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

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

module Data.Fin.Properties where

open import Algebra.FunctionProperties using (Involutive)
open import Category.Applicative using (RawApplicative)
open import Category.Functor using (RawFunctor)
open import Data.Empty using (⊥-elim)
open import Data.Fin.Base
open import Data.Nat as  using (; zero; suc; s≤s; z≤n; _∸_)
  renaming
  ( _≤_ to _ℕ≤_
  ; _<_ to _ℕ<_
  ; _+_ to _ℕ+_
  )
import Data.Nat.Properties as ℕₚ
open import Data.Unit using (tt)
open import Data.Product using (; ∃₂; ; _×_; _,_; map; proj₁)
open import Function using (_∘_; id)
open import Function.Injection using (_↣_)
open import Relation.Binary as B hiding (Decidable)
open import Relation.Binary.PropositionalEquality as P
  using (_≡_; _≢_; refl; sym; trans; cong; subst; module ≡-Reasoning)
open import Relation.Nullary using (¬_)
import Relation.Nullary.Decidable as Dec
open import Relation.Nullary.Negation using (contradiction)
open import Relation.Nullary using (Dec; yes; no; ¬_)
open import Relation.Unary as U using (U; Pred; Decidable; _⊆_)
open import Relation.Unary.Properties using (U?)

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

¬Fin0 : ¬ Fin 0
¬Fin0 ()

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

suc-injective :  {o} {m n : Fin o}  Fin.suc m  suc n  m  n
suc-injective refl = refl

infix 4 _≟_

_≟_ :  {n}  B.Decidable {A = Fin n} _≡_
zero   zero  = yes refl
zero   suc y = no λ()
suc x  zero  = no λ()
suc x  suc y with x  y
... | yes x≡y = yes (cong suc x≡y)
... | no  x≢y = no  (x≢y  suc-injective)

preorder :   Preorder _ _ _
preorder n = P.preorder (Fin n)

setoid :   Setoid _ _
setoid n = P.setoid (Fin n)

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

decSetoid :   DecSetoid _ _
decSetoid n = record
  { Carrier          = Fin n
  ; _≈_              = _≡_
  ; isDecEquivalence = isDecEquivalence
  }

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

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

toℕ-strengthen :  {n} (i : Fin n)  toℕ (strengthen i)  toℕ i
toℕ-strengthen zero    = refl
toℕ-strengthen (suc i) = cong suc (toℕ-strengthen i)

toℕ-raise :  {m} n (i : Fin m)  toℕ (raise n i)  n ℕ+ toℕ i
toℕ-raise zero    i = refl
toℕ-raise (suc n) i = cong suc (toℕ-raise n i)

toℕ<n :  {n} (i : Fin n)  toℕ i ℕ< n
toℕ<n zero    = s≤s z≤n
toℕ<n (suc i) = s≤s (toℕ<n i)

toℕ≤pred[n] :  {n} (i : Fin n)  toℕ i ℕ≤ ℕ.pred n
toℕ≤pred[n] zero                 = z≤n
toℕ≤pred[n] (suc {n = zero}  ())
toℕ≤pred[n] (suc {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]′ :  {n} (i : Fin n)  toℕ i ℕ≤ ℕ.pred n
toℕ≤pred[n]′ i = ℕₚ.<⇒≤pred (toℕ<n i)

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

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

fromℕ-toℕ :  {n} (i : Fin n)  fromℕ (toℕ i)  strengthen i
fromℕ-toℕ zero    = refl
fromℕ-toℕ (suc i) = cong suc (fromℕ-toℕ i)

------------------------------------------------------------------------
-- fromℕ≤

fromℕ≤-toℕ :  {m} (i : Fin m) (i<m : toℕ i ℕ< m)  fromℕ≤ i<m  i
fromℕ≤-toℕ zero    (s≤s z≤n)       = refl
fromℕ≤-toℕ (suc i) (s≤s (s≤s m≤n)) = cong suc (fromℕ≤-toℕ i (s≤s m≤n))

toℕ-fromℕ≤ :  {m n} (m<n : m ℕ< n)  toℕ (fromℕ≤ m<n)  m
toℕ-fromℕ≤ (s≤s z≤n)       = refl
toℕ-fromℕ≤ (s≤s (s≤s m<n)) = cong suc (toℕ-fromℕ≤ (s≤s 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≤″

