------------------------------------------------------------------------
-- A small prelude
------------------------------------------------------------------------

{-# OPTIONS --cubical-compatible --safe #-}

-- Note that parts of Agda's standard library make use of the K rule.

module Prelude where

------------------------------------------------------------------------
-- Universes

-- Basic type universes and universe levels.

open import Agda.Primitive public
  renaming (Set to Type)
  using (Prop; Level; _⊔_; lzero; lsuc)

private
  variable
    a b c ℓ p                       : Level
    @0 A A₁ A₂ B B₁ B₂ C D Whatever : Type a
    @0 P Q R                        : A → Type p

-- Lifting.

record ↑ ℓ (A : Type a) : Type (a ⊔ ℓ) where
  constructor lift
  field lower : A

open ↑ public

------------------------------------------------------------------------
-- Strings

open import Agda.Builtin.String public using (String)

------------------------------------------------------------------------
-- The unit type

-- A variant of the unit type with η-equality.

open import Agda.Builtin.Unit public using (⊤; tt)

-- A variant without η-equality.

data Unit : Type where
  unit : Unit

-- Block s is used to block unfolding (for performance reasons). The
-- string can be used to indicate what it is that is blocked.

Block : String → Type
Block _ = Unit

pattern ⊠ = unit

-- A function that can be used to locally block something.

block : (Unit → A) → A
block f = f ⊠

-- A function that can be used to unblock something.

unblock : (b : Unit) (@0 P : Unit → Type p) → P ⊠ → P b
unblock ⊠ _ p = p

------------------------------------------------------------------------
-- The empty type

-- A family of empty types.

data ⊥ {ℓ} : Type ℓ where

-- An eliminator for the empty types.

⊥-elim₀ : @0 ⊥ {ℓ = ℓ} → Whatever
⊥-elim₀ ()

-- A variant of the eliminator that takes a non-erased argument.

⊥-elim : ⊥ {ℓ = ℓ} → Whatever
⊥-elim x = ⊥-elim₀ x

-- A version of the empty type that is not universe-polymorphic.

⊥₀ : Type
⊥₀ = ⊥

-- Negation.

infix  ¬_

¬_ : Type ℓ → Type ℓ
¬ P = P → ⊥₀

------------------------------------------------------------------------
-- Natural numbers

open import Agda.Builtin.Nat public
  using (zero; suc; _+_; _*_)
  renaming (Nat to ℕ; _-_ to _∸_)

-- Dependent eliminator.

ℕ-rec : P  → (∀ n → P n → P (suc n)) → ∀ n → P n
ℕ-rec z s zero    = z
ℕ-rec z s (suc n) = s n (ℕ-rec z s n)

-- A non-recursive variant of ℕ-rec.

ℕ-case : P  → (∀ n → P (suc n)) → ∀ n → P n
ℕ-case z s = ℕ-rec z (λ n _ → s n)

-- Exponentiation.

infixr  _^_

_^_ : ℕ → ℕ → ℕ
m ^ zero  = 
m ^ suc n = m * m ^ n

-- Factorial.

infix  _!

_! : ℕ → ℕ
zero  ! = 
suc n ! = suc n * n !

-- Translation from natural numbers to levels.

# : ℕ → Level
# zero    = lzero
# (suc n) = lsuc (# n)

------------------------------------------------------------------------
-- Combinators defined using only abstraction and application

infixr  _∘_
infixl  _on_
infixr  _$_

-- The identity function.

id : A → A
id x = x

-- Composition.

_∘_ :
  {@0 B : A → Type b} {@0 C : {x : A} → B x → Type c} →
  (∀ {x} (y : B x) → C y) → (g : (x : A) → B x) →
  ((x : A) → C (g x))
f ∘ g = λ x → f (g x)

-- "Equational" reasoning combinators.

infix  -1 finally-→
infixr -2 step-→

-- For an explanation of why step-→ is defined in this way, see
-- Equality.step-≡.

step-→ : (@0 A : Type a) → (B → C) → (A → B) → A → C
step-→ _ f g = f ∘ g

syntax step-→ A B→C A→B = A →⟨ A→B ⟩ B→C

finally-→ : (@0 A : Type a) (@0 B : Type b) → (A → B) → A → B
finally-→ _ _ A→B = A→B

syntax finally-→ A B A→B = A →⟨ A→B ⟩□ B □

-- Application.

