------------------------------------------------------------------------
-- The Agda standard library
--
-- Some Vec-related properties
------------------------------------------------------------------------

module Data.Vec.Properties where

open import Algebra
open import Category.Applicative.Indexed
open import Category.Monad
open import Category.Monad.Identity
open import Data.Vec
open import Data.List.Any
  using (here; there) renaming (module Membership-≡ to List)
open import Data.Nat
open import Data.Empty using (⊥-elim)
import Data.Nat.Properties as Nat
open import Data.Fin as Fin using (Fin; zero; suc; toℕ; fromℕ)
open import Data.Fin.Properties using (_+′_)
open import Data.Empty using (⊥-elim)
open import Data.Product using (_×_ ; _,_)
open import Function
open import Function.Inverse using (_↔_)
open import Relation.Binary

module UsingVectorEquality {s₁ s₂} (S : Setoid s₁ s₂) where

  private module SS = Setoid S
  open SS using () renaming (Carrier to A)
  import Data.Vec.Equality as VecEq
  open VecEq.Equality S

  replicate-lemma :  {m} n x (xs : Vec A m) 
                    replicate {n = n}     x ++ (x  xs) 
                    replicate {n = 1 + n} x ++      xs
  replicate-lemma zero    x xs = refl (x  xs)
  replicate-lemma (suc n) x xs = SS.refl ∷-cong replicate-lemma n x xs

  xs++[]=xs :  {n} (xs : Vec A n)  xs ++ []  xs
  xs++[]=xs []       = []-cong
  xs++[]=xs (x  xs) = SS.refl ∷-cong xs++[]=xs xs

  map-++-commute :  {b m n} {B : Set b}
                   (f : B  A) (xs : Vec B m) {ys : Vec B n} 
                   map f (xs ++ ys)  map f xs ++ map f ys
  map-++-commute f []       = refl _
  map-++-commute f (x  xs) = SS.refl ∷-cong map-++-commute f xs

open import Relation.Binary.PropositionalEquality as P
  using (_≡_; _≢_; refl; _≗_)
open import Relation.Binary.HeterogeneousEquality using (_≅_; refl)

∷-injective :  {a n} {A : Set a} {x y : A} {xs ys : Vec A n} 
              (x  xs)  (y  ys)  x  y × xs  ys
∷-injective refl = refl , refl

-- lookup is an applicative functor morphism.

lookup-morphism :
   {a n} (i : Fin n) 
  Morphism (applicative {a = a})
           (RawMonad.rawIApplicative IdentityMonad)
lookup-morphism i = record
  { op      = lookup i
  ; op-pure = lookup-replicate i
  ; op-⊛    = lookup-⊛ i
  }
  where
  lookup-replicate :  {a n} {A : Set a} (i : Fin n) 
                     lookup i  replicate {A = A}  id {A = A}
  lookup-replicate zero    = λ _  refl
  lookup-replicate (suc i) = lookup-replicate i

  lookup-⊛ :  {a b n} {A : Set a} {B : Set b}
             i (fs : Vec (A  B) n) (xs : Vec A n) 
             lookup i (fs  xs)  (lookup i fs $ lookup i xs)
  lookup-⊛ zero    (f  fs) (x  xs) = refl
  lookup-⊛ (suc i) (f  fs) (x  xs) = lookup-⊛ i fs xs

-- tabulate is an inverse of flip lookup.

lookup∘tabulate :  {a n} {A : Set a} (f : Fin n  A) (i : Fin n) 
                  lookup i (tabulate f)  f i
lookup∘tabulate f zero    = refl
lookup∘tabulate f (suc i) = lookup∘tabulate (f  suc) i

tabulate∘lookup :  {a n} {A : Set a} (xs : Vec A n) 
                  tabulate (flip lookup xs)  xs
tabulate∘lookup []       = refl
tabulate∘lookup (x  xs) = P.cong (_∷_ x) $ tabulate∘lookup xs