fromℕ≤≡fromℕ≤″ :  {m n} (m<n : m ℕ< n) (m<″n : m ℕ.<″ n) 
                  fromℕ≤ m<n  fromℕ≤″ m m<″n
fromℕ≤≡fromℕ≤″ (s≤s z≤n)       (ℕ.less-than-or-equal refl) = refl
fromℕ≤≡fromℕ≤″ (s≤s (s≤s m<n)) (ℕ.less-than-or-equal refl) =
  cong suc (fromℕ≤≡fromℕ≤″ (s≤s m<n) (ℕ.less-than-or-equal refl))

toℕ-fromℕ≤″ :  {m n} (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≤″ _ m<n)) 
  toℕ (fromℕ≤ _)       ≡⟨ toℕ-fromℕ≤ (ℕₚ.≤″⇒≤ m<n) 
  m 
  where open ≡-Reasoning

------------------------------------------------------------------------
-- cast

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

------------------------------------------------------------------------
-- Properties of _≤_

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

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

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

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

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

infix 4 _≤?_ _<?_

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

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

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

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

≤-isPartialOrder :  {n}  IsPartialOrder _≡_ (_≤_ {n})
≤-isPartialOrder = record
  { isPreorder = ≤-isPreorder
  ; antisym    = ≤-antisym
  }

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

≤-isTotalOrder :  {n}  IsTotalOrder _≡_ (_≤_ {n})
≤-isTotalOrder = record
  { isPartialOrder = ≤-isPartialOrder
  ; total          = ≤-total
  }

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

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

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

-- Other properties
≤-irrelevance :  {n}  Irrelevant (_≤_ {n})
≤-irrelevance = ℕₚ.≤-irrelevance

------------------------------------------------------------------------
-- Properties of _<_

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

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

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

<-cmp :  {n}  Trichotomous _≡_ (_<_ {n})
<-cmp zero    zero    = tri≈ (λ())     refl  (λ())
<-cmp zero    (suc j) = tri< (s≤s z≤n) (λ()) (λ())
<-cmp (suc i) zero    = tri> (λ())     (λ()) (s≤s z≤n)
<-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-≡ :  {n}  (_<_ {n}) Respectsˡ _≡_
<-respˡ-≡ refl x≤y = x≤y

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

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

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

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

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

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

-- Other properties
<-irrelevance :  {n}  Irrelevant (_<_ {n})
<-irrelevance = ℕₚ.<-irrelevance

<⇒≢ :  {n} {i j : Fin n}  i < j  i  j
<⇒≢ i<i refl = ℕₚ.n≮n _ i<i

≤∧≢⇒< :  {n} {i j : Fin n}  i  j  i  j  i < j
≤∧≢⇒< {i = zero}  {zero}  _         0≢0     = contradiction refl 0≢0
≤∧≢⇒< {i = zero}  {suc j} _         _       = s≤s z≤n
≤∧≢⇒< {i = suc i} {zero}  ()
≤∧≢⇒< {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 :  {n} {i : Fin n} (j : Fin′ i) 
               toℕ (inject j)  toℕ j
toℕ-inject {i = zero}  ()
toℕ-inject {i = suc i} zero    = refl
toℕ-inject {i = suc i} (suc j) = cong suc (toℕ-inject j)

------------------------------------------------------------------------
-- inject+

toℕ-inject+ :  {m} n (i : Fin m)  toℕ i  toℕ (inject+ n i)
toℕ-inject+ n zero    = refl
toℕ-inject+ n (suc i) = cong suc (toℕ-inject+ n i)

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

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

toℕ-inject₁ :  {n} (i : Fin n)  toℕ (inject₁ i)  toℕ i
toℕ-inject₁ zero    = refl
toℕ-inject₁ (suc i) = cong suc (toℕ-inject₁ i)

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

toℕ-inject≤ :  {m n} (i : Fin m) (le : m ℕ≤ n) 
                toℕ (inject≤ i le)  toℕ i
toℕ-inject≤ zero    (s≤s le) = refl
toℕ-inject≤ (suc i) (s≤s le) = cong suc (toℕ-inject≤ i le)

inject≤-refl :  {n} (i : Fin n) (n≤n : n ℕ≤ n)  inject≤ i n≤n  i
inject≤-refl zero    (s≤s _  ) = refl
inject≤-refl (suc i) (s≤s n≤n) = cong suc (inject≤-refl i n≤n)

