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

In particular, omega is available here.

dropLast

theorem List.tail_dropLast {α : Type u_1} (l : List α) : l.dropLast.tail = l.tail.dropLast

@[simp]

theorem List.dropLast_reverse {α : Type u_1} (l : List α) : l.reverse.dropLast = l.tail.reverse

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

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

min?

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

theorem List.min?_get_le_of_mem {l : List Nat} {a : Nat} (h : a ∈ l) : l.min?.get ⋯ ≤ a

theorem List.min?_getD_le_of_mem {l : List Nat} {a : Nat} {k : Nat} (h : a ∈ l) : l.min?.getD k ≤ a

max?

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

theorem List.le_max?_get_of_mem {l : List Nat} {a : Nat} (h : a ∈ l) : a ≤ l.max?.get ⋯

theorem List.le_max?_getD_of_mem {l : List Nat} {a : Nat} {k : Nat} (h : a ∈ l) : a ≤ l.max?.getD k

@[reducible, inline, deprecated List.min?_eq_some_iff']

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

Equations

• @List.minimum?_eq_some_iff' = @List.min?_eq_some_iff'

@[reducible, inline, deprecated List.min?_cons']

abbrev List.minimum?_cons' {α : Type u_1} {x : α} [Min α] {xs : List α} : (x :: xs).min? = some (List.foldl min x xs)

Equations

• @List.minimum?_cons' = @List.min?_cons'

@[reducible, inline, deprecated List.min?_getD_le_of_mem]

abbrev List.minimum?_getD_le_of_mem {l : List Nat} {a : Nat} {k : Nat} (h : a ∈ l) : l.min?.getD k ≤ a

Equations

• @List.minimum?_getD_le_of_mem = @List.min?_getD_le_of_mem

@[reducible, inline, deprecated List.max?_eq_some_iff']

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

Equations

• @List.maximum?_eq_some_iff' = @List.max?_eq_some_iff'

@[reducible, inline, deprecated List.max?_cons']

abbrev List.maximum?_cons' {α : Type u_1} {x : α} [Max α] {xs : List α} : (x :: xs).max? = some (List.foldl max x xs)

Equations

• @List.maximum?_cons' = @List.max?_cons'

@[reducible, inline, deprecated List.le_max?_getD_of_mem]

abbrev List.le_maximum?_getD_of_mem {l : List Nat} {a : Nat} {k : Nat} (h : a ∈ l) : a ≤ l.max?.getD k

Equations

• @List.le_maximum?_getD_of_mem = @List.le_max?_getD_of_mem