```------------------------------------------------------------------------
-- The Agda standard library
--
-- Properties related to Fin, and operations making use of these
-- properties (or other properties not available in Data.Fin)
------------------------------------------------------------------------

{-# OPTIONS --cubical-compatible --safe #-}
{-# OPTIONS --warn=noUserWarning #-} -- for deprecated _≺_ (issue #1726)

module Data.Fin.Properties where

open import Axiom.Extensionality.Propositional
open import Algebra.Definitions using (Involutive)
open import Effect.Applicative using (RawApplicative)
open import Effect.Functor using (RawFunctor)
open import Data.Bool.Base using (Bool; true; false; not; _∧_; _∨_)
open import Data.Empty using (⊥; ⊥-elim)
open import Data.Fin.Base
open import Data.Fin.Patterns
open import Data.Nat.Base as ℕ
using (ℕ; zero; suc; s≤s; z≤n; z<s; s<s; _∸_; _^_)
import Data.Nat.Properties as ℕₚ
open import Data.Nat.Solver
open import Data.Unit using (⊤; tt)
open import Data.Product.Base as Prod
using (∃; ∃₂; _×_; _,_; map; proj₁; proj₂; uncurry; <_,_>)
open import Data.Product.Properties using (,-injective)
open import Data.Product.Algebra using (×-cong)
open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]; [_,_]′)
open import Data.Sum.Properties using ([,]-map; [,]-∘)
open import Function.Base using (_∘_; id; _\$_; flip)
open import Function.Bundles using (Injection; _↣_; _⇔_; _↔_; mk⇔; mk↔′)
open import Function.Definitions using (Injective; Surjective)
open import Function.Consequences.Propositional using (contraInjective)
open import Function.Construct.Composition as Comp hiding (injective)
open import Level using (Level)
open import Relation.Binary.Definitions as B hiding (Decidable)
open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_)
open import Relation.Binary.Bundles
using (Preorder; Setoid; DecSetoid; Poset; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder)
open import Relation.Binary.Structures
using (IsDecEquivalence; IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)
open import Relation.Binary.PropositionalEquality as P
using (_≡_; _≢_; refl; sym; trans; cong; cong₂; subst; _≗_; module ≡-Reasoning)
open import Relation.Nullary
using (Reflects; ofʸ; ofⁿ; Dec; _because_; does; proof; yes; no; ¬_; _×-dec_; _⊎-dec_; contradiction)
open import Relation.Nullary.Reflects
open import Relation.Nullary.Decidable as Dec using (map′)
open import Relation.Unary as U
using (U; Pred; Decidable; _⊆_; Satisfiable; Universal)
open import Relation.Unary.Properties using (U?)

private
variable
a : Level
A : Set a
m n o : ℕ
i j : Fin n

------------------------------------------------------------------------
-- Fin
------------------------------------------------------------------------

¬Fin0 : ¬ Fin 0
¬Fin0 ()

------------------------------------------------------------------------
-- Bundles

0↔⊥ : Fin 0 ↔ ⊥
0↔⊥ = mk↔′ ¬Fin0 (λ ()) (λ ()) (λ ())

1↔⊤ : Fin 1 ↔ ⊤
1↔⊤ = mk↔′ (λ { 0F → tt }) (λ { tt → 0F }) (λ { tt → refl }) λ { 0F → refl }

2↔Bool : Fin 2 ↔ Bool
2↔Bool = mk↔′ (λ { 0F → false; 1F → true }) (λ { false → 0F ; true → 1F })
(λ { false → refl ; true → refl }) (λ { 0F → refl ; 1F → refl })

------------------------------------------------------------------------
-- Properties of _≡_
------------------------------------------------------------------------

0≢1+n : zero ≢ Fin.suc i
0≢1+n ()

suc-injective : Fin.suc i ≡ suc j → i ≡ j
suc-injective refl = refl

infix 4 _≟_

_≟_ : DecidableEquality (Fin n)
zero  ≟ zero  = yes refl
zero  ≟ suc y = no λ()
suc x ≟ zero  = no λ()
suc x ≟ suc y = map′ (cong suc) suc-injective (x ≟ y)

------------------------------------------------------------------------
-- Structures

≡-isDecEquivalence : IsDecEquivalence {A = Fin n} _≡_
≡-isDecEquivalence = record
{ isEquivalence = P.isEquivalence
; _≟_           = _≟_
}

------------------------------------------------------------------------
-- Bundles

≡-preorder : ℕ → Preorder _ _ _
≡-preorder n = P.preorder (Fin n)

≡-setoid : ℕ → Setoid _ _
≡-setoid n = P.setoid (Fin n)

≡-decSetoid : ℕ → DecSetoid _ _
≡-decSetoid n = record
{ isDecEquivalence = ≡-isDecEquivalence {n}
}

------------------------------------------------------------------------
-- toℕ
------------------------------------------------------------------------

toℕ-injective : toℕ i ≡ toℕ j → i ≡ j
toℕ-injective {zero}  {}      {}      _
toℕ-injective {suc n} {zero}  {zero}  eq = refl
toℕ-injective {suc n} {suc i} {suc j} eq =
cong suc (toℕ-injective (cong ℕ.pred eq))

toℕ-strengthen : ∀ (i : Fin n) → toℕ (strengthen i) ≡ toℕ i
toℕ-strengthen zero    = refl
toℕ-strengthen (suc i) = cong suc (toℕ-strengthen i)

------------------------------------------------------------------------
-- toℕ-↑ˡ: "i" ↑ˡ n = "i" in Fin (m + n)
------------------------------------------------------------------------

toℕ-↑ˡ : ∀ (i : Fin m) n → toℕ (i ↑ˡ n) ≡ toℕ i
toℕ-↑ˡ zero    n = refl
toℕ-↑ˡ (suc i) n = cong suc (toℕ-↑ˡ i n)

↑ˡ-injective : ∀ n (i j : Fin m) → i ↑ˡ n ≡ j ↑ˡ n → i ≡ j
↑ˡ-injective n zero zero refl = refl
↑ˡ-injective n (suc i) (suc j) eq =
cong suc (↑ˡ-injective n i j (suc-injective eq))

------------------------------------------------------------------------
-- toℕ-↑ʳ: n ↑ʳ "i" = "n + i" in Fin (n + m)
------------------------------------------------------------------------

toℕ-↑ʳ : ∀ n (i : Fin m) → toℕ (n ↑ʳ i) ≡ n ℕ.+ toℕ i
toℕ-↑ʳ zero    i = refl
toℕ-↑ʳ (suc n) i = cong suc (toℕ-↑ʳ n i)

↑ʳ-injective : ∀ n (i j : Fin m) → n ↑ʳ i ≡ n ↑ʳ j → i ≡ j
↑ʳ-injective zero i i refl = refl
↑ʳ-injective (suc n) i j eq = ↑ʳ-injective n i j (suc-injective eq)

------------------------------------------------------------------------
-- toℕ and the ordering relations
------------------------------------------------------------------------

toℕ≤pred[n] : ∀ (i : Fin n) → toℕ i ℕ.≤ ℕ.pred n
toℕ≤pred[n] zero                 = z≤n
toℕ≤pred[n] (suc {n = suc n} i)  = s≤s (toℕ≤pred[n] i)

toℕ≤n : ∀ (i : Fin n) → toℕ i ℕ.≤ n
toℕ≤n {suc n} i = ℕₚ.m≤n⇒m≤1+n (toℕ≤pred[n] i)

toℕ<n : ∀ (i : Fin n) → toℕ i ℕ.< n
toℕ<n {suc n} i = s<s (toℕ≤pred[n] i)

-- A simpler implementation of toℕ≤pred[n],
-- however, with a different reduction behavior.
