------------------------------------------------------------------------
-- The Agda standard library
--
-- Bijections on finite sets (i.e. permutations).
------------------------------------------------------------------------
 
{-# OPTIONS --cubical-compatible --safe #-}
 
module Data.Fin.Permutation where
 
open import Data.Bool.Base using (true; false)
open import Data.Fin.Base using (Fin; suc; cast; opposite; punchIn; punchOut)
open import Data.Fin.Patterns using (0F; 1F)
open import Data.Fin.Properties using (punchInᵢ≢i; punchOut-punchIn;
  punchOut-cong; punchOut-cong′; punchIn-punchOut
  ; _≟_; ¬Fin0; cast-involutive; opposite-involutive)
import Data.Fin.Permutation.Components as PC
open import Data.Nat.Base using (ℕ; suc; zero; 2+)
open import Data.Product.Base using (_,_; proj₂)
open import Function.Bundles using (_↔_; Injection; Inverse; mk↔ₛ′)
open import Function.Construct.Composition using (_↔-∘_)
open import Function.Construct.Identity using (↔-id)
open import Function.Construct.Symmetry using (↔-sym)
open import Function.Definitions using (StrictlyInverseˡ; StrictlyInverseʳ)
open import Function.Properties.Inverse using (↔⇒↣)
open import Function.Base using (_∘_; _∘′_)
open import Level using (0ℓ)
open import Relation.Binary.Core using (Rel)
open import Relation.Nullary using (does; ¬_; yes; no)
open import Relation.Nullary.Decidable using (dec-yes; dec-no)
open import Relation.Nullary.Negation using (contradiction)
open import Relation.Binary.PropositionalEquality.Core
  using (_≡_; _≢_; refl; sym; trans; subst; cong; cong₂)
open import Relation.Binary.PropositionalEquality.Properties
  using (module ≡-Reasoning)
open ≡-Reasoning
 
private
  variable
    m n o : ℕ
 
------------------------------------------------------------------------
-- Types
 
-- A bijection between finite sets of potentially different sizes.
-- There only exist inhabitants of this type if in fact m = n, however
-- it is often easier to prove the existence of a bijection without
-- first proving that the sets have the same size. Indeed such a
-- bijection is a useful way to prove that the sets are in fact the same
-- size. See '↔-≡' below.
 
Permutation : ℕ → ℕ → Set
Permutation m n = Fin m ↔ Fin n
 
Permutation′ : ℕ → Set
Permutation′ n = Permutation n n
 
------------------------------------------------------------------------
-- Helper functions
 
permutation : ∀ (f : Fin m → Fin n)
              (g : Fin n → Fin m) →
              StrictlyInverseˡ _≡_ f g →
              StrictlyInverseʳ _≡_ f g →
              Permutation m n
permutation = mk↔ₛ′
 
infixl  _⟨$⟩ʳ_ _⟨$⟩ˡ_
 
_⟨$⟩ʳ_ : Permutation m n → Fin m → Fin n
_⟨$⟩ʳ_ = Inverse.to
 
_⟨$⟩ˡ_ : Permutation m n → Fin n → Fin m
_⟨$⟩ˡ_ = Inverse.from
 
inverseˡ : ∀ (π : Permutation m n) {i} → π ⟨$⟩ˡ (π ⟨$⟩ʳ i) ≡ i
inverseˡ π = Inverse.inverseʳ π refl
 
inverseʳ : ∀ (π : Permutation m n) {i} → π ⟨$⟩ʳ (π ⟨$⟩ˡ i) ≡ i
inverseʳ π = Inverse.inverseˡ π refl
 