------------------------------------------------------------------------
-- _≺_

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

<′⇒≺ : ℕ._<′_  _≺_
<′⇒≺ {n} ℕ.≤′-refl = subst (_≺ suc n) (toℕ-fromℕ n)
                              (suc n ≻toℕ fromℕ n)
<′⇒≺ (ℕ.≤′-step m≤′n) with <′⇒≺ m≤′n
... | n ≻toℕ i = subst (_≺ suc n) (toℕ-inject₁ i) (suc n ≻toℕ _)

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

<⇒≤pred :  {n} {i j : Fin n}  j < i  j  pred i
<⇒≤pred {i = zero}  {_}     ()
<⇒≤pred {i = suc i} {zero}  j<i       = z≤n
<⇒≤pred {i = suc i} {suc j} (s≤s j<i) =
  subst (_ ℕ≤_) (sym (toℕ-inject₁ i)) j<i

------------------------------------------------------------------------
-- ℕ-

toℕ‿ℕ- :  n i  toℕ (n ℕ- i)  n  toℕ i
toℕ‿ℕ- n       zero     = toℕ-fromℕ n
toℕ‿ℕ- zero    (suc ())
toℕ‿ℕ- (suc n) (suc i)  = toℕ‿ℕ- n i

------------------------------------------------------------------------
-- ℕ-ℕ

nℕ-ℕi≤n :  n i  n ℕ-ℕ i ℕ≤ n
nℕ-ℕi≤n n       zero     = ℕₚ.≤-refl
nℕ-ℕi≤n zero    (suc ())
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 :  {m} i (j k : Fin m) 
                    punchIn i j  punchIn i k  j  k
punchIn-injective zero    _       _       refl      = refl
punchIn-injective (suc i) zero    zero    _         = refl
punchIn-injective (suc i) zero    (suc k) ()
punchIn-injective (suc i) (suc j) zero    ()
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 :  {m} i (j : Fin m)  punchIn i j  i
punchInᵢ≢i zero    _    ()
punchInᵢ≢i (suc i) zero ()
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 :  {n} (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 {zero} (suc ())
punchOut-cong {suc n} (suc i) {zero} {zero} _ = refl
punchOut-cong {suc n} (suc i) {zero} {suc k} ()
punchOut-cong {suc n} (suc i) {suc j} {zero} ()
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′ :  {n} (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 :  {m} {i j k : Fin (suc m)}
                     (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 {zero}  {suc ()}
punchOut-injective {suc n} {suc i}  {zero}  {zero}  _   _    _    = refl
punchOut-injective {suc n} {suc i}  {zero}  {suc k} _   _   ()
punchOut-injective {suc n} {suc i}  {suc j} {zero}  _   _   ()
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 :  {m} {i j : Fin (suc m)} (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 {zero}  {suc ()}
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 :  {n} 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

------------------------------------------------------------------------
-- _+′_

infixl 6 _+′_

_+′_ :  {m n} (i : Fin m) (j : Fin n)  Fin (ℕ.pred m ℕ+ n)
i +′ j = inject≤ (i + j) (ℕₚ.+-mono-≤ (toℕ≤pred[n] i) ℕₚ.≤-refl)

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

∀-cons :  {n p} {P : Pred (Fin (suc n)) p} 
         P zero  (∀ i  P (suc i))  (∀ i  P i)
∀-cons z s zero    = z
∀-cons z s (suc i) = s i

decFinSubset :  {n p q} {P : Pred (Fin n) p} {Q : Pred (Fin n) q} 
               Decidable Q  (∀ {f}  Q f  Dec (P f))  Dec (Q  P)
decFinSubset {zero}  {_}     {_} _  _  = yes λ{}
decFinSubset {suc n} {P = P} {Q} Q? P? with decFinSubset (Q?  suc) P?
... | no ¬q⟶p = no  q⟶p  ¬q⟶p (q⟶p))
... | yes q⟶p with Q? zero
...   | no ¬q₀ = yes (∀-cons {P = Q U.⇒ P} (⊥-elim  ¬q₀)  _  q⟶p) _)
...   | yes q₀ with P? q₀
...     | no ¬p₀ = no  q⟶p  ¬p₀ (q⟶p q₀))
...     | yes p₀ = yes (∀-cons {P = Q U.⇒ P}  _  p₀)  _  q⟶p) _)