_$_ :
  {@0 B : A → Type b} →
  ((x : A) → B x) → ((x : A) → B x)
f $ x = f x

-- Constant functions.

const : A → (B → A)
const x = λ _ → x

{-# DISPLAY const x _ = x #-}

-- Flips the first two arguments.

flip :
  {@0 C : A → B → Type c} →
  ((x : A) (y : B) → C x y) → ((y : B) (x : A) → C x y)
flip f = λ x y → f y x

-- Applies the unary function to each argument and combines the
-- results using the binary function.

_on_ :
  {@0 C : B → B → Type c} →
  ((x y : B) → C x y) →
  (f : A → B) →
  ((x y : A) → C (f x) (f y))
_*_ on f = λ x y → f x * f y

-- A term's type.

Type-of : {A : Type a} → A → Type a
Type-of {A} _ = A

-- Type signatures.

infix  type-signature

type-signature : (@0 A : Type a) → A → A
type-signature _ a = a

syntax type-signature A a = a ⦂ A

-- The it function can be used to instantiate an argument by using
-- instance search.

it : ⦃ _ : A ⦄ → A
it ⦃ (x) ⦄ = x

-- Case expressions (to be used with pattern-matching lambdas).

infix  case_return_of_ case_of_

case_return_of_ :
  (x : A) (@0 B : A → Type b) → ((x : A) → B x) → B x
case x return B of f = f x

case_of_ : A → (A → B) → B
case x of f = case x return _ of f

------------------------------------------------------------------------
-- Σ-types

infixr  _,′_
infixr  _×_

open import Agda.Builtin.Sigma public
  using (Σ; _,_)
  hiding (module Σ)
  renaming (fst to proj₁; snd to proj₂)

module Σ where

  open Agda.Builtin.Sigma.Σ public
    using ()
    renaming (fst to proj₁; snd to proj₂)

  -- Variants of the projections with erased type arguments.

  proj₁₀ :
    {@ B : A → Type b} →
    Σ A B → A
  proj₁₀ (x , y) = x

  proj₂₀ :
    {@ B : A → Type b}
    (p : Σ A B) → B (proj₁ p)
  proj₂₀ (x , y) = y

open Σ public using (proj₁₀; proj₂₀)

-- A variant where the first argument is implicit.

∃ : {A : Type a} → (A → Type b) → Type (a ⊔ b)
∃ = Σ _

-- Binary products.

_×_ : (A : Type a) (B : Type b) → Type (a ⊔ b)
A × B = Σ A (const B)

-- A variant of _,_ that is specialised to _×_. Use of this variant
-- can make type-inference easier.

_,′_ : A → B → A × B
x ,′ y = x , y

-- A map function.

Σ-map :
  (f : A → B) → (∀ {x} → P x → Q (f x)) →
  Σ A P → Σ B Q
Σ-map f g = λ p → (f (proj₁₀ p) , g (proj₂₀ p))

-- Zip.

Σ-zip :
  (f : A → B → C) → (∀ {x y} → P x → Q y → R (f x y)) →
  Σ A P → Σ B Q → Σ C R
Σ-zip f g = λ p q → (f (proj₁₀ p) (proj₁₀ q) , g (proj₂₀ p) (proj₂₀ q))

-- Curry and uncurry.

curry :
  {@0 B : A → Type b} {@0 C : Σ A B → Type c} →
  ((p : Σ A B) → C p) →
  ((x : A) (y : B x) → C (x , y))
curry f x y = f (x , y)

uncurry :
  {@0 B : A → Type b} {@0 C : Σ A B → Type c} →
  ((x : A) (y : B x) → C (x , y)) →
  ((p : Σ A B) → C p)
uncurry f (x , y) = f x y

-- Swaps the two components of the pair.

swap : A × B → B × A
swap (x , y) = y , x

------------------------------------------------------------------------
-- W-types

data W (A : Type a) (B : A → Type b) : Type (a ⊔ b) where
  sup : (x : A) (f : B x → W A B) → W A B

-- Projections.

headᵂ :
  {@0 B : A → Type b} →
  W A B → A
headᵂ (sup x f) = x

tailᵂ :
  {@0 B : A → Type b} →
  (x : W A B) → B (headᵂ x) → W A B
tailᵂ (sup x f) = f

-- A map function.

W-map :
  (f : A → B) →
  (∀ {x} → Q (f x) → P x) →
  W A P → W B Q
W-map f g (sup x h) = sup (f x) (λ y → W-map f g (h (g y)))

-- If B is always inhabited, then W A B is empty.

opaque

  inhabited⇒W-empty :
    {@0 B : A → Type b} →
    (∀ x → B x) → ¬ W A B
  inhabited⇒W-empty b (sup x f) = inhabited⇒W-empty b (f (b x))

------------------------------------------------------------------------
-- Binary sums

infixr  _⊎_

data _⊎_ (A : Type a) (B : Type b) : Type (a ⊔ b) where
  inj₁ : (x : A) → A ⊎ B
  inj₂ : (y : B) → A ⊎ B

-- Eliminator for binary sums.

