Martin Escardo, January 2021.

It is possible to work with lists *defined* from the ingredients of
our Spartan MLTT (see the module Fin.lagda). For the moment we are
Athenian in this respect.

\begin{code}

{-# OPTIONS --safe --without-K #-} -- --exact-split

module MLTT.List where

open import MLTT.Spartan
open import MLTT.Bool
open import Naturals.Properties

data List {𝓤} (X : 𝓤 ̇ ) : 𝓤 ̇  where
[] : List X
_∷_ : X → List X → List X

infixr 3 _∷_

{-# BUILTIN LIST List #-}

length : {X : 𝓤 ̇ } → List X → ℕ
length []       = 0
length (x ∷ xs) = succ (length xs)

Vector' : 𝓤 ̇ → ℕ → 𝓤 ̇
Vector' X n = (Σ xs ꞉ List X , length xs ＝ n)

_∷'_ : {X : 𝓤 ̇ } {n : ℕ} → X → Vector' X n → Vector' X (succ n)
x ∷' (xs , p) = (x ∷ xs) , ap succ p

equal-heads : {X : 𝓤 ̇ } {x y : X} {s t : List X}
→ x ∷ s ＝ y ∷ t
→ x ＝ y

equal-tails : {X : 𝓤 ̇ } {x y : X} {s t : List X}
→ x ∷ s ＝ y ∷ t
→ s ＝ t
equal-tails {𝓤} {X} refl = refl

[_] : {X : 𝓤 ̇ } → X → List X
[ x ] = x ∷ []

[]-is-not-cons : {X : 𝓤 ̇ } (x : X) (xs : List X)
→ [] ≠ x ∷ xs
[]-is-not-cons x []        ()
[]-is-not-cons x (x₀ ∷ xs) ()

_++_ : {X : 𝓤 ̇ } → List X → List X → List X
[]      ++ t = t
(x ∷ s) ++ t = x ∷ (s ++ t)

infixr 4 _++_

[]-right-neutral : {X : 𝓤 ̇ } (s : List X) → s ＝ s ++ []
[]-right-neutral []      = refl
[]-right-neutral (x ∷ s) = ap (x ∷_) ([]-right-neutral s)

++-assoc : {X : 𝓤 ̇ } → associative (_++_ {𝓤} {X})
++-assoc []      t u = refl
++-assoc (x ∷ s) t u = ap (x ∷_) (++-assoc s t u)

foldr : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X → Y → Y) → Y → List X → Y
foldr _·_ ε []       = ε
foldr _·_ ε (x ∷ xs) = x · foldr _·_ ε xs

map : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X → Y) → List X → List Y
map f []       = []
map f (x ∷ xs) = f x ∷ map f xs

_<\$>_ = map

empty : {𝓤 : Universe} {X : 𝓤 ̇ } → List X → Bool
empty []       = true
empty (x ∷ xs) = false

data member {X : 𝓤 ̇ } : X → List X → 𝓤 ̇  where
in-head : {x : X}   {xs : List X} → member x (x ∷ xs)
in-tail : {x y : X} {xs : List X} → member x xs → member x (y ∷ xs)

member-map : {X Y : Type} (f : X → Y) (x : X) (xs : List X)
→ member x xs
→ member (f x) (map f xs)
member-map f x' (_ ∷ xs) (in-tail m) = in-tail (member-map f x' xs m)

member' : {X : Type} → X → List X → Type
member' y []       = 𝟘
member' y (x ∷ xs) = (x ＝ y) + member y xs

member'-map : {X Y : Type} (f : X → Y) (x : X) (xs : List X)
→ member' x xs
→ member' (f x) (map f xs)
member'-map f x' (x ∷ xs) (inl p) = inl (ap f p)
member'-map f x' (x ∷ xs) (inr m) = inr (member-map f x' xs m)

listed : Type → Type
listed X = Σ xs ꞉ List X , ((x : X) → member x xs)

