module Lists

-- From the "Lists" chapter of Programming Language Foundations in Agda
-- https://plfa.github.io/Lists/

-- We define a few types and functions on lists and prove a couple of properties
-- about them

-- Natural numbers are zero (Z) or the successor (S) of a natural number
-- We'll use these to represent the length of lists
data Nat : U where
    Z : Nat
    S : Nat -> Nat

-- declare a plus operator and define the corresponding function
infixl 7 _+_
_+_ : Nat -> Nat -> Nat
Z + m = m
S n + m = S (n + m)

-- A list is empty (Nil) or a value followed by a list (separated by the :: operator)
infixr 3 _::_
data List : U -> U where
    Nil : ∀ A. List A
    _::_ : ∀ A. A -> List A -> List A

-- length of a list is defined inductively
length : ∀ A . List A -> Nat
length Nil = Z
length (x :: xs) = S (length xs)

-- List concatenation
infixl 2 _++_
_++_ : ∀ A. List A -> List A -> List A
Nil ++ ys = ys
x :: xs ++ ys = x :: (xs ++ ys)

-- Equality type is the ≡ operator
-- The only constructor is Refl which says a ≡ a
-- Magic happens in the compiler when it tries to make the types
-- fit into this.
infixl 1 _≡_
data _≡_ : ∀ A . A -> A -> U where
    Refl : ∀ A . {a : A} -> a ≡ a

-- If a ≡ b then b ≡ a
sym : ∀ A. {0 a b : A} -> a ≡ b -> b ≡ a
sym Refl = Refl

-- if a ≡ b and b ≡ c then a ≡ c
trans : ∀ A. {0 a b c : A} -> a ≡ b -> b ≡ c -> a ≡ c
trans Refl x = x

-- This lets us replace a with b inside an expression if a ≡ b
replace : ∀ A a b. (P : A -> U) -> a ≡ b -> P a -> P b
replace p Refl x = x

-- if a ≡ b then f a ≡ f b
cong : ∀ A B a b. (f : A -> B) -> a ≡ b -> f a ≡ f b

-- if concatenate two lists, the length is the sum of the lengths
-- of the original lists
length-++ : ∀ A. (xs ys : List A) -> length (xs ++ ys) ≡ length xs + length ys
length-++ Nil ys = Refl
length-++ (x :: xs) ys = cong S (length-++ xs ys)

-- a function to reverse a list
reverse : ∀ A. List A -> List A
reverse Nil = Nil
reverse (x :: xs) = reverse xs ++ (x :: Nil)

-- if we add an empty list to a list, we get the original back
++-identity : ∀ A. (xs : List A) -> xs ++ Nil ≡ xs
++-identity Nil = Refl
++-identity (x :: xs) = cong (_::_ x) (++-identity xs)

-- concatenation is associative
++-associative : ∀ A. (xs ys zs : List A) -> xs ++ (ys ++ zs) ≡ (xs ++ ys) ++ zs

-- reverse distributes over ++, but switches order
reverse-++-distrib : ∀ A. (xs ys : List A) -> reverse (xs ++ ys) ≡ reverse ys ++ reverse xs
reverse-++-distrib Nil ys = sym (++-identity (reverse ys))
reverse-++-distrib (x :: xs) ys =
  trans (cong (\ z => z ++ (x :: Nil)) (reverse-++-distrib xs ys))
        (sym (++-associative (reverse ys) (reverse xs) (x :: Nil)))

-- same thing, but using `replace` in the proof
reverse-++-distrib' : ∀ A. (xs ys : List A) -> reverse (xs ++ ys) ≡ reverse ys ++ reverse xs
reverse-++-distrib' Nil ys = sym (++-identity (reverse ys))
reverse-++-distrib' {A} (x :: xs) ys =
  replace (\ z => (reverse (xs ++ ys) ++ (x :: Nil)) ≡ z)
          (sym (++-associative (reverse ys) (reverse xs) (x :: Nil)))
          (replace (\ z => (reverse (xs ++ ys)) ++ (x :: Nil) ≡ z ++ (x :: Nil)) (reverse-++-distrib' xs ys) Refl)

-- reverse of reverse gives you the original list
reverse-involutive : ∀ A. (xs : List A) -> reverse (reverse xs) ≡ xs
reverse-involutive Nil = Refl
reverse-involutive (x :: xs) =
  trans (reverse-++-distrib (reverse xs) (x :: Nil))
        (cong (_::_ x) (reverse-involutive xs))

-- helper for a different version of reverse
shunt : ∀ A. List A -> List A -> List A
shunt Nil ys = ys
shunt (x :: xs) ys = shunt xs (x :: ys)

-- lemma
shunt-reverse : ∀ A. (xs ys : List A) -> shunt xs ys ≡ reverse xs ++ ys
shunt-reverse Nil ys = Refl
shunt-reverse (x :: xs) ys =
  trans (shunt-reverse xs (x :: ys))
        (++-associative (reverse xs) (x :: Nil) ys)

-- an alternative definition of reverse
reverse' : ∀ A. List A -> List A
reverse' xs = shunt xs Nil

-- proof that the reverse and reverse' give the same results
reverses : ∀ A. (xs : List A) → reverse' xs ≡ reverse xs
reverses xs = trans (shunt-reverse xs Nil) (++-identity _)