Theorems about Array.

Preliminaries

toList

@[simp]

theorem Array.toList_inj {α : Type u_1} {a b : Array α} : a.toList = b.toList ↔ a = b

@[simp]

theorem Array.toList_eq_nil_iff {α : Type u_1} (l : Array α) : l.toList = [] ↔ l = #[]

@[simp]

theorem Array.mem_toList_iff {α : Type u_1} (a : α) (l : Array α) : a ∈ l.toList ↔ a ∈ l

empty

@[simp]

theorem Array.empty_eq {α : Type u_1} {xs : Array α} : #[] = xs ↔ xs = #[]

size

theorem Array.eq_empty_of_size_eq_zero {α✝ : Type u_1} {l : Array α✝} (h : l.size = 0) : l = #[]

theorem Array.ne_empty_of_size_eq_add_one {n : Nat} {α✝ : Type u_1} {l : Array α✝} (h : l.size = n + 1) : l ≠ #[]

theorem Array.ne_empty_of_size_pos {α✝ : Type u_1} {l : Array α✝} (h : 0 < l.size) : l ≠ #[]

theorem Array.size_eq_zero {α✝ : Type u_1} {l : Array α✝} : l.size = 0 ↔ l = #[]

theorem Array.size_pos_of_mem {α : Type u_1} {a : α} {l : Array α} (h : a ∈ l) : 0 < l.size

theorem Array.exists_mem_of_size_pos {α : Type u_1} {l : Array α} (h : 0 < l.size) : ∃ (a : α), a ∈ l

theorem Array.size_pos_iff_exists_mem {α : Type u_1} {l : Array α} : 0 < l.size ↔ ∃ (a : α), a ∈ l

theorem Array.exists_mem_of_size_eq_add_one {α : Type u_1} {n : Nat} {l : Array α} (h : l.size = n + 1) : ∃ (a : α), a ∈ l

theorem Array.size_pos {α : Type u_1} {l : Array α} : 0 < l.size ↔ l ≠ #[]

theorem Array.size_eq_one {α : Type u_1} {l : Array α} : l.size = 1 ↔ ∃ (a : α), l = #[a]

push

theorem Array.push_ne_empty {α : Type u_1} {a : α} {xs : Array α} : xs.push a ≠ #[]

@[simp]

theorem Array.push_ne_self {α : Type u_1} {a : α} {xs : Array α} : xs.push a ≠ xs

@[simp]

theorem Array.ne_push_self {α : Type u_1} {a : α} {xs : Array α} : xs ≠ xs.push a

theorem Array.back_eq_of_push_eq {α : Type u_1} {a b : α} {xs ys : Array α} (h : xs.push a = ys.push b) : a = b

theorem Array.pop_eq_of_push_eq {α : Type u_1} {a b : α} {xs ys : Array α} (h : xs.push a = ys.push b) : xs = ys

theorem Array.push_inj_left {α : Type u_1} {a : α} {xs ys : Array α} : xs.push a = ys.push a ↔ xs = ys

theorem Array.push_inj_right {α : Type u_1} {a b : α} {xs : Array α} : xs.push a = xs.push b ↔ a = b

theorem Array.push_eq_push {α : Type u_1} {a b : α} {xs ys : Array α} : xs.push a = ys.push b ↔ a = b ∧ xs = ys

theorem Array.exists_push_of_ne_empty {α : Type u_1} {xs : Array α} (h : xs ≠ #[]) : ∃ (ys : Array α), ∃ (a : α), xs = ys.push a

theorem Array.ne_empty_iff_exists_push {α : Type u_1} {xs : Array α} : xs ≠ #[] ↔ ∃ (ys : Array α), ∃ (a : α), xs = ys.push a

theorem Array.exists_push_of_size_pos {α : Type u_1} {xs : Array α} (h : 0 < xs.size) : ∃ (ys : Array α), ∃ (a : α), xs = ys.push a

theorem Array.size_pos_iff_exists_push {α : Type u_1} {xs : Array α} : 0 < xs.size ↔ ∃ (ys : Array α), ∃ (a : α), xs = ys.push a

theorem Array.exists_push_of_size_eq_add_one {α : Type u_1} {n : Nat} {xs : Array α} (h : xs.size = n + 1) : ∃ (ys : Array α), ∃ (a : α), xs = ys.push a

theorem Array.singleton_inj {α✝ : Type u_1} {a b : α✝} : #[a] = #[b] ↔ a = b

mkArray

@[simp]

theorem Array.size_mkArray {α : Type u_1} (n : Nat) (v : α) : (mkArray n v).size = n

