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

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

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

module Prelude where

------------------------------------------------------------------------
-- Support for universe polymorphism

-- Universe levels.

open import Agda.Primitive public using (Level; _⊔_; lzero; lsuc)

-- Lifting.

record  {a}  (A : Set a) : Set (a  ) where
  constructor lift
  field lower : A

open  public

------------------------------------------------------------------------
-- 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 : Set where
  unit : Unit

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

data  {} : Set  where

⊥-elim :  {w } {Whatever : Set w}   { = }  Whatever
⊥-elim ()

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

⊥₀ : Set
⊥₀ = 

-- Negation.

infix 3 ¬_

¬_ :  {}  Set   Set 
¬ P = P  ⊥₀

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

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

-- Dependent eliminator.

ℕ-rec :  {p} {P :   Set p} 
        P 0  (∀ 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)

-- Exponentiation.

infixr 8 _^_

_^_ :     
m ^ zero  = 1
m ^ suc n = m * m ^ n

-- Factorial.

infix 9 _!

_! :   
zero  ! = 1
suc n ! = suc n * n !

-- Translation from natural numbers to levels.

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

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

infixr 9 _∘_
infixl 1 _on_
infixr 0 _$_

-- The identity function.

id :  {a} {A : Set a}  A  A
id x = x

-- Composition.

_∘_ :  {a b c}
        {A : Set a} {B : A  Set b} {C : {x : A}  B x  Set c} 
      (∀ {x} (y : B x)  C y)  (g : (x : A)  B x) 
      ((x : A)  C (g x))
f  g = λ x  f (g x)

-- Application.

_$_ :  {a b} {A : Set a} {B : A  Set b} 
      ((x : A)  B x)  ((x : A)  B x)
f $ x = f x

-- Constant functions.

const :  {a b} {A : Set a} {B : Set b}  A  (B  A)
const x = λ _  x

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

-- Flips the first two arguments.

flip :  {a b c} {A : Set a} {B : Set b} {C : A  B  Set 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_ :  {a b c} {A : Set a} {B : Set b} {C : Set c} 
       (B  B  C)  (A  B)  (A  A  C)
_*_ on f = λ x y  f x * f y

-- A term's type.

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

-- Type signatures.

infix 0 type-signature

type-signature :  {a} (A : Set a)  A  A
type-signature _ a = a

syntax type-signature A a = a ⦂ A

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

infix 0 case_return_of_ case_of_

case_return_of_ :
   {a b} {A : Set a}
  (x : A) (B : A  Set b)  ((x : A)  B x)  B x
case x return B of f = f x

case_of_ :  {a b} {A : Set a} {B : Set b}  A  (A  B)  B
case x of f = case x return _ of f

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

infixr 4 _,_ _,′_
infixr 2 _×_

record Σ {a b} (A : Set a) (B : A  Set b) : Set (a  b) where
  constructor _,_
  field
    proj₁ : A
    proj₂ : B proj₁

open Σ public

-- A variant where the first argument is implicit.

 :  {a b} {A : Set a}  (A  Set b)  Set (a  b)
 = Σ _

-- Binary products.

_×_ :  {a b} (A : Set a) (B : Set b)  Set (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 : Set a} {B : Set b}  A  B  A × B
_,′_ = _,_

-- A map function.

Σ-map :  {a b p q}
          {A : Set a} {B : Set b} {P : A  Set p} {Q : B  Set q} 
        (f : A  B)  (∀ {x}  P x  Q (f x)) 
        Σ A P  Σ B Q
Σ-map f g = λ p  (f (proj₁ p) , g (proj₂ p))

-- Zip.

