Miscellaneous List lemmas, that require more Nat lemmas than are available in Init.Data.List.Lemmas.

In particular, omega is available here.

filter

theorem List.length_filter_lt_length_iff_exists : ∀ {α : Type u_1} {p : α → Bool} (l : List α), (List.filter p l).length < l.length ↔ ∃ (x : α), x ∈ l ∧ ¬p x = true

reverse

theorem List.getElem_eq_getElem_reverse {α : Type u_1} {l : List α} {i : Nat} (h : i < l.length) : l[i] = l.reverse[l.length - 1 - i]

leftpad

@[simp]

theorem List.leftpad_length {α : Type u_1} (n : Nat) (a : α) (l : List α) : (List.leftpad n a l).length = max n l.length

The length of the List returned by List.leftpad n a l is equal to the larger of n and l.length

eraseIdx

theorem List.mem_eraseIdx_iff_getElem {α : Type u_1} {x : α} {l : List α} {k : Nat} : x ∈ l.eraseIdx k ↔ ∃ (i : Nat), ∃ (h : i < l.length), i ≠ k ∧ l[i] = x

theorem List.mem_eraseIdx_iff_getElem? {α : Type u_1} {x : α} {l : List α} {k : Nat} : x ∈ l.eraseIdx k ↔ ∃ (i : Nat), i ≠ k ∧ l[i]? = some x

minimum?

theorem List.minimum?_eq_some_iff' {a : Nat} {xs : List Nat} : xs.minimum? = some a ↔ a ∈ xs ∧ ∀ (b : Nat), b ∈ xs → a ≤ b

theorem List.minimum?_cons' {a : Nat} {l : List Nat} : (a :: l).minimum? = some (match l.minimum? with | none => a | some m => min a m)

maximum?

theorem List.maximum?_eq_some_iff' {a : Nat} {xs : List Nat} : xs.maximum? = some a ↔ a ∈ xs ∧ ∀ (b : Nat), b ∈ xs → b ≤ a

theorem List.maximum?_cons' {a : Nat} {l : List Nat} : (a :: l).maximum? = some (match l.maximum? with | none => a | some m => max a m)