@[simp]

theorem Array.toList_mkArray {n : Nat} {α✝ : Type u_1} {a : α✝} : (mkArray n a).toList = List.replicate n a

@[simp]

theorem Array.mkArray_zero {α✝ : Type u_1} {a : α✝} : mkArray 0 a = #[]

theorem Array.mkArray_succ {n : Nat} {α✝ : Type u_1} {a : α✝} : mkArray (n + 1) a = (mkArray n a).push a

theorem Array.mkArray_inj {n : Nat} {α✝ : Type u_1} {a : α✝} {m : Nat} {b : α✝} : mkArray n a = mkArray m b ↔ n = m ∧ (n = 0 ∨ a = b)

@[simp]

theorem Array.getElem_mkArray {α : Type u_1} {i : Nat} (n : Nat) (v : α) (h : i < (mkArray n v).size) : (mkArray n v)[i] = v

theorem Array.getElem?_mkArray {α : Type u_1} (n : Nat) (v : α) (i : Nat) : (mkArray n v)[i]? = if i < n then some v else none

L[i] and L[i]?

@[simp]

theorem Array.getElem?_eq_none_iff {α : Type u_1} {i : Nat} {a : Array α} : a[i]? = none ↔ a.size ≤ i

@[simp]

theorem Array.none_eq_getElem?_iff {α : Type u_1} {a : Array α} {i : Nat} : none = a[i]? ↔ a.size ≤ i

theorem Array.getElem?_eq_none {α : Type u_1} {i : Nat} {a : Array α} (h : a.size ≤ i) : a[i]? = none

@[simp]

theorem Array.getElem?_eq_getElem {α : Type u_1} {a : Array α} {i : Nat} (h : i < a.size) : a[i]? = some a[i]

theorem Array.getElem?_eq_some_iff {α : Type u_1} {i : Nat} {b : α} {a : Array α} : a[i]? = some b ↔ ∃ (h : i < a.size), a[i] = b

theorem Array.some_eq_getElem?_iff {α : Type u_1} {b : α} {i : Nat} {a : Array α} : some b = a[i]? ↔ ∃ (h : i < a.size), a[i] = b

@[simp]

theorem Array.some_getElem_eq_getElem?_iff {α : Type u_1} (a : Array α) (i : Nat) (h : i < a.size) : some a[i] = a[i]? ↔ True

@[simp]

theorem Array.getElem?_eq_some_getElem_iff {α : Type u_1} (a : Array α) (i : Nat) (h : i < a.size) : a[i]? = some a[i] ↔ True

theorem Array.getElem_eq_iff {α : Type u_1} {x : α} {a : Array α} {i : Nat} {h : i < a.size} : a[i] = x ↔ a[i]? = some x

theorem Array.getElem_eq_getElem?_get {α : Type u_1} (a : Array α) (i : Nat) (h : i < a.size) : a[i] = a[i]?.get ⋯

theorem Array.getD_getElem? {α : Type u_1} (a : Array α) (i : Nat) (d : α) : a[i]?.getD d = if p : i < a.size then a[i] else d

@[simp]

theorem Array.getElem?_empty {α : Type u_1} {i : Nat} : #[][i]? = none

theorem Array.getElem_push_lt {α : Type u_1} (a : Array α) (x : α) (i : Nat) (h : i < a.size) : let_fun this := ⋯; (a.push x)[i] = a[i]

@[simp]

theorem Array.getElem_push_eq {α : Type u_1} (a : Array α) (x : α) : (a.push x)[a.size] = x

theorem Array.getElem_push {α : Type u_1} (a : Array α) (x : α) (i : Nat) (h : i < (a.push x).size) : (a.push x)[i] = if h : i < a.size then a[i] else x

theorem Array.getElem?_push {α : Type u_1} {i : Nat} {a : Array α} {x : α} : (a.push x)[i]? = if i = a.size then some x else a[i]?