-- If you look up an index in allFin n, then you get the index.

lookup-allFin :  {n} (i : Fin n)  lookup i (allFin n)  i
lookup-allFin = lookup∘tabulate id

-- Various lemmas relating lookup and _++_.

lookup-++-< :  {a} {A : Set a} {m n}
              (xs : Vec A m) (ys : Vec A n) i (i<m : toℕ i < m) 
              lookup i (xs ++ ys)  lookup (Fin.fromℕ≤ i<m) xs
lookup-++-< []       ys i       ()
lookup-++-< (x  xs) ys zero    (s≤s z≤n)       = refl
lookup-++-< (x  xs) ys (suc i) (s≤s (s≤s i<m)) =
  lookup-++-< xs ys i (s≤s i<m)

lookup-++-≥ :  {a} {A : Set a} {m n}
              (xs : Vec A m) (ys : Vec A n) i (i≥m : toℕ i  m) 
              lookup i (xs ++ ys)  lookup (Fin.reduce≥ i i≥m) ys
lookup-++-≥ []       ys i       i≥m = refl
lookup-++-≥ (x  xs) ys zero    ()
lookup-++-≥ (x  xs) ys (suc i) (s≤s i≥m) = lookup-++-≥ xs ys i i≥m

lookup-++-inject+ :  {a} {A : Set a} {m n}
                    (xs : Vec A m) (ys : Vec A n) i 
                    lookup (Fin.inject+ n i) (xs ++ ys)  lookup i xs
lookup-++-inject+ []       ys ()
lookup-++-inject+ (x  xs) ys zero    = refl
lookup-++-inject+ (x  xs) ys (suc i) = lookup-++-inject+ xs ys i

lookup-++-+′ :  {a} {A : Set a} {m n}
               (xs : Vec A m) (ys : Vec A n) i 
               lookup (fromℕ m +′ i) (xs ++ ys)  lookup i ys
lookup-++-+′ []       ys       zero    = refl
lookup-++-+′ []       (y  xs) (suc i) = lookup-++-+′ [] xs i
lookup-++-+′ (x  xs) ys       i       = lookup-++-+′ xs ys i

-- Properties relating lookup and _[_]≔_.

lookup∘update :  {a} {A : Set a} {n}
                (i : Fin n) (xs : Vec A n) x 
                lookup i (xs [ i ]≔ x)  x
lookup∘update zero    (_  xs) x = refl
lookup∘update (suc i) (_  xs) x = lookup∘update i xs x

lookup∘update′ :  {a} {A : Set a} {n} {i j : Fin n} 
                 i  j   (xs : Vec A n) y 
                 lookup i (xs [ j ]≔ y)  lookup i xs
lookup∘update′ {i = zero}  {zero}  i≢j      xs  y = ⊥-elim (i≢j refl)
lookup∘update′ {i = zero}  {suc j} i≢j (x  xs) y = refl
lookup∘update′ {i = suc i} {zero}  i≢j (x  xs) y = refl
lookup∘update′ {i = suc i} {suc j} i≢j (x  xs) y =
  lookup∘update′ (i≢j  P.cong suc) xs y

-- map is a congruence.

map-cong :  {a b n} {A : Set a} {B : Set b} {f g : A  B} 
           f  g  _≗_ {A = Vec A n} (map f) (map g)
map-cong f≗g []       = refl
map-cong f≗g (x  xs) = P.cong₂ _∷_ (f≗g x) (map-cong f≗g xs)

-- map is functorial.

map-id :  {a n} {A : Set a}  _≗_ {A = Vec A n} (map id) id
map-id []       = refl
map-id (x  xs) = P.cong (_∷_ x) (map-id xs)

map-∘ :  {a b c n} {A : Set a} {B : Set b} {C : Set c}
        (f : B  C) (g : A  B) 
        _≗_ {A = Vec A n} (map (f  g)) (map f  map g)
map-∘ f g []       = refl
map-∘ f g (x  xs) = P.cong (_∷_ (f (g x))) (map-∘ f g xs)

