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module Problem15 where
open import Data.Vec as Vec
open import Data.Fin
open import Data.Nat using (ℕ; zero; suc)
open import Data.Product hiding (map)
open import Relation.Binary.PropositionalEquality
open import Data.Vec.Properties
open import Relation.Nullary
open import Function
open ≡-Reasoning
variable n : ℕ
variable A B : Set
-- Useful properties not in stdlib 1.7.3
map-tabulate : (g : A → B) (f : Fin n → A) → map g (tabulate f) ≡ tabulate (g ∘′ f)
map-tabulate g f = begin
map g (tabulate f) ≡⟨ cong (map g) (tabulate-allFin f) ⟩
map g (map f (allFin _)) ≡˘⟨ map-∘ g f (allFin _) ⟩
map (g ∘′ f) (allFin _) ≡˘⟨ tabulate-allFin (g ∘′ f) ⟩
tabulate (g ∘′ f) ∎
Sur : ∀(v : Vec (Fin n) n) → Set
Sur {n} v = ∀(x : Fin n) → ∃[ i ] lookup v i ≡ x
Inj : ∀(v : Vec A n ) → Set
Inj {_}{n} v = (a b : Fin n) → lookup v a ≡ lookup v b → a ≡ b
record Perm (n : ℕ) : Set where
constructor P
field indices : Vec (Fin n) n
field surjective : Sur indices
field injective : Inj indices
_⟨$⟩_ : Fin n → Fin n
_⟨$⟩_ = lookup indices
open Perm
-- Agda's proof irrelevance is too janky to work here
-- And we'd need functional extensionality + UIP to prove this directly
postulate cong-Perm : ∀ (p q : Perm n) → Perm.indices p ≡ Perm.indices q → p ≡ q
permute : Perm n → Vec A n → Vec A n
permute (P p _ _) v = map (lookup v) p
_⊡_ : Perm n → Perm n → Perm n
(p ⊡ r) .indices = tabulate ((p ⟨$⟩_) ∘ (r ⟨$⟩_))
(p ⊡ r) .surjective x .proj₁ = r .surjective (p .surjective x .proj₁) .proj₁
(p ⊡ r) .surjective x .proj₂ =
let (i , prf₁) = p .surjective x in
let (j , prf₂) = r .surjective i in
begin
lookup (tabulate ((p ⟨$⟩_) ∘ (r ⟨$⟩_))) j ≡⟨ lookup∘tabulate _ j ⟩
p ⟨$⟩ (r ⟨$⟩ j) ≡⟨ cong (p ⟨$⟩_) prf₂ ⟩
p ⟨$⟩ i ≡⟨ prf₁ ⟩
x ∎
(p ⊡ r) .injective x y prf =
r .injective x y $′
p .injective (r ⟨$⟩ x) (r ⟨$⟩ y) $′
begin
p ⟨$⟩ (r ⟨$⟩ x) ≡˘⟨ lookup∘tabulate _ x ⟩
lookup (tabulate ((p ⟨$⟩_) ∘ (r ⟨$⟩_))) x ≡⟨ prf ⟩
lookup (tabulate ((p ⟨$⟩_) ∘ (r ⟨$⟩_))) y ≡⟨ lookup∘tabulate _ y ⟩
p ⟨$⟩ (r ⟨$⟩ y) ∎
infixl 20 _⊡_
composition : (v : Vec A n)(p q : Perm n) → permute (p ⊡ q) v ≡ permute q (permute p v)
composition v p q = begin
map (lookup v) ((p ⊡ q) .indices) ≡⟨⟩
map (lookup v) (tabulate ((p ⟨$⟩_) ∘′ (q ⟨$⟩_))) ≡⟨ map-tabulate (lookup v) ((p ⟨$⟩_) ∘′ (q ⟨$⟩_)) ⟩
tabulate (lookup v ∘′ (p ⟨$⟩_) ∘′ (q ⟨$⟩_)) ≡⟨⟩
tabulate (lookup v ∘′ (p ⟨$⟩_) ∘′ lookup (q .indices)) ≡˘⟨ tabulate-cong (λ i → lookup-map i (lookup v ∘′ (p ⟨$⟩_)) (q .indices)) ⟩
tabulate (lookup (map (lookup v ∘′ (p ⟨$⟩_)) (q .indices))) ≡⟨ tabulate∘lookup (map (lookup v ∘′ (p ⟨$⟩_)) (q .indices)) ⟩
map (lookup v ∘′ (p ⟨$⟩_)) (q .indices) ≡⟨⟩
map (lookup v ∘′ lookup (p .indices)) (q .indices) ≡˘⟨ map-cong (λ i → lookup-map i (lookup v) (p .indices)) (q .indices) ⟩