@[simp]

theorem Array.getElem?_push_size {α : Type u_1} {a : Array α} {x : α} : (a.push x)[a.size]? = some x

@[simp]

theorem Array.getElem_singleton {α : Type u_1} {i : Nat} (a : α) (h : i < 1) : #[a][i] = a

theorem Array.getElem?_singleton {α : Type u_1} (a : α) (i : Nat) : #[a][i]? = if i = 0 then some a else none

mem

theorem Array.not_mem_empty {α : Type u_1} (a : α) : ¬a ∈ #[]

@[simp]

theorem Array.mem_push {α : Type u_1} {a : Array α} {x y : α} : x ∈ a.push y ↔ x ∈ a ∨ x = y

theorem Array.mem_push_self {α : Type u_1} {a : Array α} {x : α} : x ∈ a.push x

theorem Array.eq_push_append_of_mem {α : Type u_1} {xs : Array α} {x : α} (h : x ∈ xs) : ∃ (as : Array α), ∃ (bs : Array α), xs = as.push x ++ bs ∧ ¬x ∈ as

theorem Array.mem_push_of_mem {α : Type u_1} {a : Array α} {x : α} (y : α) (h : x ∈ a) : x ∈ a.push y

theorem Array.exists_mem_of_ne_empty {α : Type u_1} (l : Array α) (h : l ≠ #[]) : ∃ (x : α), x ∈ l

theorem Array.eq_empty_iff_forall_not_mem {α : Type u_1} {l : Array α} : l = #[] ↔ ∀ (a : α), ¬a ∈ l

@[simp]

theorem Array.mem_dite_empty_left {α : Type u_1} {p : Prop} {x : α} [Decidable p] {l : ¬p → Array α} : (x ∈ if h : p then #[] else l h) ↔ ∃ (h : ¬p), x ∈ l h

@[simp]

theorem Array.mem_dite_empty_right {α : Type u_1} {p : Prop} {x : α} [Decidable p] {l : p → Array α} : (x ∈ if h : p then l h else #[]) ↔ ∃ (h : p), x ∈ l h

@[simp]

theorem Array.mem_ite_empty_left {α : Type u_1} {p : Prop} {x : α} [Decidable p] {l : Array α} : (x ∈ if p then #[] else l) ↔ ¬p ∧ x ∈ l

@[simp]

theorem Array.mem_ite_empty_right {α : Type u_1} {p : Prop} {x : α} [Decidable p] {l : Array α} : (x ∈ if p then l else #[]) ↔ p ∧ x ∈ l

theorem Array.eq_of_mem_singleton {α✝ : Type u_1} {b a : α✝} (h : a ∈ #[b]) : a = b

@[simp]

theorem Array.mem_singleton {α : Type u_1} {a b : α} : a ∈ #[b] ↔ a = b

theorem Array.forall_mem_push {α : Type u_1} {p : α → Prop} {xs : Array α} {a : α} : (∀ (x : α), x ∈ xs.push a → p x) ↔ p a ∧ ∀ (x : α), x ∈ xs → p x

theorem Array.forall_mem_ne {α : Type u_1} {a : α} {l : Array α} : (∀ (a' : α), a' ∈ l → ¬a = a') ↔ ¬a ∈ l

theorem Array.forall_mem_ne' {α : Type u_1} {a : α} {l : Array α} : (∀ (a' : α), a' ∈ l → ¬a' = a) ↔ ¬a ∈ l

theorem Array.exists_mem_empty {α : Type u_1} (p : α → Prop) : ¬∃ (x : α), ∃ (x_1 : x ∈ #[]), p x

theorem Array.forall_mem_empty {α : Type u_1} (p : α → Prop) (x : α) : x ∈ #[] → p x

