insertIdx

Proves various lemmas about List.insertIdx.

@[simp]

theorem List.insertIdx_zero {α : Type u} (s : List α) (x : α) : insertIdx 0 x s = x :: s

@[simp]

theorem List.insertIdx_succ_nil {α : Type u} (n : Nat) (a : α) : insertIdx (n + 1) a [] = []

@[simp]

theorem List.insertIdx_succ_cons {α : Type u} (s : List α) (hd x : α) (i : Nat) : insertIdx (i + 1) x (hd :: s) = hd :: insertIdx i x s

theorem List.length_insertIdx {α : Type u} {a : α} (i : Nat) (as : List α) : (insertIdx i a as).length = if i ≤ as.length then as.length + 1 else as.length

theorem List.length_insertIdx_of_le_length {α : Type u} {a : α} {i : Nat} {as : List α} (h : i ≤ as.length) : (insertIdx i a as).length = as.length + 1

theorem List.length_insertIdx_of_length_lt {α : Type u} {a : α} {as : List α} {i : Nat} (h : as.length < i) : (insertIdx i a as).length = as.length

@[simp]

theorem List.eraseIdx_insertIdx {α : Type u} {a : α} (i : Nat) (l : List α) : (insertIdx i a l).eraseIdx i = l

theorem List.insertIdx_eraseIdx_of_ge {α : Type u} {a : α} (i m : Nat) (as : List α) : i < as.length → i ≤ m → insertIdx m a (as.eraseIdx i) = (insertIdx (m + 1) a as).eraseIdx i

theorem List.insertIdx_eraseIdx_of_le {α : Type u} {a : α} (i j : Nat) (as : List α) : i < as.length → j ≤ i → insertIdx j a (as.eraseIdx i) = (insertIdx j a as).eraseIdx (i + 1)

theorem List.insertIdx_comm {α : Type u} (a b : α) (i j : Nat) (l : List α) : i ≤ j → j ≤ l.length → insertIdx (j + 1) b (insertIdx i a l) = insertIdx i a (insertIdx j b l)

theorem List.mem_insertIdx {α : Type u} {a b : α} {i : Nat} {l : List α} : i ≤ l.length → (a ∈ insertIdx i b l ↔ a = b ∨ a ∈ l)

theorem List.insertIdx_of_length_lt {α : Type u} (l : List α) (x : α) (i : Nat) (h : l.length < i) : insertIdx i x l = l

@[simp]

theorem List.insertIdx_length_self {α : Type u} (l : List α) (x : α) : insertIdx l.length x l = l ++ [x]

theorem List.length_le_length_insertIdx {α : Type u} (l : List α) (x : α) (i : Nat) : l.length ≤ (insertIdx i x l).length

theorem List.length_insertIdx_le_succ {α : Type u} (l : List α) (x : α) (i : Nat) : (insertIdx i x l).length ≤ l.length + 1

theorem List.getElem_insertIdx_of_lt {α : Type u} {l : List α} {x : α} {i j : Nat} (hn : j < i) (hk : j < (insertIdx i x l).length) : (insertIdx i x l)[j] = l[j]

@[simp]

theorem List.getElem_insertIdx_self {α : Type u} {l : List α} {x : α} {i : Nat} (hi : i < (insertIdx i x l).length) : (insertIdx i x l)[i] = x

theorem List.getElem_insertIdx_of_gt {α : Type u} {l : List α} {x : α} {i j : Nat} (hn : i < j) (hk : j < (insertIdx i x l).length) : (insertIdx i x l)[j] = l[j - 1]

@[reducible, inline, deprecated List.getElem_insertIdx_of_gt (since := "2025-02-04")]

abbrev List.getElem_insertIdx_of_ge {α : Type u_1} {l : List α} {x : α} {i j : Nat} (hn : i < j) (hk : j < (insertIdx i x l).length) : (insertIdx i x l)[j] = l[j - 1]

Equations

• @List.getElem_insertIdx_of_ge = @List.getElem_insertIdx_of_gt

theorem List.getElem_insertIdx {α : Type u} {l : List α} {x : α} {i j : Nat} (h : j < (insertIdx i x l).length) : (insertIdx i x l)[j] = if h₁ : j < i then l[j] else if h₂ : j = i then x else l[j - 1]

theorem List.getElem?_insertIdx {α : Type u} {l : List α} {x : α} {i j : Nat} : (insertIdx i x l)[j]? = if j < i then l[j]? else if j = i then if j ≤ l.length then some x else none else l[j - 1]?

theorem List.getElem?_insertIdx_of_lt {α : Type u} {l : List α} {x : α} {i j : Nat} (h : j < i) : (insertIdx i x l)[j]? = l[j]?

theorem List.getElem?_insertIdx_self {α : Type u} {l : List α} {x : α} {i : Nat} : (insertIdx i x l)[i]? = if i ≤ l.length then some x else none

theorem List.getElem?_insertIdx_of_gt {α : Type u} {l : List α} {x : α} {i j : Nat} (h : i < j) : (insertIdx i x l)[j]? = l[j - 1]?

@[reducible, inline, deprecated List.getElem?_insertIdx_of_gt (since := "2025-02-04")]

abbrev List.getElem?_insertIdx_of_ge {α : Type u_1} {l : List α} {x : α} {i j : Nat} (h : i < j) : (insertIdx i x l)[j]? = l[j - 1]?

Equations

• @List.getElem?_insertIdx_of_ge = @List.getElem?_insertIdx_of_gt