map (lookup (map (lookup v) (p .indices))) (q .indices) ∎
assoc : ∀ (p q r : Perm n) → p ⊡ (q ⊡ r) ≡ p ⊡ q ⊡ r
assoc p q r =
cong-Perm (p ⊡ (q ⊡ r)) (p ⊡ q ⊡ r) $′
begin
tabulate ((p ⟨$⟩_) ∘′ ((q ⊡ r) ⟨$⟩_)) ≡⟨ tabulate-cong (λ i → cong (p ⟨$⟩_) (lookup∘tabulate ((q ⟨$⟩_) ∘′ (r ⟨$⟩_)) i)) ⟩
tabulate ((p ⟨$⟩_) ∘′ (q ⟨$⟩_) ∘′ (r ⟨$⟩_)) ≡˘⟨ tabulate-cong (λ i → lookup∘tabulate ((p ⟨$⟩_) ∘′ (q ⟨$⟩_)) (r ⟨$⟩ i)) ⟩
tabulate (((p ⊡ q) ⟨$⟩_) ∘′ (r ⟨$⟩_)) ∎
ι : Perm n
ι {n} .indices = allFin n
ι {n} .surjective x = x , lookup-allFin x
ι {n} .injective x y prf = begin
x ≡˘⟨ lookup-allFin x ⟩
lookup (allFin n) x ≡⟨ prf ⟩
lookup (allFin n) y ≡⟨ lookup-allFin y ⟩
y ∎
identityˡ : ∀(p : Perm n) → ι ⊡ p ≡ p
identityˡ p =
cong-Perm (ι ⊡ p) p $′
begin
tabulate (lookup (allFin _) ∘′ (p ⟨$⟩_)) ≡⟨ tabulate-cong (λ i → lookup-allFin (p ⟨$⟩ i)) ⟩
tabulate (p ⟨$⟩_) ≡⟨ tabulate∘lookup (p .indices) ⟩
p .indices ∎
identityʳ : ∀(p : Perm n) → p ⊡ ι ≡ p
identityʳ {n} p =
cong-Perm (p ⊡ ι) p $′
begin
tabulate ((p ⟨$⟩_) ∘′ lookup (allFin n)) ≡⟨ tabulate-cong (λ i → cong (p ⟨$⟩_) (lookup-allFin i)) ⟩
tabulate (p ⟨$⟩_) ≡⟨ tabulate∘lookup (p .indices) ⟩
p .indices ∎
_⁻¹ : Perm n → Perm n
(p ⁻¹) .indices = tabulate (proj₁ ∘ p .surjective)
(p ⁻¹) .surjective x .proj₁ = p ⟨$⟩ x
(p ⁻¹) .surjective x .proj₂ =
p .injective (lookup (tabulate (proj₁ ∘ p .surjective)) (p ⟨$⟩ x)) x $′
begin
p ⟨$⟩ lookup (tabulate (proj₁ ∘ p .surjective)) (p ⟨$⟩ x) ≡⟨ cong (p ⟨$⟩_) (lookup∘tabulate (proj₁ ∘ p .surjective) (p ⟨$⟩ x)) ⟩
p ⟨$⟩ p .surjective (p ⟨$⟩ x) .proj₁ ≡⟨ p .surjective (p ⟨$⟩ x) .proj₂ ⟩
p ⟨$⟩ x ∎
(p ⁻¹) .injective x y prf =
begin
x ≡˘⟨ p .surjective x .proj₂ ⟩
p ⟨$⟩ p .surjective x .proj₁ ≡˘⟨ cong (p ⟨$⟩_) (lookup∘tabulate (proj₁ ∘ p .surjective) x) ⟩
p ⟨$⟩ lookup (tabulate (proj₁ ∘ p .surjective)) x ≡⟨ cong (p ⟨$⟩_) prf ⟩
p ⟨$⟩ lookup (tabulate (proj₁ ∘ p .surjective)) y ≡⟨ cong (p ⟨$⟩_) (lookup∘tabulate (proj₁ ∘ p .surjective) y) ⟩
p ⟨$⟩ p .surjective y .proj₁ ≡⟨ p .surjective y .proj₂ ⟩
y ∎
infixl 30 _⁻¹
inverseˡ : ∀(p : Perm n) → p ⁻¹ ⊡ p ≡ ι
inverseˡ p =
cong-Perm (p ⁻¹ ⊡ p) ι $′
begin
tabulate ((p ⁻¹ ⟨$⟩_) ∘′ (p ⟨$⟩_)) ≡⟨ tabulate-cong (λ i → lookup∘tabulate (proj₁ ∘ p .surjective) (p ⟨$⟩ i)) ⟩
tabulate (proj₁ ∘ p .surjective ∘ (p ⟨$⟩_)) ≡⟨ tabulate-cong (λ i → p .injective _ i (p .surjective (p ⟨$⟩ i) .proj₂)) ⟩
tabulate id ∎
inverseʳ : ∀(p : Perm n) → p ⊡ p ⁻¹ ≡ ι
inverseʳ p =
cong-Perm (p ⊡ p ⁻¹) ι $′
begin
tabulate ((p ⟨$⟩_) ∘′ (p ⁻¹ ⟨$⟩_)) ≡⟨ tabulate-cong (λ i → cong (p ⟨$⟩_) (lookup∘tabulate (proj₁ ∘ p .surjective) i)) ⟩
tabulate ((p ⟨$⟩_) ∘ proj₁ ∘ p .surjective) ≡⟨ tabulate-cong (λ i → p .surjective i .proj₂) ⟩
tabulate id ∎
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