theorem Array.exists_mem_push {α : Type u_1} {p : α → Prop} {a : α} {xs : Array α} : (∃ (x : α), ∃ (x_1 : x ∈ xs.push a), p x) ↔ p a ∨ ∃ (x : α), ∃ (x_1 : x ∈ xs), p x

theorem Array.forall_mem_singleton {α : Type u_1} {p : α → Prop} {a : α} : (∀ (x : α), x ∈ #[a] → p x) ↔ p a

theorem Array.mem_empty_iff {α : Type u_1} (a : α) : a ∈ #[] ↔ False

theorem Array.mem_singleton_self {α : Type u_1} (a : α) : a ∈ #[a]

theorem Array.mem_of_mem_push_of_mem {α : Type u_1} {a b : α} {l : Array α} : a ∈ l.push b → b ∈ l → a ∈ l

theorem Array.eq_or_ne_mem_of_mem {α : Type u_1} {a b : α} {l : Array α} (h' : a ∈ l.push b) : a = b ∨ a ≠ b ∧ a ∈ l

theorem Array.ne_empty_of_mem {α : Type u_1} {a : α} {l : Array α} (h : a ∈ l) : l ≠ #[]

theorem Array.mem_of_ne_of_mem {α : Type u_1} {a y : α} {l : Array α} (h₁ : a ≠ y) (h₂ : a ∈ l.push y) : a ∈ l

theorem Array.ne_of_not_mem_push {α : Type u_1} {a b : α} {l : Array α} (h : ¬a ∈ l.push b) : a ≠ b

theorem Array.not_mem_of_not_mem_push {α : Type u_1} {a b : α} {l : Array α} (h : ¬a ∈ l.push b) : ¬a ∈ l

theorem Array.not_mem_push_of_ne_of_not_mem {α : Type u_1} {a y : α} {l : Array α} : a ≠ y → ¬a ∈ l → ¬a ∈ l.push y

theorem Array.ne_and_not_mem_of_not_mem_push {α : Type u_1} {a y : α} {l : Array α} : ¬a ∈ l.push y → a ≠ y ∧ ¬a ∈ l

theorem Array.getElem_of_mem {α : Type u_1} {a : α} {l : Array α} (h : a ∈ l) : ∃ (i : Nat), ∃ (h : i < l.size), l[i] = a

theorem Array.getElem?_of_mem {α : Type u_1} {a : α} {l : Array α} (h : a ∈ l) : ∃ (i : Nat), l[i]? = some a

theorem Array.mem_of_getElem? {α : Type u_1} {l : Array α} {i : Nat} {a : α} (e : l[i]? = some a) : a ∈ l

theorem Array.mem_iff_getElem {α : Type u_1} {a : α} {l : Array α} : a ∈ l ↔ ∃ (i : Nat), ∃ (h : i < l.size), l[i] = a

theorem Array.mem_iff_getElem? {α : Type u_1} {a : α} {l : Array α} : a ∈ l ↔ ∃ (i : Nat), l[i]? = some a

theorem Array.forall_getElem {α : Type u_1} {l : Array α} {p : α → Prop} : (∀ (i : Nat) (h : i < l.size), p l[i]) ↔ ∀ (a : α), a ∈ l → p a

isEmpty

@[simp]

theorem Array.isEmpty_toList {α : Type u_1} {l : Array α} : l.toList.isEmpty = l.isEmpty

theorem Array.isEmpty_iff {α : Type u_1} {l : Array α} : l.isEmpty = true ↔ l = #[]

theorem Array.isEmpty_eq_false_iff_exists_mem {α : Type u_1} {xs : Array α} : xs.isEmpty = false ↔ ∃ (x : α), x ∈ xs

theorem Array.isEmpty_iff_size_eq_zero {α : Type u_1} {l : Array α} : l.isEmpty = true ↔ l.size = 0

@[simp]

theorem Array.isEmpty_eq_true {α : Type u_1} {l : Array α} : l.isEmpty = true ↔ l = #[]

@[simp]

theorem Array.isEmpty_eq_false {α : Type u_1} {l : Array α} : l.isEmpty = false ↔ l ≠ #[]

