Documentation

Mathlib.Data.List.GetD

getD and getI #

This file provides theorems for working with the getD and getI functions. These are used to access an element of a list by numerical index, with a default value as a fallback when the index is out of range.

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theorem List.getD_eq_getElem {α : Type u} (l : List α) (d : α) {n : ℕ} (hn : n < l.length) :
l.getD n d = l[n]
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theorem List.getD_eq_getElem? {α : Type u} (l : List α) (d : α) (i : Fin l.length) :
l.getD (↑i) d = l[i]?.get ⋯
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theorem List.getD_eq_get {α : Type u} (l : List α) (d : α) (i : Fin l.length) :
l.getD (↑i) d = l.get i
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theorem List.getD_map {α : Type u} {β : Type v} (l : List α) (d : α) {n : ℕ} (f : α → β) :
(map f l).getD n (f d) = f (l.getD n d)
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theorem List.getD_eq_default {α : Type u} (l : List α) (d : α) {n : ℕ} (hn : l.length ≤ n) :
l.getD n d = d
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theorem List.getD_reverse {α : Type u} {l : List α} (i : ℕ) (h : i < l.length) :
l.reverse.getD i = l.getD (l.length - 1 - i)
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@[instance_reducible]
def List.decidableGetDNilNe {α : Type u} (a : α) :
DecidablePred fun (i : ℕ) => [].getD i a ≠ a

An empty list can always be decidably checked for the presence of an element. Not an instance because it would clash with DecidableEq α.

Equations
Instances For
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    @[simp]
    theorem List.getElem?_getD_singleton_default_eq {α : Type u} (d : α) (n : ℕ) :
    [d][n]?.getD d = d
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    @[simp]
    theorem List.getElem?_getD_replicate_default_eq {α : Type u} (d : α) (r n : ℕ) :
    (replicate r d)[n]?.getD d = d
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    theorem List.getD_replicate {α : Type u} (x : α) {y : α} {i n : ℕ} (h : i < n) :
    (replicate n x).getD i y = x
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    theorem List.getD_append {α : Type u} (l l' : List α) (d : α) (n : ℕ) (h : n < l.length) :
    (l ++ l').getD n d = l.getD n d
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    theorem List.getD_append_right {α : Type u} (l l' : List α) (d : α) (n : ℕ) (h : l.length ≤ n) :
    (l ++ l').getD n d = l'.getD (n - l.length) d
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    theorem List.getD_surjective_iff {α : Type u} {l : List α} {d : α} :
    (Function.Surjective fun (x : ℕ) => l.getD x d) ↔ ∀ (x : α), x = d ∨ x ∈ l
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    theorem List.getD_surjective {α : Type u} {l : List α} (h : ∀ (x : α), x ∈ l) (d : α) :
    Function.Surjective fun (x : ℕ) => l.getD x d
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    @[simp]
    theorem List.getI_nil {α : Type u} (n : ℕ) [Inhabited α] :
    [].getI n = default
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    @[simp]
    theorem List.getI_cons_zero {α : Type u} (x : α) (xs : List α) [Inhabited α] :
    (x :: xs).getI 0 = x
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    @[simp]
    theorem List.getI_cons_succ {α : Type u} (x : α) (xs : List α) (n : ℕ) [Inhabited α] :
    (x :: xs).getI (n + 1) = xs.getI n
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    theorem List.getI_eq_getElem {α : Type u} (l : List α) [Inhabited α] {n : ℕ} (hn : n < l.length) :
    l.getI n = l[n]
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    theorem List.getI_eq_default {α : Type u} (l : List α) [Inhabited α] {n : ℕ} (hn : l.length ≤ n) :
    l.getI n = default
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    theorem List.getD_default_eq_getI {α : Type u} (l : List α) [Inhabited α] {n : ℕ} :
    l.getD n default = l.getI n
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    theorem List.getI_append {α : Type u} [Inhabited α] (l l' : List α) (n : ℕ) (h : n < l.length) :
    (l ++ l').getI n = l.getI n
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    theorem List.getI_append_right {α : Type u} [Inhabited α] (l l' : List α) (n : ℕ) (h : l.length ≤ n) :
    (l ++ l').getI n = l'.getI (n - l.length)
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    theorem List.getI_eq_getElem?_getD {α : Type u} (l : List α) [Inhabited α] (n : ℕ) :
    l.getI n = l[n]?.getD default
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    theorem List.getI_zero_eq_headI {α : Type u} (l : List α) [Inhabited α] :
    l.getI 0 = l.headI