listed⁺ : Type → Type
listed⁺ X = X × listed X

type-from-list : {X : Type} → List X → Type
type-from-list {X} xs = Σ x ꞉ X , member x xs

type-from-list-is-listed : {X : Type} (xs : List X)
→ listed (type-from-list xs)
type-from-list-is-listed {X} [] = [] , g
where
g : (σ : type-from-list []) → member σ []
g (x , ())
type-from-list-is-listed {X} (x ∷ xs) = g
where
h : (x : X) → type-from-list (x ∷ xs)
h x = x , in-head

t : type-from-list xs → type-from-list (x ∷ xs)
t (x , m) = x , in-tail m

α : List (type-from-list xs) → List (type-from-list (x ∷ xs))
α σs = h x ∷ map t σs

β : ((σs , μ) : listed (type-from-list xs))
→ (τ : type-from-list (x ∷ xs)) → member τ (α σs)
β (σs , μ) (y , in-tail m) = in-tail (member-map t (y , m) σs (μ (y , m)))

f : listed (type-from-list xs) → listed (type-from-list (x ∷ xs))
f (σs , μ) = α σs , β (σs , μ)

g : listed (type-from-list (x ∷ xs))
g = f (type-from-list-is-listed xs)

module _ {X : 𝓤 ̇ } where

delete : {n : ℕ}
{y : X}
((xs , _) : Vector' X (succ n))
→ member y xs
→ Vector' X n
delete {0}      _              in-head     = [] , refl
delete {0}      _              (in-tail _) = [] , refl
delete {succ _} ((_ ∷ xs) , p) in-head     = xs , succ-lc p
delete {succ n} ((x ∷ xs) , p) (in-tail m) = x ∷' delete {n} (xs , succ-lc p) m

module list-util
{𝓤 : Universe}
{X : 𝓤 ̇ }
{{_ : Eq X}}
where

_is-in_ : X → List X → Bool
x is-in []       = false
x is-in (y ∷ ys) = (x == y) || (x is-in ys)

insert : X → List X → List X
insert x xs = x ∷ xs

_contained-in_ : List X → List X → Bool
[]       contained-in ys = true
(x ∷ xs) contained-in ys = (x is-in ys) && (xs contained-in ys)

contained-lemma₀ : (x z : X) (xs ys : List X)
→ ys contained-in (x ∷ xs) ＝ true
→ ys contained-in (x ∷ z ∷ xs) ＝ true
contained-lemma₀ x z xs []       e = e
contained-lemma₀ x z xs (y ∷ ys) e = γ
where
IH : ys contained-in (x ∷ xs) ＝ true → ys contained-in (x ∷ z ∷ xs) ＝ true
IH = contained-lemma₀ x z xs ys

e₁ : (y == x) || (y is-in xs) ＝ true
e₁ = pr₁ (&&-gives-× e)

e₂ : ys contained-in (x ∷ xs) ＝ true
e₂ = pr₂ (&&-gives-× e)

a : (y == x) || ((y == z) || (y is-in xs)) ＝ true
a = Cases (||-gives-+ e₁)
(λ (e : (y == x) ＝ true)   → ||-left-intro ((y == z) || (y is-in xs)) e)
(λ (e : y is-in xs ＝ true) → ||-right-intro {y == x} ((y == z) || (y is-in xs)) (||-right-intro (y is-in xs) e))

b : ys contained-in (x ∷ z ∷ xs) ＝ true
b = IH e₂

γ : ((y == x) || ((y == z) || (y is-in xs))) && (ys contained-in (x ∷ z ∷ xs)) ＝ true
γ = &&-intro a b

contained-lemma₁ : (x : X) (ys : List X)
→ ys contained-in (x ∷ ys) ＝ true
contained-lemma₁ x []       = refl
contained-lemma₁ x (y ∷ ys) = γ
where
IH : ys contained-in (x ∷ ys) ＝ true
IH = contained-lemma₁ x ys

