Theorems about List.ofFn

def List.ofFn {α : Type u_1} {n : Nat} (f : Fin n → α) : List α

Creates a list by applying f to each potential index in order, starting at 0.

Examples:

• List.ofFn (n := 3) toString = ["0", "1", "2"]
• List.ofFn (fun i => #["red", "green", "blue"].get i.val i.isLt) = ["red", "green", "blue"]

Equations

• List.ofFn f = Fin.foldr n (fun (x1 : Fin n) (x2 : List α) => f x1 :: x2) []

def List.ofFnM {m : Type u_1 → Type u_2} {α : Type u_1} {n : Nat} [Monad m] (f : Fin n → m α) : m (List α)

Creates a list wrapped in a monad by applying the monadic function f : Fin n → m α to each potential index in order, starting at 0.

Equations

• List.ofFnM f = List.reverse <$> Fin.foldlM n (fun (xs : List α) (i : Fin n) => (fun (x : α) => x :: xs) <$> f i) []

@[simp]

theorem List.length_ofFn {n : Nat} {α : Type u_1} {f : Fin n → α} : (ofFn f).length = n

@[simp]

theorem List.getElem_ofFn {n : Nat} {α : Type u_1} {i : Nat} {f : Fin n → α} (h : i < (ofFn f).length) : (ofFn f)[i] = f ⟨i, ⋯⟩

@[simp]

theorem List.getElem?_ofFn {n : Nat} {α : Type u_1} {i : Nat} {f : Fin n → α} : (ofFn f)[i]? = if h : i < n then some (f ⟨i, h⟩) else none

@[simp]

theorem List.ofFn_zero {α : Type u_1} {f : Fin 0 → α} : ofFn f = []

ofFn on an empty domain is the empty list.

@[simp]

theorem List.ofFn_succ {α : Type u_1} {n : Nat} {f : Fin (n + 1) → α} : ofFn f = f 0 :: ofFn fun (i : Fin n) => f i.succ

theorem List.ofFn_succ_last {α : Type u_1} {n : Nat} {f : Fin (n + 1) → α} : ofFn f = (ofFn fun (i : Fin n) => f i.castSucc) ++ [f (Fin.last n)]

theorem List.ofFn_add {α : Type u_1} {n m : Nat} {f : Fin (n + m) → α} : ofFn f = (ofFn fun (i : Fin n) => f (Fin.castLE ⋯ i)) ++ ofFn fun (i : Fin m) => f (Fin.natAdd n i)

@[simp]

theorem List.ofFn_eq_nil_iff {n : Nat} {α : Type u_1} {f : Fin n → α} : ofFn f = [] ↔ n = 0

@[simp]

theorem List.ofFn_getElem {α : Type u_1} {xs : List α} : (ofFn fun (i : Fin xs.length) => xs[↑i]) = xs

@[simp]

theorem List.mem_ofFn {α : Type u_1} {n : Nat} {f : Fin n → α} {a : α} : a ∈ ofFn f ↔ ∃ (i : Fin n), f i = a

@[simp]

theorem List.map_ofFn {n : Nat} {α : Type u_1} {β : Type u_2} {f : Fin n → α} {g : α → β} : map g (ofFn f) = ofFn (g ∘ f)

theorem List.head_ofFn {α : Type u_1} {n : Nat} {f : Fin n → α} (h : ofFn f ≠ []) : (ofFn f).head h = f ⟨0, ⋯⟩

theorem List.getLast_ofFn {α : Type u_1} {n : Nat} {f : Fin n → α} (h : ofFn f ≠ []) : (ofFn f).getLast h = f ⟨n - 1, ⋯⟩

@[simp]

theorem List.ofFnM_zero {m : Type u_1 → Type u_2} {α : Type u_1} [Monad m] [LawfulMonad m] {f : Fin 0 → m α} : ofFnM f = pure []

ofFnM on an empty domain is the empty list.

See Init.Data.List.FinRange for the ofFnM_succ variant.

theorem List.ofFnM_succ_last {m : Type u_1 → Type u_2} {α : Type u_1} {n : Nat} [Monad m] [LawfulMonad m] {f : Fin (n + 1) → m α} : ofFnM f = do let as ← ofFnM fun (i : Fin n) => f i.castSucc let a ← f (Fin.last n) pure (as ++ [a])

theorem List.ofFnM_add {k : Nat} {α : Type u_1} {n : Nat} {m : Type u_1 → Type u_2} [Monad m] [LawfulMonad m] {f : Fin (n + k) → m α} : ofFnM f = do let as ← ofFnM fun (i : Fin n) => f (Fin.castLE ⋯ i) let bs ← ofFnM fun (i : Fin k) => f (Fin.natAdd n i) pure (as ++ bs)

theorem Fin.foldl_cons_eq_append {n : Nat} {α : Type u_1} {f : Fin n → α} {xs : List α} : foldl n (fun (xs : List α) (i : Fin n) => f i :: xs) xs = (List.ofFn f).reverse ++ xs

theorem Fin.foldr_cons_eq_append {n : Nat} {α : Type u_1} {f : Fin n → α} {xs : List α} : foldr n (fun (i : Fin n) (xs : List α) => f i :: xs) xs = List.ofFn f ++ xs

@[simp]

theorem List.ofFnM_pure_comp {m : Type u_1 → Type u_2} {α : Type u_1} [Monad m] [LawfulMonad m] {n : Nat} {f : Fin n → α} : ofFnM (pure ∘ f) = pure (ofFn f)

theorem List.ofFnM_pure {m : Type u_1 → Type u_2} {α : Type u_1} [Monad m] [LawfulMonad m] {n : Nat} {f : Fin n → α} : (ofFnM fun (i : Fin n) => pure (f i)) = pure (ofFn f)

@[simp]

theorem List.idRun_ofFnM {n : Nat} {α : Type u_1} {f : Fin n → Id α} : (ofFnM f).run = ofFn fun (i : Fin n) => (f i).run