[_,_] :
  {@0 C : A ⊎ B → Type c} →
  ((x : A) → C (inj₁ x)) → ((x : B) → C (inj₂ x)) →
  ((x : A ⊎ B) → C x)
[ f , g ] (inj₁ x) = f x
[ f , g ] (inj₂ y) = g y

-- A generalisation of if-then-else.

infix  if_then_else_

if_then_else_ : A ⊎ B → C → C → C
if x then t else f = [ const t , const f ] x

-- A generalisation of not.

not : A ⊎ B → B ⊎ A
not (inj₁ x) = inj₂ x
not (inj₂ x) = inj₁ x

-- A map function.

⊎-map : (A₁ → A₂) → (B₁ → B₂) → A₁ ⊎ B₁ → A₂ ⊎ B₂
⊎-map f g = [ (λ x → inj₁ (f x)) , (λ x → inj₂ (g x)) ]

-- The function from-⊎ is a safe analogue of fromJust. For an example
-- of how from-⊎ can be used, see
-- Quotient.equivalence-but-not-strong-equivalence.

From-⊎ : {A B : Type ℓ} → A ⊎ B → Type ℓ
From-⊎ {A} (inj₁ _) = A
From-⊎ {B} (inj₂ _) = B

from-⊎ : (x : A ⊎ B) → From-⊎ x
from-⊎ (inj₁ x) = x
from-⊎ (inj₂ y) = y

-- A special case of binary sums: decided predicates.

Dec : Type p → Type p
Dec P = P ⊎ ¬ P

pattern yes p = inj₁ p
pattern no  p = inj₂ p

-- Decidable relations.

Decidable : {A : Type a} {B : Type b} →
            (A → B → Type ℓ) → Type (a ⊔ b ⊔ ℓ)
Decidable _∼_ = ∀ x y → Dec (x ∼ y)

-- Exclusive or.

infixr  _Xor_

_Xor_ : Type a → Type b → Type (a ⊔ b)
A Xor B = (A × ¬ B) ⊎ (¬ A × B)

-- Maybe.

Maybe : Type a → Type a
Maybe A = ⊤ ⊎ A

pattern nothing = inj₁ tt
pattern just x  = inj₂ x

-- The truth predicate T is only inhabited when its argument is
-- inj₁ something.

T : A ⊎ B → Type
T b = if b then ⊤ else ⊥

------------------------------------------------------------------------
-- Booleans

-- Booleans.

Bool : Type
Bool = ⊤ ⊎ ⊤

pattern true  = inj₁ tt
pattern false = inj₂ tt

-- And.

infixr  _∧_

_∧_ : Bool → Bool → Bool
b₁ ∧ b₂ = if b₁ then b₂ else false

-- Or.

infixr  _∨_

_∨_ : Bool → Bool → Bool
b₁ ∨ b₂ = if b₁ then true else b₂

------------------------------------------------------------------------
-- Lists

open import Agda.Builtin.List public using (List; []; _∷_)

------------------------------------------------------------------------
-- Finite sets

Fin : ℕ → Type
Fin zero    = ⊥
Fin (suc n) = ⊤ ⊎ Fin n

pattern fzero  = inj₁ tt
pattern fsuc i = inj₂ i

------------------------------------------------------------------------
-- Some relation combinators

-- Combines two relations into a relation on functions.

_→-rel_ : {A : Type a} {C : Type c} →
          (A → C → Type ℓ) → (B → D → Type ℓ) →
          (A → B) → (C → D) → Type (a ⊔ c ⊔ ℓ)
(P →-rel Q) f g = ∀ x y → P x y → Q (f x) (g y)

-- Combines two relations into a relation on products.

_×-rel_ : (A → C → Type ℓ) → (B → D → Type ℓ) → A × B → C × D → Type ℓ
(P ×-rel Q) (x , u) (y , v) = P x y × Q u v

-- Combines two relations into a relation on sums.

_⊎-rel_ : (A → C → Type ℓ) → (B → D → Type ℓ) → A ⊎ B → C ⊎ D → Type ℓ
(P ⊎-rel Q) (inj₁ x) (inj₁ y) = P x y
(P ⊎-rel Q) (inj₁ x) (inj₂ v) = ⊥
(P ⊎-rel Q) (inj₂ u) (inj₁ y) = ⊥
(P ⊎-rel Q) (inj₂ u) (inj₂ v) = Q u v