-- If no one needs the reduction behavior of toℕ≤pred[n],
-- it can be removed in favor of toℕ≤pred[n]′.
toℕ≤pred[n]′ : ∀ (i : Fin n) → toℕ i ℕ.≤ ℕ.pred n
toℕ≤pred[n]′ i = ℕₚ.<⇒≤pred (toℕ<n i)

toℕ-mono-< : i < j → toℕ i ℕ.< toℕ j
toℕ-mono-< i<j = i<j

toℕ-mono-≤ : i ≤ j → toℕ i ℕ.≤ toℕ j
toℕ-mono-≤ i≤j = i≤j

toℕ-cancel-≤ : toℕ i ℕ.≤ toℕ j → i ≤ j
toℕ-cancel-≤ i≤j = i≤j

toℕ-cancel-< : toℕ i ℕ.< toℕ j → i < j
toℕ-cancel-< i<j = i<j

------------------------------------------------------------------------
-- fromℕ
------------------------------------------------------------------------

toℕ-fromℕ : ∀ n → toℕ (fromℕ n) ≡ n
toℕ-fromℕ zero    = refl
toℕ-fromℕ (suc n) = cong suc (toℕ-fromℕ n)

fromℕ-toℕ : ∀ (i : Fin n) → fromℕ (toℕ i) ≡ strengthen i
fromℕ-toℕ zero    = refl
fromℕ-toℕ (suc i) = cong suc (fromℕ-toℕ i)

≤fromℕ : ∀ (i : Fin (ℕ.suc n)) → i ≤ fromℕ n
≤fromℕ i = subst (toℕ i ℕ.≤_) (sym (toℕ-fromℕ _)) (toℕ≤pred[n] i)

------------------------------------------------------------------------
-- fromℕ<
------------------------------------------------------------------------

fromℕ<-toℕ : ∀ (i : Fin n) (i<n : toℕ i ℕ.< n) → fromℕ< i<n ≡ i
fromℕ<-toℕ zero    z<s       = refl
fromℕ<-toℕ (suc i) (s<s i<n) = cong suc (fromℕ<-toℕ i i<n)

toℕ-fromℕ< : ∀ (m<n : m ℕ.< n) → toℕ (fromℕ< m<n) ≡ m
toℕ-fromℕ< z<s               = refl
toℕ-fromℕ< (s<s m<n@(s≤s _)) = cong suc (toℕ-fromℕ< m<n)

-- fromℕ is a special case of fromℕ<.
fromℕ-def : ∀ n → fromℕ n ≡ fromℕ< ℕₚ.≤-refl
fromℕ-def zero    = refl
fromℕ-def (suc n) = cong suc (fromℕ-def n)

fromℕ<-cong : ∀ m n {o} → m ≡ n → (m<o : m ℕ.< o) (n<o : n ℕ.< o) →
fromℕ< m<o ≡ fromℕ< n<o
fromℕ<-cong 0       0       r z<s       z<s  = refl
fromℕ<-cong (suc _) (suc _) r (s<s m<n) (s<s n<o)
= cong suc (fromℕ<-cong _ _ (ℕₚ.suc-injective r) m<n n<o)

fromℕ<-injective : ∀ m n {o} → (m<o : m ℕ.< o) (n<o : n ℕ.< o) →
fromℕ< m<o ≡ fromℕ< n<o → m ≡ n
fromℕ<-injective 0       0       z<s               z<s r = refl
fromℕ<-injective (suc _) (suc _) (s<s m<n@(s≤s _)) (s<s n<o@(s≤s _)) r
= cong suc (fromℕ<-injective _ _ m<n n<o (suc-injective r))

------------------------------------------------------------------------
-- fromℕ<″
------------------------------------------------------------------------

fromℕ<≡fromℕ<″ : ∀ (m<n : m ℕ.< n) (m<″n : m ℕ.<″ n) →
fromℕ< m<n ≡ fromℕ<″ m m<″n
fromℕ<≡fromℕ<″ z<s               (ℕ.less-than-or-equal refl) = refl
fromℕ<≡fromℕ<″ (s<s m<n@(s≤s _)) (ℕ.less-than-or-equal refl) =
cong suc (fromℕ<≡fromℕ<″ m<n (ℕ.less-than-or-equal refl))

toℕ-fromℕ<″ : ∀ (m<n : m ℕ.<″ n) → toℕ (fromℕ<″ m m<n) ≡ m
toℕ-fromℕ<″ {m} {n} m<n = begin
toℕ (fromℕ<″ m m<n)  ≡⟨ cong toℕ (sym (fromℕ<≡fromℕ<″ (ℕₚ.≤″⇒≤ m<n) m<n)) ⟩
toℕ (fromℕ< _)       ≡⟨ toℕ-fromℕ< (ℕₚ.≤″⇒≤ m<n) ⟩
m                    ∎
where open ≡-Reasoning

------------------------------------------------------------------------
-- Properties of cast
------------------------------------------------------------------------

toℕ-cast : ∀ .(eq : m ≡ n) (k : Fin m) → toℕ (cast eq k) ≡ toℕ k
toℕ-cast {n = suc n} eq zero    = refl
toℕ-cast {n = suc n} eq (suc k) = cong suc (toℕ-cast (cong ℕ.pred eq) k)

cast-is-id : .(eq : m ≡ m) (k : Fin m) → cast eq k ≡ k
cast-is-id eq zero    = refl
cast-is-id eq (suc k) = cong suc (cast-is-id (ℕₚ.suc-injective eq) k)

subst-is-cast : (eq : m ≡ n) (k : Fin m) → subst Fin eq k ≡ cast eq k
subst-is-cast refl k = sym (cast-is-id refl k)

cast-trans : .(eq₁ : m ≡ n) .(eq₂ : n ≡ o) (k : Fin m) →
cast eq₂ (cast eq₁ k) ≡ cast (trans eq₁ eq₂) k
cast-trans {m = suc _} {n = suc _} {o = suc _} eq₁ eq₂ zero = refl
cast-trans {m = suc _} {n = suc _} {o = suc _} eq₁ eq₂ (suc k) =
cong suc (cast-trans (ℕₚ.suc-injective eq₁) (ℕₚ.suc-injective eq₂) k)

------------------------------------------------------------------------
-- Properties of _≤_
------------------------------------------------------------------------
-- Relational properties

≤-reflexive : _≡_ ⇒ (_≤_ {n})
≤-reflexive refl = ℕₚ.≤-refl

≤-refl : Reflexive (_≤_ {n})
≤-refl = ≤-reflexive refl

≤-trans : Transitive (_≤_ {n})
≤-trans = ℕₚ.≤-trans

≤-antisym : Antisymmetric _≡_ (_≤_ {n})
≤-antisym x≤y y≤x = toℕ-injective (ℕₚ.≤-antisym x≤y y≤x)

≤-total : Total (_≤_ {n})
≤-total x y = ℕₚ.≤-total (toℕ x) (toℕ y)

≤-irrelevant : Irrelevant (_≤_ {m} {n})
≤-irrelevant = ℕₚ.≤-irrelevant

infix 4 _≤?_ _<?_

_≤?_ : B.Decidable (_≤_ {m} {n})
a ≤? b = toℕ a ℕₚ.≤? toℕ b

_<?_ : B.Decidable (_<_ {m} {n})
m <? n = suc (toℕ m) ℕₚ.≤? toℕ n

------------------------------------------------------------------------
-- Structures

≤-isPreorder : IsPreorder {A = Fin n} _≡_ _≤_
≤-isPreorder = record
{ isEquivalence = P.isEquivalence
; reflexive     = ≤-reflexive
; trans         = ≤-trans
}

≤-isPartialOrder : IsPartialOrder {A = Fin n} _≡_ _≤_
≤-isPartialOrder = record
{ isPreorder = ≤-isPreorder
; antisym    = ≤-antisym
}