------------------------------------------------------------------------
-- Equality over permutations
 
infix  _≈_
 
_≈_ : Rel (Permutation m n) 0ℓ
π ≈ ρ = ∀ i → π ⟨$⟩ʳ i ≡ ρ ⟨$⟩ʳ i
 
------------------------------------------------------------------------
-- Permutation properties
 
id : Permutation n n
id = ↔-id _
 
flip : Permutation m n → Permutation n m
flip = ↔-sym
 
infixr  _∘ₚ_
 
_∘ₚ_ : Permutation m n → Permutation n o → Permutation m o
π₁ ∘ₚ π₂ = π₂ ↔-∘ π₁
 
------------------------------------------------------------------------
-- Non-trivial identity
 
cast-id : .(m ≡ n) → Permutation m n
cast-id m≡n = permutation
  (cast m≡n)
  (cast (sym m≡n))
  (cast-involutive m≡n (sym m≡n))
  (cast-involutive (sym m≡n) m≡n)
 
------------------------------------------------------------------------
-- Transposition
 
-- Transposes two elements in the permutation, keeping the remainder
-- of the permutation the same
transpose : Fin n → Fin n → Permutation n n
transpose i j = permutation
  (PC.transpose i j)
  (PC.transpose j i)
  (λ _ → PC.transpose-inverse _ _)
  (λ _ → PC.transpose-inverse _ _)
 
------------------------------------------------------------------------
-- Reverse
 
-- Reverses a permutation
reverse : Permutation n n
reverse = permutation
  opposite
  opposite
  opposite-involutive
  opposite-involutive
 
------------------------------------------------------------------------
-- Element removal
--
-- `remove k [0 ↦ i₀, …, k ↦ iₖ, …, n ↦ iₙ]` yields
--
-- [0 ↦ i₀, …, k-1 ↦ iₖ₋₁, k ↦ iₖ₊₁, k+1 ↦ iₖ₊₂, …, n-1 ↦ iₙ]
 
remove : Fin (suc m) → Permutation (suc m) (suc n) → Permutation m n
remove {m} {n} i π = permutation to from inverseˡ′ inverseʳ′
  where
  πʳ = π ⟨$⟩ʳ_
  πˡ = π ⟨$⟩ˡ_
 
  permute-≢ : ∀ {i j} → i ≢ j → πʳ i ≢ πʳ j
  permute-≢ p = p ∘ Injection.injective (↔⇒↣ π)
 
  to-punchOut : ∀ {j : Fin m} → πʳ i ≢ πʳ (punchIn i j)
  to-punchOut = permute-≢ (punchInᵢ≢i _ _ ∘ sym)
 
  from-punchOut : ∀ {j : Fin n} → i ≢ πˡ (punchIn (πʳ i) j)
  from-punchOut {j} p = punchInᵢ≢i (πʳ i) j (sym (begin
    πʳ i                        ≡⟨ cong πʳ p ⟩
    πʳ (πˡ (punchIn (πʳ i) j))  ≡⟨ inverseʳ π ⟩
    punchIn (πʳ i) j            ∎))
 
  to : Fin m → Fin n
  to j = punchOut (to-punchOut {j})
 
  from : Fin n → Fin m
  from j = punchOut {j = πˡ (punchIn (πʳ i) j)} from-punchOut
 
  inverseʳ′ : StrictlyInverseʳ _≡_ to from
  inverseʳ′ j = begin
    from (to j)                                                      ≡⟨⟩
    punchOut {i = i} {πˡ (punchIn (πʳ i) (punchOut to-punchOut))} _  ≡⟨ punchOut-cong′ i (cong πˡ (punchIn-punchOut _)) ⟩
    punchOut {i = i} {πˡ (πʳ (punchIn i j))}                      _  ≡⟨ punchOut-cong i (inverseˡ π) ⟩
    punchOut {i = i} {punchIn i j}                                _  ≡⟨ punchOut-punchIn i ⟩
    j                                                                ∎
 
  inverseˡ′ : StrictlyInverseˡ _≡_ to from
  inverseˡ′ j = begin
    to (from j)                                                       ≡⟨⟩
    punchOut {i = πʳ i} {πʳ (punchIn i (punchOut from-punchOut))}  _  ≡⟨ punchOut-cong′ (πʳ i) (cong πʳ (punchIn-punchOut _)) ⟩
    punchOut {i = πʳ i} {πʳ (πˡ (punchIn (πʳ i) j))}               _  ≡⟨ punchOut-cong (πʳ i) (inverseʳ π) ⟩
    punchOut {i = πʳ i} {punchIn (πʳ i) j}                         _  ≡⟨ punchOut-punchIn (πʳ i) ⟩
    j                                                                 ∎
 