-- tabulate can be expressed using map and allFin.

tabulate-∘ :  {n a b} {A : Set a} {B : Set b}
             (f : A  B) (g : Fin n  A) 
             tabulate (f  g)  map f (tabulate g)
tabulate-∘ {zero}  f g = refl
tabulate-∘ {suc n} f g =
  P.cong (_∷_ (f (g zero))) (tabulate-∘ f (g  suc))

tabulate-allFin :  {n a} {A : Set a} (f : Fin n  A) 
                  tabulate f  map f (allFin n)
tabulate-allFin f = tabulate-∘ f id

-- The positive case of allFin can be expressed recursively using map.

allFin-map :  n  allFin (suc n)  zero  map suc (allFin n)
allFin-map n = P.cong (_∷_ zero) $ tabulate-∘ suc id

-- If you look up every possible index, in increasing order, then you
-- get back the vector you started with.

map-lookup-allFin :  {a} {A : Set a} {n} (xs : Vec A n) 
                    map  x  lookup x xs) (allFin n)  xs
map-lookup-allFin {n = n} xs = begin
  map  x  lookup x xs) (allFin n) ≡⟨ P.sym $ tabulate-∘  x  lookup x xs) id 
  tabulate  x  lookup x xs)       ≡⟨ tabulate∘lookup xs 
  xs                                 
  where open P.≡-Reasoning

-- tabulate f contains f i.

∈-tabulate :  {n a} {A : Set a} (f : Fin n  A) i  f i  tabulate f
∈-tabulate f zero    = here
∈-tabulate f (suc i) = there (∈-tabulate (f  suc) i)

-- allFin n contains all elements in Fin n.

∈-allFin :  {n} (i : Fin n)  i  allFin n
∈-allFin = ∈-tabulate id

-- sum commutes with _++_.

sum-++-commute :  {m n} (xs : Vec  m) {ys : Vec  n} 
                 sum (xs ++ ys)  sum xs + sum ys
sum-++-commute []            = refl
sum-++-commute (x  xs) {ys} = begin
  x + sum (xs ++ ys)
    ≡⟨ P.cong  p  x + p) (sum-++-commute xs) 
  x + (sum xs + sum ys)
    ≡⟨ P.sym (+-assoc x (sum xs) (sum ys)) 
  sum (x  xs) + sum ys
    
  where
  open P.≡-Reasoning
  open CommutativeSemiring Nat.commutativeSemiring hiding (_+_; sym)

-- foldr is a congruence.

foldr-cong :  {a b} {A : Set a}
               {B₁ :   Set b}
               {f₁ :  {n}  A  B₁ n  B₁ (suc n)} {e₁}
               {B₂ :   Set b}
               {f₂ :  {n}  A  B₂ n  B₂ (suc n)} {e₂} 
             (∀ {n x} {y₁ : B₁ n} {y₂ : B₂ n} 
                y₁  y₂  f₁ x y₁  f₂ x y₂) 
             e₁  e₂ 
              {n} (xs : Vec A n) 
             foldr B₁ f₁ e₁ xs  foldr B₂ f₂ e₂ xs
foldr-cong           _     e₁=e₂ []       = e₁=e₂
foldr-cong {B₁ = B₁} f₁=f₂ e₁=e₂ (x  xs) =
  f₁=f₂ (foldr-cong {B₁ = B₁} f₁=f₂ e₁=e₂ xs)

-- foldl is a congruence.