≤-isTotalOrder : IsTotalOrder {A = Fin n} _≡_ _≤_
≤-isTotalOrder = record
{ isPartialOrder = ≤-isPartialOrder
; total          = ≤-total
}

≤-isDecTotalOrder : IsDecTotalOrder {A = Fin n} _≡_ _≤_
≤-isDecTotalOrder = record
{ isTotalOrder = ≤-isTotalOrder
; _≟_          = _≟_
; _≤?_         = _≤?_
}

------------------------------------------------------------------------
-- Bundles

≤-preorder : ℕ → Preorder _ _ _
≤-preorder n = record
{ isPreorder = ≤-isPreorder {n}
}

≤-poset : ℕ → Poset _ _ _
≤-poset n = record
{ isPartialOrder = ≤-isPartialOrder {n}
}

≤-totalOrder : ℕ → TotalOrder _ _ _
≤-totalOrder n = record
{ isTotalOrder = ≤-isTotalOrder {n}
}

≤-decTotalOrder : ℕ → DecTotalOrder _ _ _
≤-decTotalOrder n = record
{ isDecTotalOrder = ≤-isDecTotalOrder {n}
}

------------------------------------------------------------------------
-- Properties of _<_
------------------------------------------------------------------------
-- Relational properties

<-irrefl : Irreflexive _≡_ (_<_ {n})
<-irrefl refl = ℕₚ.<-irrefl refl

<-asym : Asymmetric (_<_ {n})
<-asym = ℕₚ.<-asym

<-trans : Transitive (_<_ {n})
<-trans = ℕₚ.<-trans

<-cmp : Trichotomous _≡_ (_<_ {n})
<-cmp zero    zero    = tri≈ (λ()) refl  (λ())
<-cmp zero    (suc j) = tri< z<s   (λ()) (λ())
<-cmp (suc i) zero    = tri> (λ()) (λ()) z<s
<-cmp (suc i) (suc j) with <-cmp i j
... | tri< i<j i≢j j≮i = tri< (s<s i<j)         (i≢j ∘ suc-injective) (j≮i ∘ ℕₚ.≤-pred)
... | tri> i≮j i≢j j<i = tri> (i≮j ∘ ℕₚ.≤-pred) (i≢j ∘ suc-injective) (s<s j<i)
... | tri≈ i≮j i≡j j≮i = tri≈ (i≮j ∘ ℕₚ.≤-pred) (cong suc i≡j)        (j≮i ∘ ℕₚ.≤-pred)

<-respˡ-≡ : (_<_ {m} {n}) Respectsˡ _≡_
<-respˡ-≡ refl x≤y = x≤y

<-respʳ-≡ : (_<_ {m} {n}) Respectsʳ _≡_
<-respʳ-≡ refl x≤y = x≤y

<-resp₂-≡ : (_<_ {n}) Respects₂ _≡_
<-resp₂-≡ = <-respʳ-≡ , <-respˡ-≡

<-irrelevant : Irrelevant (_<_ {m} {n})
<-irrelevant = ℕₚ.<-irrelevant

------------------------------------------------------------------------
-- Structures

<-isStrictPartialOrder : IsStrictPartialOrder {A = Fin n} _≡_ _<_
<-isStrictPartialOrder = record
{ isEquivalence = P.isEquivalence
; irrefl        = <-irrefl
; trans         = <-trans
; <-resp-≈      = <-resp₂-≡
}

<-isStrictTotalOrder : IsStrictTotalOrder {A = Fin n} _≡_ _<_
<-isStrictTotalOrder = record
{ isEquivalence = P.isEquivalence
; trans         = <-trans
; compare       = <-cmp
}

------------------------------------------------------------------------
-- Bundles

<-strictPartialOrder : ℕ → StrictPartialOrder _ _ _
<-strictPartialOrder n = record
{ isStrictPartialOrder = <-isStrictPartialOrder {n}
}

<-strictTotalOrder : ℕ → StrictTotalOrder _ _ _
<-strictTotalOrder n = record
{ isStrictTotalOrder = <-isStrictTotalOrder {n}
}

------------------------------------------------------------------------
-- Other properties

i<1+i : ∀ (i : Fin n) → i < suc i
i<1+i = ℕₚ.n<1+n ∘ toℕ

<⇒≢ : i < j → i ≢ j
<⇒≢ i<i refl = ℕₚ.n≮n _ i<i

≤∧≢⇒< : i ≤ j → i ≢ j → i < j
≤∧≢⇒< {i = zero}  {zero}  _         0≢0     = contradiction refl 0≢0
≤∧≢⇒< {i = zero}  {suc j} _         _       = z<s
≤∧≢⇒< {i = suc i} {suc j} (s≤s i≤j) 1+i≢1+j =
s<s (≤∧≢⇒< i≤j (1+i≢1+j ∘ (cong suc)))

------------------------------------------------------------------------
-- inject
------------------------------------------------------------------------

toℕ-inject : ∀ {i : Fin n} (j : Fin′ i) → toℕ (inject j) ≡ toℕ j
toℕ-inject {i = suc i} zero    = refl
toℕ-inject {i = suc i} (suc j) = cong suc (toℕ-inject j)

------------------------------------------------------------------------
-- inject₁
------------------------------------------------------------------------

fromℕ≢inject₁ : fromℕ n ≢ inject₁ i
fromℕ≢inject₁ {i = suc i} eq = fromℕ≢inject₁ {i = i} (suc-injective eq)

inject₁-injective : inject₁ i ≡ inject₁ j → i ≡ j
inject₁-injective {i = zero}  {zero}  i≡j = refl
inject₁-injective {i = suc i} {suc j} i≡j =
cong suc (inject₁-injective (suc-injective i≡j))

toℕ-inject₁ : ∀ (i : Fin n) → toℕ (inject₁ i) ≡ toℕ i
toℕ-inject₁ zero    = refl
toℕ-inject₁ (suc i) = cong suc (toℕ-inject₁ i)

toℕ-inject₁-≢ : ∀ (i : Fin n) → n ≢ toℕ (inject₁ i)
toℕ-inject₁-≢ (suc i) = toℕ-inject₁-≢ i ∘ ℕₚ.suc-injective

inject₁ℕ< : ∀ (i : Fin n) → toℕ (inject₁ i) ℕ.< n
inject₁ℕ< i rewrite toℕ-inject₁ i = toℕ<n i

inject₁ℕ≤ : ∀ (i : Fin n) → toℕ (inject₁ i) ℕ.≤ n
inject₁ℕ≤ = ℕₚ.<⇒≤ ∘ inject₁ℕ<

≤̄⇒inject₁< : i ≤ j → inject₁ i < suc j
≤̄⇒inject₁< {i = i} i≤j rewrite sym (toℕ-inject₁ i) = s<s i≤j

ℕ<⇒inject₁< : ∀ {i : Fin (ℕ.suc n)} {j : Fin n} → j < i → inject₁ j < i
ℕ<⇒inject₁< {i = suc i} (s≤s j≤i) = ≤̄⇒inject₁< j≤i

i≤inject₁[j]⇒i≤1+j : i ≤ inject₁ j → i ≤ suc j
i≤inject₁[j]⇒i≤1+j {i = zero} i≤j = z≤n
i≤inject₁[j]⇒i≤1+j {i = suc i} {j = suc j} (s≤s i≤j) = s≤s (ℕₚ.m≤n⇒m≤1+n (subst (toℕ i ℕ.≤_) (toℕ-inject₁ j) i≤j))

------------------------------------------------------------------------
-- lower₁