Decidability of bounded quantifiers

instance Array.instDecidableForallForallMemOfDecidablePred {α : Type u_1} {xs : Array α} {p : α → Prop} [DecidablePred p] : Decidable (∀ (x : α), x ∈ xs → p x)

Equations

• Array.instDecidableForallForallMemOfDecidablePred = decidable_of_iff (∀ (i : Nat) (h : i < xs.size), p xs[i]) ⋯

instance Array.instDecidableExistsAndMemOfDecidablePred {α : Type u_1} {xs : Array α} {p : α → Prop} [DecidablePred p] : Decidable (∃ (x : α), x ∈ xs ∧ p x)

Equations

• Array.instDecidableExistsAndMemOfDecidablePred = decidable_of_iff (∃ (i : Nat), ∃ (h : i < xs.size), p xs[i]) ⋯

any / all

theorem Array.anyM_eq_anyM_loop {m : Type → Type u_1} {α : Type u_2} [Monad m] (p : α → m Bool) (as : Array α) (start stop : Nat) : Array.anyM p as start stop = Array.anyM.loop p as (min stop as.size) ⋯ start

theorem Array.anyM_stop_le_start {m : Type → Type u_1} {α : Type u_2} [Monad m] (p : α → m Bool) (as : Array α) (start stop : Nat) (h : min stop as.size ≤ start) : Array.anyM p as start stop = pure false

@[irreducible]

theorem Array.anyM_loop_cons {m : Type → Type u_1} {α : Type u_2} [Monad m] (p : α → m Bool) (a : α) (as : List α) (stop start : Nat) (h : stop + 1 ≤ (a :: as).length) : Array.anyM.loop p { toList := a :: as } (stop + 1) h (start + 1) = Array.anyM.loop p { toList := as } stop ⋯ start

@[simp, irreducible]

theorem Array.anyM_toList {m : Type → Type u_1} {α : Type u_2} [Monad m] (p : α → m Bool) (as : Array α) : List.anyM p as.toList = Array.anyM p as

@[irreducible]

theorem Array.anyM_loop_iff_exists {α : Type u_1} {p : α → Bool} {as : Array α} {start stop : Nat} (h : stop ≤ as.size) : Array.anyM.loop p as stop h start = true ↔ ∃ (i : Nat), ∃ (x : i < as.size), start ≤ i ∧ i < stop ∧ p as[i] = true

theorem Array.any_iff_exists {α : Type u_1} {p : α → Bool} {as : Array α} {start stop : Nat} : as.any p start stop = true ↔ ∃ (i : Nat), ∃ (x : i < as.size), start ≤ i ∧ i < stop ∧ p as[i] = true

@[simp]

theorem Array.any_eq_true {α : Type u_1} {p : α → Bool} {as : Array α} : as.any p = true ↔ ∃ (i : Nat), ∃ (x : i < as.size), p as[i] = true

@[simp]

theorem Array.any_eq_false {α : Type u_1} {p : α → Bool} {as : Array α} : as.any p = false ↔ ∀ (i : Nat) (x : i < as.size), ¬p as[i] = true

@[simp]

theorem Array.any_toList {α : Type u_1} {p : α → Bool} (as : Array α) : as.toList.any p = as.any p

theorem Array.allM_eq_not_anyM_not {m : Type → Type u_1} {α : Type u_2} [Monad m] [LawfulMonad m] (p : α → m Bool) (as : Array α) : Array.allM p as = (fun (x : Bool) => !x) <$> Array.anyM (fun (x : α) => (fun (x : Bool) => !x) <$> p x) as

@[simp]

theorem Array.allM_toList {m : Type → Type u_1} {α : Type u_2} [Monad m] [LawfulMonad m] (p : α → m Bool) (as : Array α) : List.allM p as.toList = Array.allM p as

theorem Array.all_eq_not_any_not {α : Type u_1} (p : α → Bool) (as : Array α) (start stop : Nat) : as.all p start stop = !as.any (fun (x : α) => !p x) start stop

theorem Array.all_iff_forall {α : Type u_1} {p : α → Bool} {as : Array α} {start stop : Nat} : as.all p start stop = true ↔ ∀ (i : Nat) (x : i < as.size), start ≤ i ∧ i < stop → p as[i] = true