------------------------------------------------------------------------
-- Lifting
 
-- Takes a permutation m → n and creates a permutation (suc m) → (suc n)
-- by mapping 0 to 0 and applying the input permutation to everything else
--
-- Note: should be refactored as a special-case when we add the
-- concatenation of two permutations
lift₀ : Permutation m n → Permutation (suc m) (suc n)
lift₀ {m} {n} π = permutation to from inverseˡ′ inverseʳ′
  where
  to : Fin (suc m) → Fin (suc n)
  to 0F      = 0F
  to (suc i) = suc (π ⟨$⟩ʳ i)
 
  from : Fin (suc n) → Fin (suc m)
  from 0F      = 0F
  from (suc i) = suc (π ⟨$⟩ˡ i)
 
  inverseʳ′ : StrictlyInverseʳ _≡_ to from
  inverseʳ′ 0F      = refl
  inverseʳ′ (suc j) = cong suc (inverseˡ π)
 
  inverseˡ′ : StrictlyInverseˡ _≡_ to from
  inverseˡ′ 0F      = refl
  inverseˡ′ (suc j) = cong suc (inverseʳ π)
 
------------------------------------------------------------------------
-- Insertion
 
-- insert i j π is the permutation that maps i to j and otherwise looks like π
-- it's roughly an inverse of remove
insert : ∀ {m n} → Fin (suc m) → Fin (suc n) → Permutation m n → Permutation (suc m) (suc n)
insert {m} {n} i j π = permutation to from inverseˡ′ inverseʳ′
  where
  to : Fin (suc m) → Fin (suc n)
  to k with i ≟ k
  ... | yes i≡k = j
  ... | no  i≢k = punchIn j (π ⟨$⟩ʳ punchOut i≢k)
 
  from : Fin (suc n) → Fin (suc m)
  from k with j ≟ k
  ... | yes j≡k = i
  ... | no  j≢k = punchIn i (π ⟨$⟩ˡ punchOut j≢k)
 
  inverseʳ′ : StrictlyInverseʳ _≡_ to from
  inverseʳ′ k with i ≟ k
  ... | yes i≡k rewrite proj₂ (dec-yes (j ≟ j) refl) = i≡k
  ... | no  i≢k
    with j≢punchInⱼπʳpunchOuti≢k ← punchInᵢ≢i j (π ⟨$⟩ʳ punchOut i≢k) ∘ sym
    rewrite dec-no (j ≟ punchIn j (π ⟨$⟩ʳ punchOut i≢k)) j≢punchInⱼπʳpunchOuti≢k
    = begin
    punchIn i (π ⟨$⟩ˡ punchOut j≢punchInⱼπʳpunchOuti≢k)                    ≡⟨ cong (λ l → punchIn i (π ⟨$⟩ˡ l)) (punchOut-cong j refl) ⟩
    punchIn i (π ⟨$⟩ˡ punchOut (punchInᵢ≢i j (π ⟨$⟩ʳ punchOut i≢k) ∘ sym)) ≡⟨ cong (λ l → punchIn i (π ⟨$⟩ˡ l)) (punchOut-punchIn j) ⟩
    punchIn i (π ⟨$⟩ˡ (π ⟨$⟩ʳ punchOut i≢k))                              ≡⟨ cong (punchIn i) (inverseˡ π) ⟩
    punchIn i (punchOut i≢k)                                              ≡⟨ punchIn-punchOut i≢k ⟩
    k                                                                     ∎
 