------------------------------------------------------------------------

toℕ-lower₁ : ∀ i (p : n ≢ toℕ i) → toℕ (lower₁ i p) ≡ toℕ i
toℕ-lower₁ {ℕ.zero}  zero    p = contradiction refl p
toℕ-lower₁ {ℕ.suc m} zero    p = refl
toℕ-lower₁ {ℕ.suc m} (suc i) p = cong ℕ.suc (toℕ-lower₁ i (p ∘ cong ℕ.suc))

lower₁-injective : ∀ {n≢i : n ≢ toℕ i} {n≢j : n ≢ toℕ j} →
lower₁ i n≢i ≡ lower₁ j n≢j → i ≡ j
lower₁-injective {zero}  {zero}  {_}     {n≢i} {_}   _    = ⊥-elim (n≢i refl)
lower₁-injective {zero}  {_}     {zero}  {_}   {n≢j} _    = ⊥-elim (n≢j refl)
lower₁-injective {suc n} {zero}  {zero}  {_}   {_}   refl = refl
lower₁-injective {suc n} {suc i} {suc j} {n≢i} {n≢j} eq   =
cong suc (lower₁-injective (suc-injective eq))

------------------------------------------------------------------------
-- inject₁ and lower₁

inject₁-lower₁ : ∀ (i : Fin (suc n)) (n≢i : n ≢ toℕ i) →
inject₁ (lower₁ i n≢i) ≡ i
inject₁-lower₁ {zero}  zero     0≢0     = contradiction refl 0≢0
inject₁-lower₁ {suc n} zero     _       = refl
inject₁-lower₁ {suc n} (suc i)  n+1≢i+1 =
cong suc (inject₁-lower₁ i  (n+1≢i+1 ∘ cong suc))

lower₁-inject₁′ : ∀ (i : Fin n) (n≢i : n ≢ toℕ (inject₁ i)) →
lower₁ (inject₁ i) n≢i ≡ i
lower₁-inject₁′ zero    _       = refl
lower₁-inject₁′ (suc i) n+1≢i+1 =
cong suc (lower₁-inject₁′ i (n+1≢i+1 ∘ cong suc))

lower₁-inject₁ : ∀ (i : Fin n) →
lower₁ (inject₁ i) (toℕ-inject₁-≢ i) ≡ i
lower₁-inject₁ i = lower₁-inject₁′ i (toℕ-inject₁-≢ i)

lower₁-irrelevant : ∀ (i : Fin (suc n)) (n≢i₁ n≢i₂ : n ≢ toℕ i) →
lower₁ i n≢i₁ ≡ lower₁ i n≢i₂
lower₁-irrelevant {zero}  zero     0≢0 _ = contradiction refl 0≢0
lower₁-irrelevant {suc n} zero     _   _ = refl
lower₁-irrelevant {suc n} (suc i)  _   _ =
cong suc (lower₁-irrelevant i _ _)

inject₁≡⇒lower₁≡ : ∀ {i : Fin n} {j : Fin (ℕ.suc n)} →
(n≢j : n ≢ toℕ j) → inject₁ i ≡ j → lower₁ j n≢j ≡ i
inject₁≡⇒lower₁≡ n≢j i≡j = inject₁-injective (trans (inject₁-lower₁ _ n≢j) (sym i≡j))

------------------------------------------------------------------------
-- inject≤
------------------------------------------------------------------------

toℕ-inject≤ : ∀ i (m≤n : m ℕ.≤ n) → toℕ (inject≤ i m≤n) ≡ toℕ i
toℕ-inject≤ {_} {suc n} zero    _         = refl
toℕ-inject≤ {_} {suc n} (suc i) (s≤s m≤n) = cong suc (toℕ-inject≤ i m≤n)

inject≤-refl : ∀ i (n≤n : n ℕ.≤ n) → inject≤ i n≤n ≡ i
inject≤-refl {suc n} zero    _         = refl
inject≤-refl {suc n} (suc i) (s≤s n≤n) = cong suc (inject≤-refl i n≤n)

inject≤-idempotent : ∀ (i : Fin m)
(m≤n : m ℕ.≤ n) (n≤o : n ℕ.≤ o) (m≤o : m ℕ.≤ o) →
inject≤ (inject≤ i m≤n) n≤o ≡ inject≤ i m≤o
inject≤-idempotent {_} {suc n} {suc o} zero    _   _   _ = refl
inject≤-idempotent {_} {suc n} {suc o} (suc i) (s≤s m≤n) (s≤s n≤o) (s≤s m≤o) =
cong suc (inject≤-idempotent i m≤n n≤o m≤o)

inject≤-injective : ∀ (m≤n m≤n′ : m ℕ.≤ n) i j →
inject≤ i m≤n ≡ inject≤ j m≤n′ → i ≡ j
inject≤-injective (s≤s p) (s≤s q) zero    zero    eq = refl
inject≤-injective (s≤s p) (s≤s q) (suc i) (suc j) eq =
cong suc (inject≤-injective p q i j (suc-injective eq))

------------------------------------------------------------------------
-- pred
------------------------------------------------------------------------

pred< : ∀ (i : Fin (suc n)) → i ≢ zero → pred i < i
pred< zero    i≢0 = contradiction refl i≢0
pred< (suc i) _   = ≤̄⇒inject₁< ℕₚ.≤-refl

------------------------------------------------------------------------
-- splitAt
------------------------------------------------------------------------

-- Fin (m + n) ↔ Fin m ⊎ Fin n

splitAt-↑ˡ : ∀ m i n → splitAt m (i ↑ˡ n) ≡ inj₁ i
splitAt-↑ˡ (suc m) zero    n = refl
splitAt-↑ˡ (suc m) (suc i) n rewrite splitAt-↑ˡ m i n = refl

splitAt⁻¹-↑ˡ : ∀ {m} {n} {i} {j} → splitAt m {n} i ≡ inj₁ j → j ↑ˡ n ≡ i
splitAt⁻¹-↑ˡ {suc m} {n} {0F} {.0F} refl = refl
splitAt⁻¹-↑ˡ {suc m} {n} {suc i} {j} eq with splitAt m i in splitAt[m][i]≡inj₁[j]
... | inj₁ k with refl ← eq = cong suc (splitAt⁻¹-↑ˡ {i = i} {j = k} splitAt[m][i]≡inj₁[j])

splitAt-↑ʳ : ∀ m n i → splitAt m (m ↑ʳ i) ≡ inj₂ {B = Fin n} i
splitAt-↑ʳ zero    n i = refl
splitAt-↑ʳ (suc m) n i rewrite splitAt-↑ʳ m n i = refl

splitAt⁻¹-↑ʳ : ∀ {m} {n} {i} {j} → splitAt m {n} i ≡ inj₂ j → m ↑ʳ j ≡ i
splitAt⁻¹-↑ʳ {zero}  {n} {i} {j} refl = refl
splitAt⁻¹-↑ʳ {suc m} {n} {suc i} {j} eq with splitAt m i in splitAt[m][i]≡inj₂[k]
... | inj₂ k with refl ← eq = cong suc (splitAt⁻¹-↑ʳ {i = i} {j = k} splitAt[m][i]≡inj₂[k])

splitAt-join : ∀ m n i → splitAt m (join m n i) ≡ i
splitAt-join m n (inj₁ x) = splitAt-↑ˡ m x n
splitAt-join m n (inj₂ y) = splitAt-↑ʳ m n y

join-splitAt : ∀ m n i → join m n (splitAt m i) ≡ i
join-splitAt zero    n i       = refl
join-splitAt (suc m) n zero    = refl
join-splitAt (suc m) n (suc i) = begin
[ _↑ˡ n , (suc m) ↑ʳ_ ]′ (splitAt (suc m) (suc i)) ≡⟨ [,]-map (splitAt m i) ⟩
[ suc ∘ (_↑ˡ n) , suc ∘ (m ↑ʳ_) ]′ (splitAt m i)   ≡˘⟨ [,]-∘ suc (splitAt m i) ⟩