@[simp]

theorem Array.all_eq_true {α : Type u_1} {p : α → Bool} {as : Array α} : as.all p = true ↔ ∀ (i : Nat) (x : i < as.size), p as[i] = true

@[simp]

theorem Array.all_eq_false {α : Type u_1} {p : α → Bool} {as : Array α} : as.all p = false ↔ ∃ (i : Nat), ∃ (x : i < as.size), ¬p as[i] = true

@[simp]

theorem Array.all_toList {α : Type u_1} {p : α → Bool} (as : Array α) : as.toList.all p = as.all p

theorem Array.all_eq_true_iff_forall_mem {α : Type u_1} {p : α → Bool} {l : Array α} : l.all p = true ↔ ∀ (x : α), x ∈ l → p x = true

@[simp]

theorem List.anyM_toArray' {m : Type → Type u_1} {α : Type u_2} {stop : Nat} [Monad m] [LawfulMonad m] (p : α → m Bool) (l : List α) (h : stop = l.toArray.size) : Array.anyM p l.toArray 0 stop = List.anyM p l

Variant of anyM_toArray with a side condition on stop.

@[simp]

theorem List.any_toArray' {α : Type u_1} {stop : Nat} (p : α → Bool) (l : List α) (h : stop = l.toArray.size) : l.toArray.any p 0 stop = l.any p

Variant of any_toArray with a side condition on stop.

@[simp]

theorem List.allM_toArray' {m : Type → Type u_1} {α : Type u_2} {stop : Nat} [Monad m] [LawfulMonad m] (p : α → m Bool) (l : List α) (h : stop = l.toArray.size) : Array.allM p l.toArray 0 stop = List.allM p l

Variant of allM_toArray with a side condition on stop.

@[simp]

theorem List.all_toArray' {α : Type u_1} {stop : Nat} (p : α → Bool) (l : List α) (h : stop = l.toArray.size) : l.toArray.all p 0 stop = l.all p

Variant of all_toArray with a side condition on stop.

theorem List.anyM_toArray {m : Type → Type u_1} {α : Type u_2} [Monad m] [LawfulMonad m] (p : α → m Bool) (l : List α) : Array.anyM p l.toArray = List.anyM p l

theorem List.any_toArray {α : Type u_1} (p : α → Bool) (l : List α) : l.toArray.any p = l.any p

theorem List.allM_toArray {m : Type → Type u_1} {α : Type u_2} [Monad m] [LawfulMonad m] (p : α → m Bool) (l : List α) : Array.allM p l.toArray = List.allM p l

theorem List.all_toArray {α : Type u_1} (p : α → Bool) (l : List α) : l.toArray.all p = l.all p

theorem Array.any_eq_true' {α : Type u_1} {p : α → Bool} {as : Array α} : as.any p = true ↔ ∃ (x : α), x ∈ as ∧ p x = true

Variant of any_eq_true in terms of membership rather than an array index.

theorem Array.any_eq_false' {α : Type u_1} {p : α → Bool} {as : Array α} : as.any p = false ↔ ∀ (x : α), x ∈ as → ¬p x = true

Variant of any_eq_false in terms of membership rather than an array index.

theorem Array.all_eq_true' {α : Type u_1} {p : α → Bool} {as : Array α} : as.all p = true ↔ ∀ (x : α), x ∈ as → p x = true

Variant of all_eq_true in terms of membership rather than an array index.

theorem Array.all_eq_false' {α : Type u_1} {p : α → Bool} {as : Array α} : as.all p = false ↔ ∃ (x : α), x ∈ as ∧ ¬p x = true

Variant of all_eq_false in terms of membership rather than an array index.

theorem Array.any_eq {α : Type u_1} {xs : Array α} {p : α → Bool} : xs.any p = decide (∃ (i : Nat), ∃ (h : i < xs.size), p xs[i] = true)

theorem Array.any_eq' {α : Type u_1} {xs : Array α} {p : α → Bool} : xs.any p = decide (∃ (x : α), x ∈ xs ∧ p x = true)