any? :  {n p} {P : Fin n  Set p}  Decidable P  Dec ( P)
any? {zero}  {_} P? = no λ { (() , _) }
any? {suc n} {P} P? with P? zero | any? (P?  suc)
... | yes P₀ | _              = yes (_ , P₀)
... | no  _  | yes (_ , P₁₊ᵢ) = yes (_ , P₁₊ᵢ)
... | no ¬P₀ | no ¬P₁₊ᵢ       = no λ
  { (zero  , P₀)    ¬P₀ P₀
  ; (suc f , P₁₊ᵢ)  ¬P₁₊ᵢ (_ , P₁₊ᵢ)
  }

all? :  {n p} {P : Pred (Fin n) p} 
       Decidable P  Dec (∀ f  P f)
all? P? with decFinSubset U?  {f} _  P? f)
... | yes ∀p = yes  f  ∀p tt)
... | no ¬∀p = no   ∀p  ¬∀p  _  ∀p _))

-- 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
... | no ¬P₀ = (zero , ¬P₀ , λ ())
... | yes P₀ = map suc (map id (∀-cons P₀))
  (¬∀⟶∃¬-smallest n (P  suc) (P?  suc) (¬∀P  (∀-cons 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)

-- The pigeonhole principle.

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

------------------------------------------------------------------------
-- Categorical

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 =  f  f i) <$> F∀iPi

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

module _ {a} {A : Set a} where

  eq? :  {n}  A  Fin n  B.Decidable {A = A} _≡_
  eq? inj = Dec.via-injection inj _≟_

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

-- Version 0.15

cmp              = <-cmp
{-# WARNING_ON_USAGE cmp
"Warning: cmp was deprecated in v0.15.
Please use <-cmp instead."
#-}
strictTotalOrder = <-strictTotalOrder
{-# WARNING_ON_USAGE strictTotalOrder
"Warning: strictTotalOrder was deprecated in v0.15.
Please use <-strictTotalOrder instead."
#-}

-- Version 0.16

to-from = toℕ-fromℕ
{-# WARNING_ON_USAGE to-from
"Warning: to-from was deprecated in v0.16.
Please use toℕ-fromℕ instead."
#-}
from-to          = fromℕ-toℕ
{-# WARNING_ON_USAGE from-to
"Warning: from-to was deprecated in v0.16.
Please use fromℕ-toℕ instead."
#-}
bounded = toℕ<n
{-# WARNING_ON_USAGE bounded
"Warning: bounded was deprecated in v0.16.
Please use toℕ<n instead."
#-}
prop-toℕ-≤ = toℕ≤pred[n]
{-# WARNING_ON_USAGE prop-toℕ-≤
"Warning: prop-toℕ-≤ was deprecated in v0.16.
Please use toℕ≤pred[n] instead."
#-}
prop-toℕ-≤′ = toℕ≤pred[n]′
{-# WARNING_ON_USAGE prop-toℕ-≤′
"Warning: prop-toℕ-≤′ was deprecated in v0.16.
Please use toℕ≤pred[n]′ instead."
#-}
inject-lemma = toℕ-inject
{-# WARNING_ON_USAGE inject-lemma
"Warning: inject-lemma was deprecated in v0.16.
Please use toℕ-inject instead."
#-}
inject+-lemma = toℕ-inject+
{-# WARNING_ON_USAGE inject+-lemma
"Warning: inject+-lemma was deprecated in v0.16.
Please use toℕ-inject+ instead."
#-}
inject₁-lemma = toℕ-inject₁
{-# WARNING_ON_USAGE inject₁-lemma
"Warning: inject₁-lemma was deprecated in v0.16.
Please use toℕ-inject₁ instead."
#-}
inject≤-lemma = toℕ-inject≤
{-# WARNING_ON_USAGE inject≤-lemma
"Warning: inject≤-lemma was deprecated in v0.16.
Please use toℕ-inject≤ instead."
#-}

-- Version 0.17

≤+≢⇒< = ≤∧≢⇒<
{-# WARNING_ON_USAGE ≤+≢⇒<
"Warning: ≤+≢⇒< was deprecated in v0.17.
Please use ≤∧≢⇒< instead."
#-}