ ↝⟨ ¬⊥↔⊤ ext ⟩
⊤ ↝⟨ inverse ⊎-right-identity ⟩
⊤ ⊎ ⊥ ↝⟨ id ⟩□
Fin 1 □
[Fin↔Fin]↔Fin! ext (suc n) =
Fin (suc n) ↔ Fin (suc n) ↝⟨ [⊤⊎↔⊤⊎]↔[⊤⊎×↔] ext Fin._≟_ ⟩
Fin (suc n) × (Fin n ↔ Fin n) ↝⟨ id ×-cong [Fin↔Fin]↔Fin! ext n ⟩
Fin (suc n) × Fin (n !) ↝⟨ Fin×Fin↔Fin* _ _ ⟩□
Fin (suc n !) □
[Fin≡Fin]↔Fin! :
Extensionality (# 0) (# 0) →
Univalence (# 0) →
∀ n → (Fin n ≡ Fin n) ↔ Fin (n !)
[Fin≡Fin]↔Fin! ext univ n =
Fin n ≡ Fin n ↔⟨ ≡≃≃ univ ⟩
Fin n ≃ Fin n ↝⟨ inverse $ Eq.↔↔≃ ext (Fin-set n) ⟩
Fin n ↔ Fin n ↝⟨ [Fin↔Fin]↔Fin! ext n ⟩□
Fin (n !) □