suc ([ _↑ˡ n , m ↑ʳ_ ]′ (splitAt m i))             ≡⟨ cong suc (join-splitAt m n i) ⟩
suc i                                                         ∎
where open ≡-Reasoning

-- splitAt "m" "i" ≡ inj₁ "i" if i < m

splitAt-< : ∀ m {n} (i : Fin (m ℕ.+ n)) (i<m : toℕ i ℕ.< m) →
splitAt m i ≡ inj₁ (fromℕ< i<m)
splitAt-< (suc m) zero    z<s               = refl
splitAt-< (suc m) (suc i) (s<s i<m) = cong (Sum.map suc id) (splitAt-< m i i<m)

-- splitAt "m" "i" ≡ inj₂ "i - m" if i ≥ m

splitAt-≥ : ∀ m {n} (i : Fin (m ℕ.+ n)) (i≥m : toℕ i ℕ.≥ m) →
splitAt m i ≡ inj₂ (reduce≥ i i≥m)
splitAt-≥ zero    i       _         = refl
splitAt-≥ (suc m) (suc i) (s≤s i≥m) = cong (Sum.map suc id) (splitAt-≥ m i i≥m)

------------------------------------------------------------------------
-- Bundles

+↔⊎ : Fin (m ℕ.+ n) ↔ (Fin m ⊎ Fin n)
+↔⊎ {m} {n} = mk↔′ (splitAt m {n}) (join m n) (splitAt-join m n) (join-splitAt m n)

------------------------------------------------------------------------
-- remQuot
------------------------------------------------------------------------

-- Fin (m * n) ↔ Fin m × Fin n

remQuot-combine : ∀ {n k} (i : Fin n) j → remQuot k (combine i j) ≡ (i , j)
remQuot-combine {suc n} {k} zero    j rewrite splitAt-↑ˡ k j (n ℕ.* k) = refl
remQuot-combine {suc n} {k} (suc i) j rewrite splitAt-↑ʳ k   (n ℕ.* k) (combine i j) =
cong (Prod.map₁ suc) (remQuot-combine i j)

combine-remQuot : ∀ {n} k (i : Fin (n ℕ.* k)) → uncurry combine (remQuot {n} k i) ≡ i
combine-remQuot {suc n} k i with splitAt k i in eq
... | inj₁ j = begin
join k (n ℕ.* k) (inj₁ j)      ≡˘⟨ cong (join k (n ℕ.* k)) eq ⟩
join k (n ℕ.* k) (splitAt k i) ≡⟨ join-splitAt k (n ℕ.* k) i ⟩
i                              ∎
where open ≡-Reasoning
... | inj₂ j = begin
k ↑ʳ (uncurry combine (remQuot {n} k j)) ≡⟨ cong (k ↑ʳ_) (combine-remQuot {n} k j) ⟩
join k (n ℕ.* k) (inj₂ j)                ≡˘⟨ cong (join k (n ℕ.* k)) eq ⟩
join k (n ℕ.* k) (splitAt k i)           ≡⟨ join-splitAt k (n ℕ.* k) i ⟩
i                                        ∎
where open ≡-Reasoning

toℕ-combine : ∀ (i : Fin m) (j : Fin n) → toℕ (combine i j) ≡ n ℕ.* toℕ i ℕ.+ toℕ j
toℕ-combine {suc m} {n} i@0F j = begin
toℕ (combine i j)          ≡⟨⟩
toℕ (j ↑ˡ (m ℕ.* n))       ≡⟨ toℕ-↑ˡ j (m ℕ.* n) ⟩
toℕ j                      ≡⟨⟩
0 ℕ.+ toℕ j                ≡˘⟨ cong (ℕ._+ toℕ j) (ℕₚ.*-zeroʳ n) ⟩
n ℕ.* toℕ i ℕ.+ toℕ j      ∎
where open ≡-Reasoning
toℕ-combine {suc m} {n} (suc i) j = begin
toℕ (combine (suc i) j)        ≡⟨⟩
toℕ (n ↑ʳ combine i j)         ≡⟨ toℕ-↑ʳ n (combine i j) ⟩
n ℕ.+ toℕ (combine i j)        ≡⟨ cong (n ℕ.+_) (toℕ-combine i j) ⟩
n ℕ.+ (n ℕ.* toℕ i ℕ.+ toℕ j)  ≡⟨ solve 3 (λ n i j → n :+ (n :* i :+ j) := n :* (con 1 :+ i) :+ j) refl n (toℕ i) (toℕ j) ⟩
n ℕ.* toℕ (suc i) ℕ.+ toℕ j    ∎
where open ≡-Reasoning; open +-*-Solver

combine-monoˡ-< : ∀ {i j : Fin m} (k l : Fin n) →
i < j → combine i k < combine j l
combine-monoˡ-< {m} {n} {i} {j} k l i<j = begin-strict
toℕ (combine i k)      ≡⟨ toℕ-combine i k ⟩
n ℕ.* toℕ i ℕ.+ toℕ k  <⟨ ℕₚ.+-monoʳ-< (n ℕ.* toℕ i) (toℕ<n k) ⟩
n ℕ.* toℕ i ℕ.+ n      ≡⟨ ℕₚ.+-comm _ n ⟩
n ℕ.+ n ℕ.* toℕ i      ≡⟨ cong (n ℕ.+_) (ℕₚ.*-comm n _) ⟩
n ℕ.+ toℕ i ℕ.* n      ≡⟨ ℕₚ.*-comm (suc (toℕ i)) n ⟩
n ℕ.* suc (toℕ i)      ≤⟨ ℕₚ.*-monoʳ-≤ n (toℕ-mono-< i<j) ⟩
n ℕ.* toℕ j            ≤⟨ ℕₚ.m≤m+n (n ℕ.* toℕ j) (toℕ l) ⟩
n ℕ.* toℕ j ℕ.+ toℕ l  ≡˘⟨ toℕ-combine j l ⟩
toℕ (combine j l)      ∎
where open ℕₚ.≤-Reasoning; open +-*-Solver

combine-injectiveˡ : ∀ (i : Fin m) (j : Fin n) (k : Fin m) (l : Fin n) →
combine i j ≡ combine k l → i ≡ k
combine-injectiveˡ i j k l cᵢⱼ≡cₖₗ with <-cmp i k
... | tri< i<k _ _ = contradiction cᵢⱼ≡cₖₗ (<⇒≢ (combine-monoˡ-< j l i<k))
... | tri≈ _ i≡k _ = i≡k
... | tri> _ _ i>k = contradiction (sym cᵢⱼ≡cₖₗ) (<⇒≢ (combine-monoˡ-< l j i>k))

combine-injectiveʳ : ∀ (i : Fin m) (j : Fin n) (k : Fin m) (l : Fin n) →
combine i j ≡ combine k l → j ≡ l
combine-injectiveʳ {m} {n} i j k l cᵢⱼ≡cₖₗ with combine-injectiveˡ i j k l cᵢⱼ≡cₖₗ
... | refl = toℕ-injective (ℕₚ.+-cancelˡ-≡ (n ℕ.* toℕ i) _ _ (begin
n ℕ.* toℕ i ℕ.+ toℕ j ≡˘⟨ toℕ-combine i j ⟩
toℕ (combine i j)     ≡⟨ cong toℕ cᵢⱼ≡cₖₗ ⟩
toℕ (combine i l)     ≡⟨ toℕ-combine i l ⟩
n ℕ.* toℕ i ℕ.+ toℕ l ∎))
where open ≡-Reasoning

combine-injective : ∀ (i : Fin m) (j : Fin n) (k : Fin m) (l : Fin n) →
combine i j ≡ combine k l → i ≡ k × j ≡ l
combine-injective i j k l cᵢⱼ≡cₖₗ =
combine-injectiveˡ i j k l cᵢⱼ≡cₖₗ ,
combine-injectiveʳ i j k l cᵢⱼ≡cₖₗ

combine-surjective : ∀ (i : Fin (m ℕ.* n)) → ∃₂ λ j k → combine j k ≡ i
combine-surjective {m} {n} i with remQuot {m} n i in eq
... | j , k = j , k , (begin
combine j k                       ≡˘⟨ uncurry (cong₂ combine) (,-injective eq) ⟩
uncurry combine (remQuot {m} n i) ≡⟨ combine-remQuot {m} n i ⟩
i                                 ∎)
where open ≡-Reasoning

------------------------------------------------------------------------