Variant of any_eq in terms of membership rather than an array index.

theorem Array.all_eq {α : Type u_1} {xs : Array α} {p : α → Bool} : xs.all p = decide (∀ (i : Nat) (x : i < xs.size), p xs[i] = true)

theorem Array.all_eq' {α : Type u_1} {xs : Array α} {p : α → Bool} : xs.all p = decide (∀ (x : α), x ∈ xs → p x = true)

Variant of all_eq in terms of membership rather than an array index.

theorem Array.decide_exists_mem {α : Type u_1} {xs : Array α} {p : α → Prop} [DecidablePred p] : decide (∃ (x : α), x ∈ xs ∧ p x) = xs.any fun (b : α) => decide (p b)

theorem Array.decide_forall_mem {α : Type u_1} {xs : Array α} {p : α → Prop} [DecidablePred p] : decide (∀ (x : α), x ∈ xs → p x) = xs.all fun (b : α) => decide (p b)

@[simp]

theorem List.contains_toArray {α : Type u_1} [BEq α] {l : List α} {a : α} : l.toArray.contains a = l.contains a

theorem List.elem_toArray {α : Type u_1} [BEq α] {l : List α} {a : α} : Array.elem a l.toArray = List.elem a l

theorem Array.any_beq {α : Type u_1} [BEq α] {xs : Array α} {a : α} : (xs.any fun (x : α) => a == x) = xs.contains a

theorem Array.any_beq' {α : Type u_1} {a : α} [BEq α] [PartialEquivBEq α] {xs : Array α} : (xs.any fun (x : α) => x == a) = xs.contains a

Variant of any_beq with == reversed.

theorem Array.all_bne {α : Type u_1} {a : α} [BEq α] {xs : Array α} : (xs.all fun (x : α) => a != x) = !xs.contains a

theorem Array.all_bne' {α : Type u_1} {a : α} [BEq α] [PartialEquivBEq α] {xs : Array α} : (xs.all fun (x : α) => x != a) = !xs.contains a

Variant of all_bne with != reversed.

theorem Array.mem_of_contains_eq_true {α : Type u_1} [BEq α] [LawfulBEq α] {a : α} {as : Array α} : as.contains a = true → a ∈ as

@[reducible, inline, deprecated Array.mem_of_contains_eq_true (since := "2024-12-12")]

abbrev Array.mem_of_elem_eq_true {α : Type u_1} [BEq α] [LawfulBEq α] {a : α} {as : Array α} : as.contains a = true → a ∈ as

Equations

• @Array.mem_of_elem_eq_true = @Array.mem_of_contains_eq_true

theorem Array.contains_eq_true_of_mem {α : Type u_1} [BEq α] [LawfulBEq α] {a : α} {as : Array α} (h : a ∈ as) : as.contains a = true

@[reducible, inline, deprecated Array.contains_eq_true_of_mem (since := "2024-12-12")]

abbrev Array.elem_eq_true_of_mem {α : Type u_1} [BEq α] [LawfulBEq α] {a : α} {as : Array α} (h : a ∈ as) : as.contains a = true

Equations

• @Array.elem_eq_true_of_mem = @Array.contains_eq_true_of_mem

instance Array.instDecidableMemOfLawfulBEq {α : Type u_1} [BEq α] [LawfulBEq α] (a : α) (as : Array α) : Decidable (a ∈ as)

Equations

• Array.instDecidableMemOfLawfulBEq a as = decidable_of_decidable_of_iff ⋯

@[simp]

theorem Array.elem_eq_contains {α : Type u_1} [BEq α] {a : α} {l : Array α} : Array.elem a l = l.contains a

theorem Array.elem_iff {α : Type u_1} [BEq α] [LawfulBEq α] {a : α} {as : Array α} : Array.elem a as = true ↔ a ∈ as

theorem Array.contains_iff {α : Type u_1} [BEq α] [LawfulBEq α] {a : α} {as : Array α} : as.contains a = true ↔ a ∈ as

theorem Array.elem_eq_mem {α : Type u_1} [BEq α] [LawfulBEq α] (a : α) (as : Array α) : Array.elem a as = decide (a ∈ as)