a : y == x || (y == y || (y is-in ys)) ＝ true
a = ||-right-intro {y == x} ((y == y) || (y is-in ys)) (||-left-intro (y is-in ys) (==-refl y))

b : ys contained-in (x ∷ y ∷ ys) ＝ true
b = contained-lemma₀ x y ys ys IH

γ : (y == x || (y == y || (y is-in ys))) && (ys contained-in (x ∷ y ∷ ys)) ＝ true
γ = &&-intro a b

some-contained : List (List X) → List X → Bool
some-contained []         ys = false
some-contained (xs ∷ xss) ys = xs contained-in ys || some-contained xss ys

remove-first : X → List X → List X
remove-first x []       = []
remove-first x (y ∷ ys) = if x == y then ys else (y ∷ remove-first x ys)

remove-all : X → List X → List X
remove-all x []       = []
remove-all x (y ∷ ys) = if x == y then remove-all x ys else (y ∷ remove-all x ys)

_minus_ : List X → List X → List X
xs minus []       = xs
xs minus (y ∷ ys) = (remove-all y xs) minus ys

\end{code}

Remove first occurrence:

\begin{code}

remove : X → List X → List X
remove x []       = []
remove x (y ∷ ys) = if x == y then ys else (y ∷ remove x ys)

remove-head : (x y : X) (ys : List X)
→ (x == y) ＝ true
→ remove x (y ∷ ys) ＝ ys
remove-head x y ys q =
remove x (y ∷ ys)                          ＝⟨ refl ⟩
(if x == y then ys else (y ∷ remove x ys)) ＝⟨ I ⟩
(if true then ys else (y ∷ remove x ys))   ＝⟨ refl ⟩
ys                                         ∎
where
I = ap (λ - → if - then ys else (y ∷ remove x ys)) q

remove-tail : (x y : X) (ys : List X)
→ (x == y) ＝ false
→ remove x (y ∷ ys) ＝ y ∷ remove x ys
remove-tail x y ys q =
remove x (y ∷ ys)                        ＝⟨ refl ⟩
if x == y then ys else (y ∷ remove x ys) ＝⟨ I ⟩
if false then ys else (y ∷ remove x ys)  ＝⟨ refl ⟩
y ∷ remove x ys                          ∎
where
I  = ap (λ - → if - then ys else (y ∷ remove x ys)) q

remove-length : (x : X) (ys : List X)
→ member x ys
→ (n : ℕ)
→ length ys ＝ succ n
→ length (remove x ys) ＝ n
remove-length x ys@(z ∷ zs) m n p = h m n p (x == z) refl
where
h : member x ys
→ (n : ℕ)
→ length ys ＝ succ n
→ (b : Bool) → (x == z) ＝ b → length (remove x ys) ＝ n
h _ n p true q =
length (remove x (z ∷ zs)) ＝⟨ ap length (remove-head x z zs q) ⟩
length zs                  ＝⟨ succ-lc p ⟩
n                          ∎

h in-head n p false q =
𝟘-elim (true-is-not-false
(true    ＝⟨ (==-refl x)⁻¹ ⟩
(x == x) ＝⟨ q ⟩
false    ∎))
h (in-tail in-head)     0        () false q
h (in-tail (in-tail m)) 0        () false q
h (in-tail m)           (succ n) p  false q =
length (remove x (z ∷ zs))  ＝⟨ I ⟩
length (z ∷ remove x zs)    ＝⟨ refl ⟩
succ (length (remove x zs)) ＝⟨ II ⟩
succ n                      ∎
where
I  = ap length (remove-tail x z zs q)
II = ap succ (remove-length x zs m n (succ-lc p))

delete' : {n : ℕ}
(x : X)
((xs , _) : Vector' X (succ n))
→ member x xs
→ Vector' X n
delete' {n} x (xs , p) m = remove x xs , remove-length x xs m n p

\end{code}