insertNth

Proves various lemmas about List.insertNth.

@[simp]

theorem List.insertNth_zero {α : Type u} (s : List α) (x : α) : List.insertNth 0 x s = x :: s

@[simp]

theorem List.insertNth_succ_nil {α : Type u} (n : ℕ) (a : α) : List.insertNth (n + 1) a [] = []

@[simp]

theorem List.insertNth_succ_cons {α : Type u} (s : List α) (hd : α) (x : α) (n : ℕ) : List.insertNth (n + 1) x (hd :: s) = hd :: List.insertNth n x s

theorem List.length_insertNth {α : Type u} {a : α} (n : ℕ) (as : List α) : n ≤ as.length → (List.insertNth n a as).length = as.length + 1

theorem List.eraseIdx_insertNth {α : Type u} {a : α} (n : ℕ) (l : List α) : (List.insertNth n a l).eraseIdx n = l

@[deprecated List.eraseIdx_insertNth]

theorem List.removeNth_insertNth {α : Type u} {a : α} (n : ℕ) (l : List α) : (List.insertNth n a l).eraseIdx n = l

Alias of List.eraseIdx_insertNth.

theorem List.insertNth_eraseIdx_of_ge {α : Type u} {a : α} (n : ℕ) (m : ℕ) (as : List α) : n < as.length → n ≤ m → List.insertNth m a (as.eraseIdx n) = (List.insertNth (m + 1) a as).eraseIdx n

@[deprecated List.insertNth_eraseIdx_of_ge]

theorem List.insertNth_removeNth_of_ge {α : Type u} {a : α} (n : ℕ) (m : ℕ) (as : List α) : n < as.length → n ≤ m → List.insertNth m a (as.eraseIdx n) = (List.insertNth (m + 1) a as).eraseIdx n

Alias of List.insertNth_eraseIdx_of_ge.

theorem List.insertNth_eraseIdx_of_le {α : Type u} {a : α} (n : ℕ) (m : ℕ) (as : List α) : n < as.length → m ≤ n → List.insertNth m a (as.eraseIdx n) = (List.insertNth m a as).eraseIdx (n + 1)

@[deprecated List.insertNth_eraseIdx_of_le]

theorem List.insertNth_removeNth_of_le {α : Type u} {a : α} (n : ℕ) (m : ℕ) (as : List α) : n < as.length → m ≤ n → List.insertNth m a (as.eraseIdx n) = (List.insertNth m a as).eraseIdx (n + 1)

Alias of List.insertNth_eraseIdx_of_le.

theorem List.insertNth_comm {α : Type u} (a : α) (b : α) (i : ℕ) (j : ℕ) (l : List α) : i ≤ j → j ≤ l.length → List.insertNth (j + 1) b (List.insertNth i a l) = List.insertNth i a (List.insertNth j b l)

theorem List.mem_insertNth {α : Type u} {a : α} {b : α} {n : ℕ} {l : List α} : n ≤ l.length → (a ∈ List.insertNth n b l ↔ a = b ∨ a ∈ l)

theorem List.insertNth_of_length_lt {α : Type u} (l : List α) (x : α) (n : ℕ) (h : l.length < n) : List.insertNth n x l = l

@[simp]

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

theorem List.length_le_length_insertNth {α : Type u} (l : List α) (x : α) (n : ℕ) : l.length ≤ (List.insertNth n x l).length

theorem List.length_insertNth_le_succ {α : Type u} (l : List α) (x : α) (n : ℕ) : (List.insertNth n x l).length ≤ l.length + 1

theorem List.getElem_insertNth_of_lt {α : Type u} (l : List α) (x : α) (n : ℕ) (k : ℕ) (hn : k < n) (hk : k < l.length) (hk' : optParam (k < (List.insertNth n x l).length) ⋯) : (List.insertNth n x l)[k] = l[k]

theorem List.get_insertNth_of_lt {α : Type u} (l : List α) (x : α) (n : ℕ) (k : ℕ) (hn : k < n) (hk : k < l.length) (hk' : optParam (k < (List.insertNth n x l).length) ⋯) : (List.insertNth n x l).get ⟨k, hk'⟩ = l.get ⟨k, hk⟩

@[simp]

theorem List.getElem_insertNth_self {α : Type u} (l : List α) (x : α) (n : ℕ) (hn : n ≤ l.length) (hn' : optParam (n < (List.insertNth n x l).length) ⋯) : (List.insertNth n x l)[n] = x

theorem List.get_insertNth_self {α : Type u} (l : List α) (x : α) (n : ℕ) (hn : n ≤ l.length) (hn' : optParam (n < (List.insertNth n x l).length) ⋯) : (List.insertNth n x l).get ⟨n, hn'⟩ = x

theorem List.getElem_insertNth_add_succ {α : Type u} (l : List α) (x : α) (n : ℕ) (k : ℕ) (hk' : n + k < l.length) (hk : optParam (n + k + 1 < (List.insertNth n x l).length) ⋯) : (List.insertNth n x l)[n + k + 1] = l[n + k]

theorem List.get_insertNth_add_succ {α : Type u} (l : List α) (x : α) (n : ℕ) (k : ℕ) (hk' : n + k < l.length) (hk : optParam (n + k + 1 < (List.insertNth n x l).length) ⋯) : (List.insertNth n x l).get ⟨n + k + 1, hk⟩ = l.get ⟨n + k, hk'⟩

theorem List.insertNth_injective {α : Type u} (n : ℕ) (x : α) : Function.Injective (List.insertNth n x)