foldl-cong :  {a b} {A : Set a}
               {B₁ :   Set b}
               {f₁ :  {n}  B₁ n  A  B₁ (suc n)} {e₁}
               {B₂ :   Set b}
               {f₂ :  {n}  B₂ n  A  B₂ (suc n)} {e₂} 
             (∀ {n x} {y₁ : B₁ n} {y₂ : B₂ n} 
                y₁  y₂  f₁ y₁ x  f₂ y₂ x) 
             e₁  e₂ 
              {n} (xs : Vec A n) 
             foldl B₁ f₁ e₁ xs  foldl B₂ f₂ e₂ xs
foldl-cong           _     e₁=e₂ []       = e₁=e₂
foldl-cong {B₁ = B₁} f₁=f₂ e₁=e₂ (x  xs) =
  foldl-cong {B₁ = B₁  suc} f₁=f₂ (f₁=f₂ e₁=e₂) xs

-- The (uniqueness part of the) universality property for foldr.

foldr-universal :  {a b} {A : Set a} (B :   Set b)
                  (f :  {n}  A  B n  B (suc n)) {e}
                  (h :  {n}  Vec A n  B n) 
                  h []  e 
                  (∀ {n} x  h  _∷_ x  f {n} x  h) 
                   {n}  h  foldr B {n} f e
foldr-universal B f     h base step []       = base
foldr-universal B f {e} h base step (x  xs) = begin
  h (x  xs)
    ≡⟨ step x xs 
  f x (h xs)
    ≡⟨ P.cong (f x) (foldr-universal B f h base step xs) 
  f x (foldr B f e xs)
    
  where open P.≡-Reasoning

-- A fusion law for foldr.

foldr-fusion :  {a b c} {A : Set a}
                 {B :   Set b} {f :  {n}  A  B n  B (suc n)} e
                 {C :   Set c} {g :  {n}  A  C n  C (suc n)}
               (h :  {n}  B n  C n) 
               (∀ {n} x  h  f {n} x  g x  h) 
                {n}  h  foldr B {n} f e  foldr C g (h e)
foldr-fusion {B = B} {f} e {C} h fuse =
  foldr-universal C _ _ refl  x xs  fuse x (foldr B f e xs))

-- The identity function is a fold.

idIsFold :  {a n} {A : Set a}  id  foldr (Vec A) {n} _∷_ []
idIsFold = foldr-universal _ _ id refl  _ _  refl)

-- The _∈_ predicate is equivalent (in the following sense) to the
-- corresponding predicate for lists.

∈⇒List-∈ :  {a} {A : Set a} {n x} {xs : Vec A n} 
           x  xs  x List.∈ toList xs
∈⇒List-∈ here       = here P.refl
∈⇒List-∈ (there x∈) = there (∈⇒List-∈ x∈)

List-∈⇒∈ :  {a} {A : Set a} {x : A} {xs} 
           x List.∈ xs  x  fromList xs
List-∈⇒∈ (here P.refl) = here
List-∈⇒∈ (there x∈)    = there (List-∈⇒∈ x∈)

-- Proof irrelevance for _[_]=_.

proof-irrelevance-[]= :  {a} {A : Set a} {n} {xs : Vec A n} {i x} 
                        (p q : xs [ i ]= x)  p  q
proof-irrelevance-[]= here            here             = refl
proof-irrelevance-[]= (there xs[i]=x) (there xs[i]=x') =
  P.cong there (proof-irrelevance-[]= xs[i]=x xs[i]=x')

-- _[_]=_ can be expressed using lookup and _≡_.

[]=↔lookup :  {a n i} {A : Set a} {x} {xs : Vec A n} 
             xs [ i ]= x  lookup i xs  x
[]=↔lookup {i = i} {x = x} {xs} = record
  { to         = P.→-to-⟶ to
  ; from       = P.→-to-⟶ (from i xs)
  ; inverse-of = record
    { left-inverse-of  = λ _  proof-irrelevance-[]= _ _
    ; right-inverse-of = λ _  P.proof-irrelevance _ _
    }
  }
  where
  to :  {n xs} {i : Fin n}  xs [ i ]= x  lookup i xs  x
  to here            = refl
  to (there xs[i]=x) = to xs[i]=x

  from :  {n} (i : Fin n) xs  lookup i xs  x  xs [ i ]= x
  from zero    (.x  _)  refl = here
  from (suc i) (_   xs) p    = there (from i xs p)

