@[inline]

def Fin.foldl {α : Sort u_1} (n : Nat) (f : α → Fin n → α) (init : α) : α

Combine all the values that can be represented by Fin n with an initial value, starting at 0 and nesting to the left.

Example:

• Fin.foldl 3 (· + ·.val) (0 : Nat) = ((0 + (0 : Fin 3).val) + (1 : Fin 3).val) + (2 : Fin 3).val

Equations

• Fin.foldl n f init = Fin.foldl.loop n f init 0

@[specialize #[]]

def Fin.foldl.loop {α : Sort u_1} (n : Nat) (f : α → Fin n → α) (x : α) (i : Nat) : α

Inner loop for Fin.foldl. Fin.foldl.loop n f x i = f (f (f x i) ...) (n-1)

Equations

• Fin.foldl.loop n f x i = if h : i < n then Fin.foldl.loop n f (f x ⟨i, h⟩) (i + 1) else x

@[inline]

def Fin.foldr {α : Sort u_1} (n : Nat) (f : Fin n → α → α) (init : α) : α

Combine all the values that can be represented by Fin n with an initial value, starting at n - 1 and nesting to the right.

Example:

• Fin.foldr 3 (·.val + ·) (0 : Nat) = (0 : Fin 3).val + ((1 : Fin 3).val + ((2 : Fin 3).val + 0))

Equations

• Fin.foldr n f init = Fin.foldr.loop n f n ⋯ init

@[specialize #[]]

def Fin.foldr.loop {α : Sort u_1} (n : Nat) (f : Fin n → α → α) (i : Nat) : i ≤ n → α → α

Inner loop for Fin.foldr. Fin.foldr.loop n f i x = f 0 (f ... (f (i-1) x))

Equations

• Fin.foldr.loop n f 0 x_3 x = x
• Fin.foldr.loop n f i.succ h x = Fin.foldr.loop n f i ⋯ (f ⟨i, h⟩ x)

@[inline]

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

Folds a monadic function over all the values in Fin n from left to right, starting with 0.

It is the sequence of steps:

Fin.foldlM n f x₀ = do
  let x₁ ← f x₀ 0
  let x₂ ← f x₁ 1
  ...
  let xₙ ← f xₙ₋₁ (n-1)
  pure xₙ

Equations

• Fin.foldlM n f init = Fin.foldlM.loop n f init 0

@[specialize #[]]

def Fin.foldlM.loop {m : Type u_1 → Type u_2} {α : Type u_1} [Monad m] (n : Nat) (f : α → Fin n → m α) (x : α) (i : Nat) : m α

Inner loop for Fin.foldlM.

Fin.foldlM.loop n f xᵢ i = do
  let xᵢ₊₁ ← f xᵢ i
  ...
  let xₙ ← f xₙ₋₁ (n-1)
  pure xₙ

Equations

• Fin.foldlM.loop n f x i = if h : i < n then do let x ← f x ⟨i, h⟩ Fin.foldlM.loop n f x (i + 1) else pure x

@[inline]

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

Folds a monadic function over Fin n from right to left, starting with n-1.

It is the sequence of steps:

Fin.foldrM n f xₙ = do
  let xₙ₋₁ ← f (n-1) xₙ
  let xₙ₋₂ ← f (n-2) xₙ₋₁
  ...
  let x₀ ← f 0 x₁
  pure x₀

Equations

• Fin.foldrM n f init = Fin.foldrM.loop n f ⟨n, ⋯⟩ init

@[specialize #[]]

def Fin.foldrM.loop {m : Type u_1 → Type u_2} {α : Type u_1} [Monad m] (n : Nat) (f : Fin n → α → m α) : { i : Nat // i ≤ n } → α → m α

Inner loop for Fin.foldrM.

Fin.foldrM.loop n f i xᵢ = do
  let xᵢ₋₁ ← f (i-1) xᵢ
  ...
  let x₁ ← f 1 x₂
  let x₀ ← f 0 x₁
  pure x₀

Equations

• Fin.foldrM.loop n f ⟨0, property⟩ x✝ = pure x✝
• Fin.foldrM.loop n f ⟨i.succ, h⟩ x✝ = f ⟨i, h⟩ x✝ >>= Fin.foldrM.loop n f ⟨i, ⋯⟩

foldlM

theorem Fin.foldlM_loop_lt {m : Type u_1 → Type u_2} {α : Type u_1} {n i : Nat} [Monad m] (f : α → Fin n → m α) (x : α) (h : i < n) : foldlM.loop n f x i = do let x ← f x ⟨i, h⟩ foldlM.loop n f x (i + 1)

theorem Fin.foldlM_loop_eq {m : Type u_1 → Type u_2} {α : Type u_1} {n : Nat} [Monad m] (f : α → Fin n → m α) (x : α) : foldlM.loop n f x n = pure x

@[irreducible]

theorem Fin.foldlM_loop {m : Type u_1 → Type u_2} {α : Type u_1} {n i : Nat} [Monad m] (f : α → Fin (n + 1) → m α) (x : α) (h : i < n + 1) : foldlM.loop (n + 1) f x i = do let x ← f x ⟨i, h⟩ foldlM.loop n (fun (x : α) (j : Fin n) => f x j.succ) x i