-- Bundles

*↔× : Fin (m ℕ.* n) ↔ (Fin m × Fin n)
*↔× {m} {n} = mk↔′ (remQuot {m} n) (uncurry combine)
(uncurry remQuot-combine)
(combine-remQuot {m} n)

------------------------------------------------------------------------
-- fin→fun
------------------------------------------------------------------------

funToFin-finToFin : funToFin {m} {n} ∘ finToFun ≗ id
funToFin-finToFin {zero}  {n} zero = refl
funToFin-finToFin {suc m} {n} k    =
begin
combine (finToFun {n} {suc m} k zero) (funToFin (finToFun {n} {suc m} k ∘ suc))
≡⟨⟩
combine (quotient {n} (n ^ m) k)
(funToFin (finToFun {n} {m} (remainder {n} (n ^ m) k)))
≡⟨ cong (combine (quotient {n} (n ^ m) k))
(funToFin-finToFin {m} (remainder {n} (n ^ m) k)) ⟩
combine (quotient {n} (n ^ m) k) (remainder {n} (n ^ m) k)
≡⟨⟩
uncurry combine (remQuot {n} (n ^ m) k)
≡⟨ combine-remQuot {n = n} (n ^ m) k ⟩
k
∎ where open ≡-Reasoning

finToFun-funToFin : (f : Fin m → Fin n) → finToFun (funToFin f) ≗ f
finToFun-funToFin {suc m} {n} f  zero   =
begin
quotient (n ^ m) (combine (f zero) (funToFin (f ∘ suc)))
≡⟨ cong proj₁ (remQuot-combine _ _) ⟩
proj₁ (f zero , funToFin (f ∘ suc))
≡⟨⟩
f zero
∎ where open ≡-Reasoning
finToFun-funToFin {suc m} {n} f (suc i) =
begin
finToFun (remainder {n} (n ^ m) (combine (f zero) (funToFin (f ∘ suc)))) i
≡⟨ cong (λ rq → finToFun (proj₂ rq) i) (remQuot-combine {n} _ _) ⟩
finToFun (proj₂ (f zero , funToFin (f ∘ suc))) i
≡⟨⟩
finToFun (funToFin (f ∘ suc)) i
≡⟨ finToFun-funToFin (f ∘ suc) i ⟩
(f ∘ suc) i
≡⟨⟩
f (suc i)
∎ where open ≡-Reasoning

------------------------------------------------------------------------
-- Bundles

^↔→ : Extensionality _ _ → Fin (m ^ n) ↔ (Fin n → Fin m)
^↔→ {m} {n} ext = mk↔′ finToFun funToFin
(ext ∘ finToFun-funToFin)
(funToFin-finToFin {n} {m})

------------------------------------------------------------------------
-- lift
------------------------------------------------------------------------

lift-injective : ∀ (f : Fin m → Fin n) → Injective _≡_ _≡_ f →
∀ k → Injective _≡_ _≡_ (lift k f)
lift-injective f inj zero    {_}     {_}     eq = inj eq
lift-injective f inj (suc k) {zero}  {zero}  eq = refl
lift-injective f inj (suc k) {suc _} {suc _} eq =
cong suc (lift-injective f inj k (suc-injective eq))

------------------------------------------------------------------------
-- pred
------------------------------------------------------------------------

<⇒≤pred : i < j → i ≤ pred j
<⇒≤pred {i = zero}  {j = suc j} z<s       = z≤n
<⇒≤pred {i = suc i} {j = suc j} (s<s i<j) rewrite toℕ-inject₁ j = i<j

------------------------------------------------------------------------
-- _ℕ-_
------------------------------------------------------------------------

toℕ‿ℕ- : ∀ n i → toℕ (n ℕ- i) ≡ n ∸ toℕ i
toℕ‿ℕ- n       zero     = toℕ-fromℕ n
toℕ‿ℕ- (suc n) (suc i)  = toℕ‿ℕ- n i

------------------------------------------------------------------------
-- _ℕ-ℕ_
------------------------------------------------------------------------

ℕ-ℕ≡toℕ‿ℕ- : ∀ n i → n ℕ-ℕ i ≡ toℕ (n ℕ- i)
ℕ-ℕ≡toℕ‿ℕ- n       zero    = sym (toℕ-fromℕ n)
ℕ-ℕ≡toℕ‿ℕ- (suc n) (suc i) = ℕ-ℕ≡toℕ‿ℕ- n i

nℕ-ℕi≤n : ∀ n i → n ℕ-ℕ i ℕ.≤ n
nℕ-ℕi≤n n       zero     = ℕₚ.≤-refl
nℕ-ℕi≤n (suc n) (suc i)  = begin
n ℕ-ℕ i  ≤⟨ nℕ-ℕi≤n n i ⟩
n        ≤⟨ ℕₚ.n≤1+n n ⟩
suc n    ∎
where open ℕₚ.≤-Reasoning

------------------------------------------------------------------------
-- punchIn
------------------------------------------------------------------------

punchIn-injective : ∀ i (j k : Fin n) →
punchIn i j ≡ punchIn i k → j ≡ k
punchIn-injective zero    _       _       refl      = refl
punchIn-injective (suc i) zero    zero    _         = refl
punchIn-injective (suc i) (suc j) (suc k) ↑j+1≡↑k+1 =
cong suc (punchIn-injective i j k (suc-injective ↑j+1≡↑k+1))

punchInᵢ≢i : ∀ i (j : Fin n) → punchIn i j ≢ i
punchInᵢ≢i (suc i) (suc j) = punchInᵢ≢i i j ∘ suc-injective

------------------------------------------------------------------------
-- punchOut
------------------------------------------------------------------------

-- A version of 'cong' for 'punchOut' in which the inequality argument
-- can be changed out arbitrarily (reflecting the proof-irrelevance of
-- that argument).

punchOut-cong : ∀ (i : Fin (suc n)) {j k} {i≢j : i ≢ j} {i≢k : i ≢ k} →
j ≡ k → punchOut i≢j ≡ punchOut i≢k
punchOut-cong {_}     zero    {zero}         {i≢j = 0≢0} = contradiction refl 0≢0
punchOut-cong {_}     zero    {suc j} {zero} {i≢k = 0≢0} = contradiction refl 0≢0
punchOut-cong {_}     zero    {suc j} {suc k}            = suc-injective
punchOut-cong {suc n} (suc i) {zero}  {zero}   _ = refl
punchOut-cong {suc n} (suc i) {suc j} {suc k}    = cong suc ∘ punchOut-cong i ∘ suc-injective

-- An alternative to 'punchOut-cong' in the which the new inequality
-- argument is specific. Useful for enabling the omission of that
-- argument during equational reasoning.