Σ-zip :  {a b c p q r}
          {A : Set a} {B : Set b} {C : Set c}
          {P : A  Set p} {Q : B  Set q} {R : C  Set r} 
        (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 (proj₁ p) (proj₁ q) , g (proj₂ p) (proj₂ q))

-- Curry and uncurry.

curry :  {a b c} {A : Set a} {B : A  Set b} {C : Σ A B  Set c} 
        ((p : Σ A B)  C p) 
        ((x : A) (y : B x)  C (x , y))
curry f x y = f (x , y)

uncurry :  {a b c} {A : Set a} {B : A  Set b} {C : Σ A B  Set 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} {A : Set a} {B : Set b}  A × B  B × A
swap (x , y) = y , x

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

data W {a b} (A : Set a) (B : A  Set b) : Set (a  b) where
  sup : (x : A) (f : B x  W A B)  W A B

-- Projections.

head :  {a b} {A : Set a} {B : A  Set b} 
       W A B  A
head (sup x f) = x

tail :  {a b} {A : Set a} {B : A  Set b} 
       (x : W A B)  B (head x)  W A B
tail (sup x f) = f

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

abstract

  inhabited⇒W-empty :  {a b} {A : Set a} {B : A  Set b} 
                      (∀ x  B x)  ¬ W A B
  inhabited⇒W-empty b (sup x f) = inhabited⇒W-empty b (f (b x))

------------------------------------------------------------------------
-- Support for coinduction

open import Agda.Builtin.Size public
  using (Size; Size<_)
  renaming (SizeU to Size-universe; ↑_ to ssuc; ω to )

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

infixr 1 _⊎_

data _⊎_ {a b} (A : Set a) (B : Set b) : Set (a  b) where
  inj₁ : (x : A)  A  B
  inj₂ : (y : B)  A  B

-- Eliminator for binary sums.

[_,_] :  {a b c} {A : Set a} {B : Set b} {C : A  B  Set 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 5 if_then_else_

if_then_else_ :  {a b c} {A : Set a} {B : Set b} {C : Set c} 
                A  B  C  C  C
if x then t else f = [ const t , const f ] x

-- A map function.

⊎-map :  {a₁ a₂ b₁ b₂}
          {A₁ : Set a₁} {A₂ : Set a₂} {B₁ : Set b₁} {B₂ : Set b₂} 
        (A₁  A₂)  (B₁  B₂)  A₁  B₁  A₂  B₂
⊎-map f g = [ inj₁  f , inj₂  g ]

-- 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 : Set }  A  B  Set 
From-⊎ {A = A} (inj₁ _) = A
From-⊎ {B = B} (inj₂ _) = B

from-⊎ :  {} {A B : Set } (x : A  B)  From-⊎ x
from-⊎ (inj₁ x) = x
from-⊎ (inj₂ y) = y

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

Dec :  {p}  Set p  Set p
Dec P = P  ¬ P

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

-- Decidable relations.

Decidable :  {a b } {A : Set a} {B : Set b} 
            (A  B  Set )  Set (a  b  )
Decidable _∼_ =  x y  Dec (x  y)

-- Exclusive or.

infixr 1 _Xor_

_Xor_ :  {a b}  Set a  Set b  Set (a  b)
A Xor B = (A × ¬ B)  (¬ A × B)

-- Maybe.

Maybe :  {a}  Set a  Set 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} {A : Set a} {B : Set b}  A  B  Set
T b = if b then  else 

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

-- Booleans.

Bool : Set
Bool =   

pattern true  = inj₁ tt
pattern false = inj₂ tt

-- Not.

not : Bool  Bool
not b = if b then false else true

-- And.

infixr 6 _∧_

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

-- Or.

infixr 5 _∨_

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

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

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

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

Fin :   Set
Fin zero    = 
Fin (suc n) =   Fin n

pattern fzero  = inj₁ tt
pattern fsuc i = inj₂ i

------------------------------------------------------------------------
-- Vectors

Vec :  {a}  Set a    Set a
Vec A zero    =  _ 
Vec A (suc n) = A × Vec A n

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

-- Combines two relations into a relation on functions.

_→-rel_ :  {a b c d }
            {A : Set a} {B : Set b} {C : Set c} {D : Set d} 
          (A  C  Set )  (B  D  Set ) 
          (A  B)  (C  D)  Set (a  c  )
(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 b c d }
            {A : Set a} {B : Set b} {C : Set c} {D : Set d} 
          (A  C  Set )  (B  D  Set )  A × B  C × D  Set 
(P ×-rel Q) (x , u) (y , v) = P x y × Q u v

-- Combines two relations into a relation on sums.

_⊎-rel_ :  {a b c d }
            {A : Set a} {B : Set b} {C : Set c} {D : Set d} 
          (A  C  Set )  (B  D  Set )  A  B  C  D  Set 
(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