@[simp]

theorem Fin.foldlM_zero {m : Type u_1 → Type u_2} {α : Type u_1} [Monad m] (f : α → Fin 0 → m α) (x : α) : foldlM 0 f x = pure x

theorem Fin.foldlM_succ {m : Type u_1 → Type u_2} {α : Type u_1} {n : Nat} [Monad m] (f : α → Fin (n + 1) → m α) (x : α) : foldlM (n + 1) f x = f x 0 >>= foldlM n fun (x : α) (j : Fin n) => f x j.succ

foldrM

theorem Fin.foldrM_loop_zero {m : Type u_1 → Type u_2} {n : Nat} {α : Type u_1} [Monad m] (f : Fin n → α → m α) (x : α) : foldrM.loop n f ⟨0, ⋯⟩ x = pure x

theorem Fin.foldrM_loop_succ {m : Type u_1 → Type u_2} {n : Nat} {α : Type u_1} {i : Nat} [Monad m] (f : Fin n → α → m α) (x : α) (h : i < n) : foldrM.loop n f ⟨i + 1, h⟩ x = f ⟨i, h⟩ x >>= foldrM.loop n f ⟨i, ⋯⟩

theorem Fin.foldrM_loop {m : Type u_1 → Type u_2} {n : Nat} {α : Type u_1} {i : Nat} [Monad m] [LawfulMonad m] (f : Fin (n + 1) → α → m α) (x : α) (h : i + 1 ≤ n + 1) : foldrM.loop (n + 1) f ⟨i + 1, h⟩ x = foldrM.loop n (fun (j : Fin n) => f j.succ) ⟨i, ⋯⟩ x >>= f 0

@[simp]

theorem Fin.foldrM_zero {m : Type u_1 → Type u_2} {α : Type u_1} [Monad m] (f : Fin 0 → α → m α) (x : α) : foldrM 0 f x = pure x

theorem Fin.foldrM_succ {m : Type u_1 → Type u_2} {n : Nat} {α : Type u_1} [Monad m] [LawfulMonad m] (f : Fin (n + 1) → α → m α) (x : α) : foldrM (n + 1) f x = foldrM n (fun (i : Fin n) => f i.succ) x >>= f 0

foldl

theorem Fin.foldl_loop_lt {α : Sort u_1} {n i : Nat} (f : α → Fin n → α) (x : α) (h : i < n) : foldl.loop n f x i = foldl.loop n f (f x ⟨i, h⟩) (i + 1)

theorem Fin.foldl_loop_eq {α : Sort u_1} {n : Nat} (f : α → Fin n → α) (x : α) : foldl.loop n f x n = x

@[irreducible]

theorem Fin.foldl_loop {α : Sort u_1} {n i : Nat} (f : α → Fin (n + 1) → α) (x : α) (h : i < n + 1) : foldl.loop (n + 1) f x i = foldl.loop n (fun (x : α) (j : Fin n) => f x j.succ) (f x ⟨i, h⟩) i

@[simp]

theorem Fin.foldl_zero {α : Sort u_1} (f : α → Fin 0 → α) (x : α) : foldl 0 f x = x

theorem Fin.foldl_succ {α : Sort u_1} {n : Nat} (f : α → Fin (n + 1) → α) (x : α) : foldl (n + 1) f x = foldl n (fun (x : α) (i : Fin n) => f x i.succ) (f x 0)

theorem Fin.foldl_succ_last {α : Sort u_1} {n : Nat} (f : α → Fin (n + 1) → α) (x : α) : foldl (n + 1) f x = f (foldl n (fun (x1 : α) (x2 : Fin n) => f x1 x2.castSucc) x) (last n)

theorem Fin.foldl_eq_foldlM {α : Type u_1} {n : Nat} (f : α → Fin n → α) (x : α) : foldl n f x = foldlM n f x

foldr

theorem Fin.foldr_loop_zero {n : Nat} {α : Sort u_1} (f : Fin n → α → α) (x : α) : foldr.loop n f 0 ⋯ x = x

theorem Fin.foldr_loop_succ {n : Nat} {α : Sort u_1} {i : Nat} (f : Fin n → α → α) (x : α) (h : i < n) : foldr.loop n f (i + 1) h x = foldr.loop n f i ⋯ (f ⟨i, h⟩ x)

theorem Fin.foldr_loop {n : Nat} {α : Sort u_1} {i : Nat} (f : Fin (n + 1) → α → α) (x : α) (h : i + 1 ≤ n + 1) : foldr.loop (n + 1) f (i + 1) h x = f 0 (foldr.loop n (fun (j : Fin n) => f j.succ) i ⋯ x)

@[simp]

theorem Fin.foldr_zero {α : Sort u_1} (f : Fin 0 → α → α) (x : α) : foldr 0 f x = x

theorem Fin.foldr_succ {n : Nat} {α : Sort u_1} (f : Fin (n + 1) → α → α) (x : α) : foldr (n + 1) f x = f 0 (foldr n (fun (i : Fin n) => f i.succ) x)

theorem Fin.foldr_succ_last {n : Nat} {α : Sort u_1} (f : Fin (n + 1) → α → α) (x : α) : foldr (n + 1) f x = foldr n (fun (x : Fin n) => f x.castSucc) (f (last n) x)

theorem Fin.foldr_eq_foldrM {n : Nat} {α : Type u_1} (f : Fin n → α → α) (x : α) : foldr n f x = foldrM n f x

theorem Fin.foldl_rev {n : Nat} {α : Sort u_1} (f : Fin n → α → α) (x : α) : foldl n (fun (x : α) (i : Fin n) => f i.rev x) x = foldr n f x

theorem Fin.foldr_rev {α : Sort u_1} {n : Nat} (f : α → Fin n → α) (x : α) : foldr n (fun (i : Fin n) (x : α) => f x i.rev) x = foldl n f x