punchOut-cong′ : ∀ (i : Fin (suc n)) {j k} {p : i ≢ j} (q : j ≡ k) →
punchOut p ≡ punchOut (p ∘ sym ∘ trans q ∘ sym)
punchOut-cong′ i q = punchOut-cong i q

punchOut-injective : ∀ {i j k : Fin (suc n)}
(i≢j : i ≢ j) (i≢k : i ≢ k) →
punchOut i≢j ≡ punchOut i≢k → j ≡ k
punchOut-injective {_}     {zero}   {zero}  {_}     0≢0 _   _     = contradiction refl 0≢0
punchOut-injective {_}     {zero}   {_}     {zero}  _   0≢0 _     = contradiction refl 0≢0
punchOut-injective {_}     {zero}   {suc j} {suc k} _   _   pⱼ≡pₖ = cong suc pⱼ≡pₖ
punchOut-injective {suc n} {suc i}  {zero}  {zero}  _   _    _    = refl
punchOut-injective {suc n} {suc i}  {suc j} {suc k} i≢j i≢k pⱼ≡pₖ =
cong suc (punchOut-injective (i≢j ∘ cong suc) (i≢k ∘ cong suc) (suc-injective pⱼ≡pₖ))

punchIn-punchOut : ∀ {i j : Fin (suc n)} (i≢j : i ≢ j) →
punchIn i (punchOut i≢j) ≡ j
punchIn-punchOut {_}     {zero}   {zero}  0≢0 = contradiction refl 0≢0
punchIn-punchOut {_}     {zero}   {suc j} _   = refl
punchIn-punchOut {suc m} {suc i}  {zero}  i≢j = refl
punchIn-punchOut {suc m} {suc i}  {suc j} i≢j =
cong suc (punchIn-punchOut (i≢j ∘ cong suc))

punchOut-punchIn : ∀ i {j : Fin n} → punchOut {i = i} {j = punchIn i j} (punchInᵢ≢i i j ∘ sym) ≡ j
punchOut-punchIn zero    {j}     = refl
punchOut-punchIn (suc i) {zero}  = refl
punchOut-punchIn (suc i) {suc j} = cong suc (begin
punchOut (punchInᵢ≢i i j ∘ suc-injective ∘ sym ∘ cong suc)  ≡⟨ punchOut-cong i refl ⟩
punchOut (punchInᵢ≢i i j ∘ sym)                             ≡⟨ punchOut-punchIn i ⟩
j                                                           ∎)
where open ≡-Reasoning

------------------------------------------------------------------------
-- pinch
------------------------------------------------------------------------

pinch-surjective : ∀ (i : Fin n) → Surjective _≡_ _≡_ (pinch i)
pinch-surjective _       zero    = zero , λ { refl → refl }
pinch-surjective zero    (suc j) = suc (suc j) , λ { refl → refl }
pinch-surjective (suc i) (suc j) = map suc (λ {f refl → cong suc (f refl)}) (pinch-surjective i j)

pinch-mono-≤ : ∀ (i : Fin n) → (pinch i) Preserves _≤_ ⟶ _≤_
pinch-mono-≤ 0F      {0F}    {k}     0≤n       = z≤n
pinch-mono-≤ 0F      {suc j} {suc k} (s≤s j≤k) = j≤k
pinch-mono-≤ (suc i) {0F}    {k}     0≤n       = z≤n
pinch-mono-≤ (suc i) {suc j} {suc k} (s≤s j≤k) = s≤s (pinch-mono-≤ i j≤k)

pinch-injective : ∀ {i : Fin n} {j k : Fin (ℕ.suc n)} →
suc i ≢ j → suc i ≢ k → pinch i j ≡ pinch i k → j ≡ k
pinch-injective {i = i}     {zero}  {zero}  _     _     _  = refl
pinch-injective {i = zero}  {zero}  {suc k} _     1+i≢k eq =
pinch-injective {i = zero}  {suc j} {zero}  1+i≢j _     eq =
contradiction (cong suc (sym eq)) 1+i≢j
pinch-injective {i = zero}  {suc j} {suc k} _     _     eq =
cong suc eq
pinch-injective {i = suc i} {suc j} {suc k} 1+i≢j 1+i≢k eq =
cong suc
(pinch-injective (1+i≢j ∘ cong suc) (1+i≢k ∘ cong suc)
(suc-injective eq))

------------------------------------------------------------------------
-- Quantification
------------------------------------------------------------------------

module _ {p} {P : Pred (Fin (suc n)) p} where

∀-cons : P zero → Π[ P ∘ suc ] → Π[ P ]
∀-cons z s zero    = z
∀-cons z s (suc i) = s i

∀-cons-⇔ : (P zero × Π[ P ∘ suc ]) ⇔ Π[ P ]
∀-cons-⇔ = mk⇔ (uncurry ∀-cons) < _\$ zero , _∘ suc >

∃-here : P zero → ∃⟨ P ⟩
∃-here = zero ,_

∃-there : ∃⟨ P ∘ suc ⟩ → ∃⟨ P ⟩
∃-there = map suc id

∃-toSum : ∃⟨ P ⟩ → P zero ⊎ ∃⟨ P ∘ suc ⟩
∃-toSum ( zero , P₀ ) = inj₁ P₀
∃-toSum (suc f , P₁₊) = inj₂ (f , P₁₊)

⊎⇔∃ : (P zero ⊎ ∃⟨ P ∘ suc ⟩) ⇔ ∃⟨ P ⟩
⊎⇔∃ = mk⇔ [ ∃-here , ∃-there ] ∃-toSum

decFinSubset : ∀ {p q} {P : Pred (Fin n) p} {Q : Pred (Fin n) q} →
Decidable Q → (∀ {i} → Q i → Dec (P i)) → Dec (Q ⊆ P)
decFinSubset {zero}  {_}     {_} Q? P? = yes λ {}
decFinSubset {suc n} {P = P} {Q} Q? P?
with Q? zero | ∀-cons {P = λ x → Q x → P x}
... | false because [¬Q0] | cons =
map′ (λ f {x} → cons (⊥-elim ∘ invert [¬Q0]) (λ x → f {x}) x)
(λ f {x} → f {suc x})
(decFinSubset (Q? ∘ suc) P?)
... | true  because  [Q0] | cons =
map′ (uncurry λ P0 rec {x} → cons (λ _ → P0) (λ x → rec {x}) x)
< _\$ invert [Q0] , (λ f {x} → f {suc x}) >
(P? (invert [Q0]) ×-dec decFinSubset (Q? ∘ suc) P?)

any? : ∀ {p} {P : Pred (Fin n) p} → Decidable P → Dec (∃ P)
any? {zero}  {P = _} P? = no λ { (() , _) }
any? {suc n} {P = P} P? = Dec.map ⊎⇔∃ (P? zero ⊎-dec any? (P? ∘ suc))

all? : ∀ {p} {P : Pred (Fin n) p} → Decidable P → Dec (∀ f → P f)
all? P? = map′ (λ ∀p f → ∀p tt) (λ ∀p {x} _ → ∀p x)
(decFinSubset U? (λ {f} _ → P? f))

private
-- A nice computational property of `all?`:
-- The boolean component of the result is exactly the
-- obvious fold of boolean tests (`foldr _∧_ true`).
note : ∀ {p} {P : Pred (Fin 3) p} (P? : Decidable P) →
∃ λ z → does (all? P?) ≡ z
note P? = does (P? 0F) ∧ does (P? 1F) ∧ does (P? 2F) ∧ true
, refl

-- If a decidable predicate P over a finite set is sometimes false,
-- then we can find the smallest value for which this is the case.

¬∀⟶∃¬-smallest : ∀ n {p} (P : Pred (Fin n) p) → Decidable P →
¬ (∀ i → P i) → ∃ λ i → ¬ P i × ((j : Fin′ i) → P (inject j))
¬∀⟶∃¬-smallest zero    P P? ¬∀P = contradiction (λ()) ¬∀P
¬∀⟶∃¬-smallest (suc n) P P? ¬∀P with P? zero
... | false because [¬P₀] = (zero , invert [¬P₀] , λ ())
... | true  because  [P₀] = map suc (map id (∀-cons (invert [P₀])))
(¬∀⟶∃¬-smallest n (P ∘ suc) (P? ∘ suc) (¬∀P ∘ (∀-cons (invert [P₀]))))

-- When P is a decidable predicate over a finite set the following
-- lemma can be proved.

¬∀⟶∃¬ : ∀ n {p} (P : Pred (Fin n) p) → Decidable P →
¬ (∀ i → P i) → (∃ λ i → ¬ P i)
¬∀⟶∃¬ n P P? ¬P = map id proj₁ (¬∀⟶∃¬-smallest n P P? ¬P)

------------------------------------------------------------------------
-- Properties of functions to and from Fin
------------------------------------------------------------------------

-- The pigeonhole principle.