  inverseˡ′ : StrictlyInverseˡ _≡_ to from
  inverseˡ′ k with j ≟ k
  ... | yes j≡k rewrite proj₂ (dec-yes (i ≟ i) refl) = j≡k
  ... | no  j≢k
    with i≢punchInᵢπˡpunchOutj≢k ← punchInᵢ≢i i (π ⟨$⟩ˡ punchOut j≢k) ∘ sym
    rewrite dec-no (i ≟ punchIn i (π ⟨$⟩ˡ punchOut j≢k)) i≢punchInᵢπˡpunchOutj≢k
    = begin
    punchIn j (π ⟨$⟩ʳ punchOut i≢punchInᵢπˡpunchOutj≢k)                    ≡⟨ cong (λ l → punchIn j (π ⟨$⟩ʳ l)) (punchOut-cong i refl) ⟩
    punchIn j (π ⟨$⟩ʳ punchOut (punchInᵢ≢i i (π ⟨$⟩ˡ punchOut j≢k) ∘ sym)) ≡⟨ cong (λ l → punchIn j (π ⟨$⟩ʳ l)) (punchOut-punchIn i) ⟩
    punchIn j (π ⟨$⟩ʳ (π ⟨$⟩ˡ punchOut j≢k))                               ≡⟨ cong (punchIn j) (inverseʳ π) ⟩
    punchIn j (punchOut j≢k)                                               ≡⟨ punchIn-punchOut j≢k ⟩
    k                                                                      ∎
 
------------------------------------------------------------------------
-- Swapping
 
-- Takes a permutation m → n and creates a permutation
-- suc (suc m) → suc (suc n) by mapping 0 to 1 and 1 to 0 and
-- then applying the input permutation to everything else
--
-- Note: should be refactored as a special-case when we add the
-- concatenation of two permutations
swap : Permutation m n → Permutation (2+ m) (2+ n)
swap π = transpose 0F 1F ∘ₚ lift₀ (lift₀ π)
 
------------------------------------------------------------------------
-- Other properties
 
module _ (π : Permutation (suc m) (suc n)) where
  private
    πʳ = π ⟨$⟩ʳ_
    πˡ = π ⟨$⟩ˡ_
 
  punchIn-permute : ∀ i j → πʳ (punchIn i j) ≡ punchIn (πʳ i) (remove i π ⟨$⟩ʳ j)
  punchIn-permute i j = sym (punchIn-punchOut _)
 
  punchIn-permute′ : ∀ i j → πʳ (punchIn (πˡ i) j) ≡ punchIn i (remove (πˡ i) π ⟨$⟩ʳ j)
  punchIn-permute′ i j = begin
    πʳ (punchIn (πˡ i) j)                         ≡⟨ punchIn-permute _ _ ⟩
    punchIn (πʳ (πˡ i)) (remove (πˡ i) π ⟨$⟩ʳ j)  ≡⟨ cong₂ punchIn (inverseʳ π) refl ⟩
    punchIn i (remove (πˡ i) π ⟨$⟩ʳ j)            ∎
 
  lift₀-remove : πʳ 0F ≡ 0F → lift₀ (remove 0F π) ≈ π
  lift₀-remove p 0F      = sym p
  lift₀-remove p (suc i) = punchOut-zero (πʳ (suc i)) p
    where
    punchOut-zero : ∀ {i} (j : Fin (suc n)) {neq} → i ≡ 0F → suc (punchOut {i = i} {j} neq) ≡ j
    punchOut-zero 0F {neq} p = contradiction p neq
    punchOut-zero (suc j) refl = refl
 
