# Documentation

## Init.Data.List.BasicAux

The following functions can't be defined at Init.Data.List.Basic, because they depend on Init.Util, and Init.Util depends on Init.Data.List.Basic.

## Alternative getters #

### get! #

def List.get! {α : Type u_1} [] (as : List α) (i : Nat) :
α

Returns the i-th element in the list (zero-based).

If the index is out of bounds (i ≥ as.length), this function panics when executed, and returns default. See get? and getD for safer alternatives.

Equations
• (a :: tail).get! 0 = a
• (head :: as).get! n.succ = as.get! n
• x✝.get! x = panicWithPosWithDecl "Init.Data.List.BasicAux" "List.get!" 28 18 "invalid index"
Instances For
theorem List.get!_nil {α : Type u_1} [] (n : Nat) :
[].get! n = default
theorem List.get!_cons_succ {α : Type u_1} [] (l : List α) (a : α) (n : Nat) :
(a :: l).get! (n + 1) = l.get! n
theorem List.get!_cons_zero {α : Type u_1} [] (l : List α) (a : α) :
(a :: l).get! 0 = a

### getLast! #

def List.getLast! {α : Type u_1} [] :
List αα

Returns the last element in the list.

If the list is empty, this function panics when executed, and returns default. See getLast and getLastD for safer alternatives.

Equations
• x.getLast! = match x with | [] => panicWithPosWithDecl "Init.Data.List.BasicAux" "List.getLast!" 44 13 "empty list" | a :: as => (a :: as).getLast
Instances For

def List.head! {α : Type u_1} [] :
List αα

Returns the first element in the list.

If the list is empty, this function panics when executed, and returns default. See head and headD for safer alternatives.

Equations
• x.head! = match x with | [] => panicWithPosWithDecl "Init.Data.List.BasicAux" "List.head!" 58 12 "empty list" | a :: tail => a
Instances For

### tail! #

def List.tail! {α : Type u_1} :
List αList α

Drops the first element of the list.

If the list is empty, this function panics when executed, and returns the empty list. See tail and tailD for safer alternatives.

Equations
• x.tail! = match x with | [] => panicWithPosWithDecl "Init.Data.List.BasicAux" "List.tail!" 70 13 "empty list" | head :: as => as
Instances For
@[simp]
theorem List.tail!_cons {α : Type u_1} {a : α} {l : List α} :
(a :: l).tail! = l

### partitionM #

@[inline]
def List.partitionM {m : TypeType u_1} {α : Type} [] (p : αm Bool) (l : List α) :
m (List α × List α)

Monadic generalization of List.partition.

This uses Array.toList and which isn't imported by Init.Data.List.Basic or Init.Data.List.Control.

def posOrNeg (x : Int) : Except String Bool :=
if x > 0 then pure true
else if x < 0 then pure false
else throw "Zero is not positive or negative"

partitionM posOrNeg [-1, 2, 3] = Except.ok ([2, 3], [-1])
partitionM posOrNeg [0, 2, 3] = Except.error "Zero is not positive or negative"

Equations
Instances For
@[specialize #[]]
def List.partitionM.go {m : TypeType u_1} {α : Type} [] (p : αm Bool) :
List αm (List α × List α)

Auxiliary for partitionM: partitionM.go p l acc₁ acc₂ returns (acc₁.toList ++ left, acc₂.toList ++ right) if partitionM p l returns (left, right).

Equations
Instances For

### partitionMap #

@[inline]
def List.partitionMap {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : αβ γ) (l : List α) :
List β × List γ

Given a function f : α → β ⊕ γ, partitionMap f l maps the list by f whilst partitioning the result into a pair of lists, List β × List γ, partitioning the .inl _ into the left list, and the .inr _ into the right List.

partitionMap (id : Nat ⊕ Nat → Nat ⊕ Nat) [inl 0, inr 1, inl 2] = ([0, 2], [1])

Equations
Instances For
@[specialize #[]]
def List.partitionMap.go {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : αβ γ) :
List αList β × List γ

Auxiliary for partitionMap: partitionMap.go f l acc₁ acc₂ = (acc₁.toList ++ left, acc₂.toList ++ right) if partitionMap f l = (left, right).

Equations
Instances For

### mapMono #

This is a performance optimization for List.mapM that avoids allocating a new list when the result of each f a is a pointer equal value a.

For verification purposes, List.mapMono = List.map.

@[implemented_by _private.Init.Data.List.BasicAux.0.List.mapMonoMImp]
def List.mapMonoM {m : Type u_1 → Type u_2} {α : Type u_1} [] (as : List α) (f : αm α) :
m (List α)

Monomorphic List.mapM. The internal implementation uses pointer equality, and does not allocate a new list if the result of each f a is a pointer equal value a.

Equations
• [].mapMonoM f = pure []
• (a :: as_1).mapMonoM f = do let __do_liftf a let __do_lift_1as_1.mapMonoM f pure (__do_lift :: __do_lift_1)
Instances For
def List.mapMono {α : Type u_1} (as : List α) (f : αα) :
List α
Equations
• as.mapMono f = (as.mapMonoM f).run
Instances For

## Additional lemmas required for bootstrapping Array. #

theorem List.getElem_append_left {α : Type u_1} {i : Nat} (as : List α) (bs : List α) (h : i < as.length) {h' : i < (as ++ bs).length} :
(as ++ bs)[i] = as[i]
theorem List.getElem_append_right {α : Type u_1} {i : Nat} (as : List α) (bs : List α) (h : ¬i < as.length) {h' : i < (as ++ bs).length} {h'' : i - as.length < bs.length} :
(as ++ bs)[i] = bs[i - as.length]
theorem List.get_last {α : Type u_1} {a : α} {as : List α} {i : Fin (as ++ [a]).length} (h : ¬i < as.length) :
(as ++ [a]).get i = a
theorem List.sizeOf_lt_of_mem {α : Type u_1} {a : α} [] {as : List α} (h : a as) :
< sizeOf as

This tactic, added to the decreasing_trivial toolbox, proves that sizeOf a < sizeOf as when a ∈ as, which is useful for well founded recursions over a nested inductive like inductive T | mk : List T → T.

Equations
Instances For
theorem List.append_cancel_left {α : Type u_1} {as : List α} {bs : List α} {cs : List α} (h : as ++ bs = as ++ cs) :
bs = cs
theorem List.append_cancel_right {α : Type u_1} {as : List α} {bs : List α} {cs : List α} (h : as ++ bs = cs ++ bs) :
as = cs
@[simp]
theorem List.append_cancel_left_eq {α : Type u_1} (as : List α) (bs : List α) (cs : List α) :
(as ++ bs = as ++ cs) = (bs = cs)
@[simp]
theorem List.append_cancel_right_eq {α : Type u_1} (as : List α) (bs : List α) (cs : List α) :
(as ++ bs = cs ++ bs) = (as = cs)
@[simp]
theorem List.sizeOf_get {α : Type u_1} [] (as : List α) (i : Fin as.length) :
sizeOf (as.get i) < sizeOf as
theorem List.le_antisymm {α : Type u_1} [LT α] [s : Antisymm fun (x x_1 : α) => ¬x < x_1] {as : List α} {bs : List α} (h₁ : as bs) (h₂ : bs as) :
as = bs
instance List.instAntisymmLeOfNotLt {α : Type u_1} [LT α] [Antisymm fun (x x_1 : α) => ¬x < x_1] :
Antisymm fun (x x_1 : List α) => x x_1
Equations
• =