pigeonhole : m ℕ.< n → (f : Fin n → Fin m) → ∃₂ λ i j → i < j × f i ≡ f j
pigeonhole z<s               f = contradiction (f zero) λ()
pigeonhole (s<s m<n@(s≤s _)) f with any? (λ k → f zero ≟ f (suc k))
... | yes (j , f₀≡fⱼ) = zero , suc j , z<s , f₀≡fⱼ
... | no  f₀≢fₖ with pigeonhole m<n (λ j → punchOut (f₀≢fₖ ∘ (j ,_ )))
...   | (i , j , i<j , fᵢ≡fⱼ) =
suc i , suc j , s<s i<j ,
punchOut-injective (f₀≢fₖ ∘ (i ,_)) _ fᵢ≡fⱼ

injective⇒≤ : ∀ {f : Fin m → Fin n} → Injective _≡_ _≡_ f → m ℕ.≤ n
injective⇒≤ {zero}  {_}     {f} _   = z≤n
injective⇒≤ {suc _} {zero}  {f} _   = contradiction (f zero) ¬Fin0
injective⇒≤ {suc _} {suc _} {f} inj = s≤s (injective⇒≤ (λ eq →
suc-injective (inj (punchOut-injective
(contraInjective inj 0≢1+n)
(contraInjective inj 0≢1+n) eq))))

<⇒notInjective : ∀ {f : Fin m → Fin n} → n ℕ.< m → ¬ (Injective _≡_ _≡_ f)
<⇒notInjective n<m inj = ℕₚ.≤⇒≯ (injective⇒≤ inj) n<m

ℕ→Fin-notInjective : ∀ (f : ℕ → Fin n) → ¬ (Injective _≡_ _≡_ f)
ℕ→Fin-notInjective f inj = ℕₚ.<-irrefl refl
(injective⇒≤ (Comp.injective _≡_ _≡_ _≡_ toℕ-injective inj))

-- Cantor-Schröder-Bernstein for finite sets

cantor-schröder-bernstein : ∀ {f : Fin m → Fin n} {g : Fin n → Fin m} →
Injective _≡_ _≡_ f → Injective _≡_ _≡_ g →
m ≡ n
cantor-schröder-bernstein f-inj g-inj = ℕₚ.≤-antisym
(injective⇒≤ f-inj) (injective⇒≤ g-inj)

------------------------------------------------------------------------
-- Effectful
------------------------------------------------------------------------

module _ {f} {F : Set f → Set f} (RA : RawApplicative F) where

open RawApplicative RA

sequence : ∀ {n} {P : Pred (Fin n) f} →
(∀ i → F (P i)) → F (∀ i → P i)
sequence {zero}  ∀iPi = pure λ()
sequence {suc n} ∀iPi = ∀-cons <\$> ∀iPi zero <*> sequence (∀iPi ∘ suc)

module _ {f} {F : Set f → Set f} (RF : RawFunctor F) where

open RawFunctor RF

sequence⁻¹ : ∀ {A : Set f} {P : Pred A f} →
F (∀ i → P i) → (∀ i → F (P i))
sequence⁻¹ F∀iPi i = (λ f → f i) <\$> F∀iPi

------------------------------------------------------------------------
-- If there is an injection from a type A to a finite set, then the type
-- has decidable equality.

module _ {ℓ} {S : Setoid a ℓ} (inj : Injection S (≡-setoid n)) where
open Setoid S

inj⇒≟ : B.Decidable _≈_
inj⇒≟ = Dec.via-injection inj _≟_

inj⇒decSetoid : DecSetoid a ℓ
inj⇒decSetoid = record
{ isDecEquivalence = record
{ isEquivalence = isEquivalence
; _≟_           = inj⇒≟
}
}

------------------------------------------------------------------------
-- Opposite
------------------------------------------------------------------------

opposite-prop : ∀ (i : Fin n) → toℕ (opposite i) ≡ n ∸ suc (toℕ i)
opposite-prop {suc n} zero    = toℕ-fromℕ n
opposite-prop {suc n} (suc i) = begin
toℕ (inject₁ (opposite i)) ≡⟨ toℕ-inject₁ (opposite i) ⟩
toℕ (opposite i)           ≡⟨ opposite-prop i ⟩
n ∸ suc (toℕ i)            ∎
where open ≡-Reasoning

opposite-involutive : Involutive {A = Fin n} _≡_ opposite
opposite-involutive {suc n} i = toℕ-injective (begin
toℕ (opposite (opposite i)) ≡⟨ opposite-prop (opposite i) ⟩
n ∸ (toℕ (opposite i))      ≡⟨ cong (n ∸_) (opposite-prop i) ⟩
n ∸ (n ∸ (toℕ i))           ≡⟨ ℕₚ.m∸[m∸n]≡n (toℕ≤pred[n] i) ⟩
toℕ i                       ∎)
where open ≡-Reasoning

opposite-suc : ∀ (i : Fin n) → toℕ (opposite (suc i)) ≡ toℕ (opposite i)
opposite-suc {n} i = begin
toℕ (opposite (suc i))     ≡⟨ opposite-prop (suc i) ⟩
suc n ∸ suc (toℕ (suc i))  ≡⟨⟩
n ∸ toℕ (suc i)            ≡⟨⟩
n ∸ suc (toℕ i)            ≡⟨ sym (opposite-prop i) ⟩
toℕ (opposite i)           ∎
where open ≡-Reasoning

------------------------------------------------------------------------
-- DEPRECATED NAMES
------------------------------------------------------------------------
-- Please use the new names as continuing support for the old names is
-- not guaranteed.

-- Version 1.5

inject+-raise-splitAt = join-splitAt
{-# WARNING_ON_USAGE inject+-raise-splitAt
"Warning: inject+-raise-splitAt was deprecated in v1.5.
#-}

-- Version 2.0

toℕ-raise = toℕ-↑ʳ
{-# WARNING_ON_USAGE toℕ-raise
"Warning: toℕ-raise was deprecated in v2.0.
#-}
toℕ-inject+ : ∀ {m} n (i : Fin m) → toℕ i ≡ toℕ (i ↑ˡ n)
toℕ-inject+ n i = sym (toℕ-↑ˡ i n)
{-# WARNING_ON_USAGE toℕ-inject+
"Warning: toℕ-inject+ was deprecated in v2.0.
NB argument order has been flipped:
the left-hand argument is the Fin m
the right-hand is the Nat index increment."
#-}
splitAt-inject+ : ∀ m n i → splitAt m (i ↑ˡ n) ≡ inj₁ i
splitAt-inject+ m n i = splitAt-↑ˡ m i n
{-# WARNING_ON_USAGE splitAt-inject+
"Warning: splitAt-inject+ was deprecated in v2.0.
NB argument order has been flipped."
#-}
splitAt-raise : ∀ m n i → splitAt m (m ↑ʳ i) ≡ inj₂ {B = Fin n} i
splitAt-raise = splitAt-↑ʳ
{-# WARNING_ON_USAGE splitAt-raise
"Warning: splitAt-raise was deprecated in v2.0.
#-}
Fin0↔⊥ : Fin 0 ↔ ⊥
Fin0↔⊥ = 0↔⊥
{-# WARNING_ON_USAGE Fin0↔⊥
"Warning: Fin0↔⊥ was deprecated in v2.0.
#-}
eq? : A ↣ Fin n → DecidableEquality A
eq? = inj⇒≟
{-# WARNING_ON_USAGE eq?
"Warning: eq? was deprecated in v2.0.
#-}

private

z≺s : ∀ {n} → zero ≺ suc n
z≺s = _ ≻toℕ zero

s≺s : ∀ {m n} → m ≺ n → suc m ≺ suc n
s≺s (n ≻toℕ i) = (suc n) ≻toℕ (suc i)

<⇒≺ : ℕ._<_ ⇒ _≺_
<⇒≺ {zero}  z<s      = z≺s
<⇒≺ {suc m} (s<s lt) = s≺s (<⇒≺ lt)

≺⇒< : _≺_ ⇒ ℕ._<_
≺⇒< (n ≻toℕ i) = toℕ<n i

≺⇒<′ : _≺_ ⇒ ℕ._<′_
≺⇒<′ lt = ℕₚ.<⇒<′ (≺⇒< lt)
{-# WARNING_ON_USAGE ≺⇒<′
"Warning: ≺⇒<′ was deprecated in v2.0.