↔⇒≡ : Permutation m n → m ≡ n
↔⇒≡ {zero}  {zero}  π = refl
↔⇒≡ {zero}  {suc n} π = contradiction (π ⟨$⟩ˡ 0F) ¬Fin0
↔⇒≡ {suc m} {zero}  π = contradiction (π ⟨$⟩ʳ 0F) ¬Fin0
↔⇒≡ {suc m} {suc n} π = cong suc (↔⇒≡ (remove 0F π))
 
fromPermutation : Permutation m n → Permutation′ m
fromPermutation π = subst (Permutation _) (sym (↔⇒≡ π)) π
 
refute : m ≢ n → ¬ Permutation m n
refute m≢n π = contradiction (↔⇒≡ π) m≢n
 
lift₀-id : (i : Fin (suc n)) → lift₀ id ⟨$⟩ʳ i ≡ i
lift₀-id 0F      = refl
lift₀-id (suc i) = refl
 
lift₀-comp : ∀ (π : Permutation m n) (ρ : Permutation n o) →
             lift₀ π ∘ₚ lift₀ ρ ≈ lift₀ (π ∘ₚ ρ)
lift₀-comp π ρ 0F      = refl
lift₀-comp π ρ (suc i) = refl
 
lift₀-cong : ∀ (π ρ : Permutation m n) → π ≈ ρ → lift₀ π ≈ lift₀ ρ
lift₀-cong π ρ f 0F      = refl
lift₀-cong π ρ f (suc i) = cong suc (f i)
 
lift₀-transpose : ∀ (i j : Fin n) → transpose (suc i) (suc j) ≈ lift₀ (transpose i j)
lift₀-transpose i j 0F      = refl
lift₀-transpose i j (suc k) with does (k ≟ i)
... | true = refl
... | false with does (k ≟ j)
...   | false = refl
...   | true = refl
 
insert-punchIn : ∀ i j (π : Permutation m n) k → insert i j π ⟨$⟩ʳ punchIn i k ≡ punchIn j (π ⟨$⟩ʳ k)
insert-punchIn i j π k with i ≟ punchIn i k
... | yes i≡punchInᵢk = contradiction (sym i≡punchInᵢk) (punchInᵢ≢i i k)
... | no  i≢punchInᵢk = begin
  punchIn j (π ⟨$⟩ʳ punchOut i≢punchInᵢk)            ≡⟨ cong (λ l → punchIn j (π ⟨$⟩ʳ l)) (punchOut-cong i refl) ⟩
  punchIn j (π ⟨$⟩ʳ punchOut (punchInᵢ≢i i k ∘ sym)) ≡⟨ cong (λ l → punchIn j (π ⟨$⟩ʳ l)) (punchOut-punchIn i) ⟩
  punchIn j (π ⟨$⟩ʳ k)                               ∎
 
insert-remove : ∀ i (π : Permutation (suc m) (suc n)) → insert i (π ⟨$⟩ʳ i) (remove i π) ≈ π
insert-remove {m = m} {n = n} i π j with i ≟ j
... | yes i≡j = cong (π ⟨$⟩ʳ_) i≡j
... | no  i≢j = begin
  punchIn (π ⟨$⟩ʳ i) (punchOut (punchInᵢ≢i i (punchOut i≢j) ∘ sym ∘ Injection.injective (↔⇒↣ π))) ≡⟨ punchIn-punchOut _ ⟩
  π ⟨$⟩ʳ punchIn i (punchOut i≢j) ≡⟨ cong (π ⟨$⟩ʳ_) (punchIn-punchOut i≢j) ⟩
  π ⟨$⟩ʳ j ∎
 
remove-insert : ∀ i j (π : Permutation m n) → remove i (insert i j π) ≈ π
remove-insert i j π k with i ≟ i
... | no i≢i = contradiction refl i≢i
... | yes _ = begin
  punchOut {i = j} _
    ≡⟨ punchOut-cong j (insert-punchIn i j π k) ⟩
  punchOut {i = j} (punchInᵢ≢i j (π ⟨$⟩ʳ k) ∘ sym)
    ≡⟨ punchOut-punchIn j ⟩
  π ⟨$⟩ʳ k
    ∎