------------------------------------------------------------------------
-- Some properties related to _[_]≔_

[]≔-idempotent :
   {n a} {A : Set a} (xs : Vec A n) (i : Fin n) {x₁ x₂ : A} 
  (xs [ i ]≔ x₁) [ i ]≔ x₂  xs [ i ]≔ x₂
[]≔-idempotent []       ()
[]≔-idempotent (x  xs) zero    = refl
[]≔-idempotent (x  xs) (suc i) = P.cong (_∷_ x) $ []≔-idempotent xs i

[]≔-commutes :
   {n a} {A : Set a} (xs : Vec A n) (i j : Fin n) {x y : A} 
  i  j  (xs [ i ]≔ x) [ j ]≔ y  (xs [ j ]≔ y) [ i ]≔ x
[]≔-commutes []       ()      ()      _
[]≔-commutes (x  xs) zero    zero    0≢0 = ⊥-elim $ 0≢0 refl
[]≔-commutes (x  xs) zero    (suc i) _   = refl
[]≔-commutes (x  xs) (suc i) zero    _   = refl
[]≔-commutes (x  xs) (suc i) (suc j) i≢j =
  P.cong (_∷_ x) $ []≔-commutes xs i j (i≢j  P.cong suc)

[]≔-updates :  {n a} {A : Set a} (xs : Vec A n) (i : Fin n) {x : A} 
              (xs [ i ]≔ x) [ i ]= x
[]≔-updates []       ()
[]≔-updates (x  xs) zero    = here
[]≔-updates (x  xs) (suc i) = there ([]≔-updates xs i)

[]≔-minimal :
   {n a} {A : Set a} (xs : Vec A n) (i j : Fin n) {x y : A} 
  i  j  xs [ i ]= x  (xs [ j ]≔ y) [ i ]= x
[]≔-minimal []       ()      ()      _   _
[]≔-minimal (x  xs) .zero   zero    0≢0 here        = ⊥-elim $ 0≢0 refl
[]≔-minimal (x  xs) .zero   (suc j) _   here        = here
[]≔-minimal (x  xs) (suc i) zero    _   (there loc) = there loc
[]≔-minimal (x  xs) (suc i) (suc j) i≢j (there loc) =
  there ([]≔-minimal xs i j (i≢j  P.cong suc) loc)

map-[]≔ :  {n a b} {A : Set a} {B : Set b}
          (f : A  B) (xs : Vec A n) (i : Fin n) {x : A} 
          map f (xs [ i ]≔ x)  map f xs [ i ]≔ f x
map-[]≔ f []       ()
map-[]≔ f (x  xs) zero    = refl
map-[]≔ f (x  xs) (suc i) = P.cong (_∷_ _) $ map-[]≔ f xs i

[]≔-lookup :  {a} {A : Set a} {n} (xs : Vec A n) (i : Fin n) 
             xs [ i ]≔ lookup i xs  xs
[]≔-lookup []       ()
[]≔-lookup (x  xs) zero    = refl
[]≔-lookup (x  xs) (suc i) = P.cong (_∷_ x) $ []≔-lookup xs i

[]≔-++-inject+ :  {a} {A : Set a} {m n x}
                 (xs : Vec A m) (ys : Vec A n) i 
                 (xs ++ ys) [ Fin.inject+ n i ]≔ x  xs [ i ]≔ x ++ ys
[]≔-++-inject+ []       ys ()
[]≔-++-inject+ (x  xs) ys zero    = refl
[]≔-++-inject+ (x  xs) ys (suc i) =
  P.cong (_∷_ x) $ []≔-++-inject+ xs ys i