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

module Data.Fin.Props where

open import Data.Fin
open import Data.Nat as N
using (; zero; suc; s≤s; z≤n)
renaming (_≤_ to _ℕ≤_; _<_ to _ℕ<_; _+_ to _ℕ+_)
open N.≤-Reasoning
import Data.Nat.Properties as N
open import Function
open import Function.Equality as FunS using (_⟨\$⟩_)
open import Function.Injection
using (Injection; module Injection)
open import Relation.Nullary
open import Relation.Binary
open import Relation.Binary.PropositionalEquality as PropEq
using (_≡_; refl; cong; subst)
open import Category.Functor
open import Category.Applicative

------------------------------------------------------------------------
-- Properties

private
drop-suc :  {o} {m n : Fin o}
suc m  (suc n  Fin (suc o))  m  n
drop-suc refl = refl

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

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

strictTotalOrder :   StrictTotalOrder _ _ _
strictTotalOrder n = record
{ Carrier            = Fin n
; _≈_                = _≡_
; _<_                = _<_
; isStrictTotalOrder = record
{ isEquivalence = PropEq.isEquivalence
; trans         = N.<-trans
; compare       = cmp
; <-resp-≈      = PropEq.resp₂ _<_
}
}
where
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<  lt ¬eq ¬gt = tri< (s≤s lt)         (¬eq  drop-suc) (¬gt  N.≤-pred)
... | tri> ¬lt ¬eq  gt = tri> (¬lt  N.≤-pred) (¬eq  drop-suc) (s≤s gt)
... | tri≈ ¬lt  eq ¬gt = tri≈ (¬lt  N.≤-pred) (cong suc eq)    (¬gt  N.≤-pred)

decSetoid :   DecSetoid _ _
decSetoid n = StrictTotalOrder.decSetoid (strictTotalOrder n)

infix 4 _≟_

_≟_ : {n : }  Decidable {A = Fin n} _≡_
_≟_ {n} = DecSetoid._≟_ (decSetoid n)

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

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

prop-toℕ-≤ :  {n} (x : Fin n)  toℕ x ℕ≤ N.pred n
prop-toℕ-≤ zero                 = z≤n
prop-toℕ-≤ (suc {n = zero}  ())
prop-toℕ-≤ (suc {n = suc n} i)  = s≤s (prop-toℕ-≤ i)

nℕ-ℕi≤n :  n i  n ℕ-ℕ i ℕ≤ n
nℕ-ℕi≤n n       zero     = begin n
nℕ-ℕi≤n zero    (suc ())
nℕ-ℕi≤n (suc n) (suc i)  = begin
n ℕ-ℕ i ≤⟨ nℕ-ℕi≤n n i
n       ≤⟨ N.n≤1+n n
suc n

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

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

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

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

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

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

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))

------------------------------------------------------------------------
-- Operations

infixl 6 _+′_

_+′_ :  {m n} (i : Fin m) (j : Fin n)  Fin (N.pred m ℕ+ n)
i +′ j = inject≤ (i + j) (N._+-mono_ (prop-toℕ-≤ i) ≤-refl)
where open DecTotalOrder N.decTotalOrder renaming (refl to ≤-refl)

-- reverse {n} "i" = "n ∸ 1 ∸ i".

reverse :  {n}  Fin n  Fin n
reverse {zero}  ()
reverse {suc n} i  = inject≤ (n ℕ- i) (N.n∸m≤n (toℕ i) (suc n))

-- If there is an injection from a set to a finite set, then equality
-- of the set can be decided.

eq? :  {s₁ s₂ n} {S : Setoid s₁ s₂}
Injection S (PropEq.setoid (Fin n))  Decidable (Setoid._≈_ S)
eq? inj x y with to ⟨\$⟩ x  to ⟨\$⟩ y where open Injection inj
... | yes tox≡toy = yes (Injection.injective inj tox≡toy)
... | no  tox≢toy = no  (tox≢toy  FunS.cong (Injection.to inj))

-- Quantification over finite sets commutes with applicative functors.

sequence :  {F n} {P : Fin n  Set}  RawApplicative F
(∀ i  F (P i))  F (∀ i  P i)
sequence {F} RA = helper _ _
where
open RawApplicative RA

helper :  n (P : Fin n  Set)  (∀ i  F (P i))  F (∀ i  P i)
helper zero    P ∀iPi = pure (λ())
helper (suc n) P ∀iPi =
combine <\$> ∀iPi zero  helper n  n  P (suc n)) (∀iPi  suc)
where
combine : P zero  (∀ i  P (suc i))   i  P i
combine z s zero    = z
combine z s (suc i) = s i

private

-- Included just to show that sequence above has an inverse (under
-- an equivalence relation with two equivalence classes, one with
-- all inhabited sets and the other with all uninhabited sets).

sequence⁻¹ :  {F}{A} {P : A  Set}  RawFunctor F
F (∀ i  P i)   i  F (P i)
sequence⁻¹ RF F∀iPi i =  f  f i) <\$> F∀iPi
where open RawFunctor RF