@[simp]

theorem Array.contains_eq_mem {α : Type u_1} [BEq α] [LawfulBEq α] (a : α) (as : Array α) : as.contains a = decide (a ∈ as)

@[simp]

theorem Array.any_push' {α : Type u_1} {stop : Nat} [BEq α] {as : Array α} {a : α} {p : α → Bool} (h : stop = as.size + 1) : (as.push a).any p 0 stop = (as.any p || p a)

Variant of any_push with a side condition on stop.

theorem Array.any_push {α : Type u_1} [BEq α] {as : Array α} {a : α} {p : α → Bool} : (as.push a).any p = (as.any p || p a)

@[simp]

theorem Array.all_push' {α : Type u_1} {stop : Nat} [BEq α] {as : Array α} {a : α} {p : α → Bool} (h : stop = as.size + 1) : (as.push a).all p 0 stop = (as.all p && p a)

Variant of all_push with a side condition on stop.

theorem Array.all_push {α : Type u_1} [BEq α] {as : Array α} {a : α} {p : α → Bool} : (as.push a).all p = (as.all p && p a)

@[simp]

theorem Array.contains_push {α : Type u_1} [BEq α] {l : Array α} {a b : α} : (l.push a).contains b = (l.contains b || b == a)

set

@[simp]

theorem Array.getElem_set_self {α : Type u_1} (a : Array α) (i : Nat) (h : i < a.size) (v : α) {j : Nat} (eq : i = j) (p : j < (a.set i v h).size) : (a.set i v h)[j] = v

@[reducible, inline, deprecated Array.getElem_set_self (since := "2024-12-11")]

abbrev Array.getElem_set_eq {α : Type u_1} (a : Array α) (i : Nat) (h : i < a.size) (v : α) {j : Nat} (eq : i = j) (p : j < (a.set i v h).size) : (a.set i v h)[j] = v

Equations

• @Array.getElem_set_eq = @Array.getElem_set_self

@[simp]

theorem Array.getElem?_set_self {α : Type u_1} (a : Array α) (i : Nat) (h : i < a.size) (v : α) : (a.set i v h)[i]? = some v

@[reducible, inline, deprecated Array.getElem?_set_self (since := "2024-12-11")]

abbrev Array.getElem?_set_eq {α : Type u_1} (a : Array α) (i : Nat) (h : i < a.size) (v : α) : (a.set i v h)[i]? = some v

Equations

• @Array.getElem?_set_eq = @Array.getElem?_set_self

@[simp]

theorem Array.getElem_set_ne {α : Type u_1} (a : Array α) (i : Nat) (h' : i < a.size) (v : α) {j : Nat} (pj : j < (a.set i v h').size) (h : i ≠ j) : (a.set i v h')[j] = a[j]

@[simp]

theorem Array.getElem?_set_ne {α : Type u_1} (a : Array α) (i : Nat) (h : i < a.size) {j : Nat} (v : α) (ne : i ≠ j) : (a.set i v h)[j]? = a[j]?

theorem Array.getElem_set {α : Type u_1} (a : Array α) (i : Nat) (h' : i < a.size) (v : α) (j : Nat) (h : j < (a.set i v h').size) : (a.set i v h')[j] = if i = j then v else a[j]

theorem Array.getElem?_set {α : Type u_1} (a : Array α) (i : Nat) (h : i < a.size) (v : α) (j : Nat) : (a.set i v h)[j]? = if i = j then some v else a[j]?

@[simp]

theorem Array.set_getElem_self {α : Type u_1} {as : Array α} {i : Nat} (h : i < as.size) : as.set i as[i] h = as

@[simp]

theorem Array.set_eq_empty_iff {α : Type u_1} {as : Array α} (n : Nat) (a : α) (h : n < as.size) : as.set n a h = #[] ↔ as = #[]

theorem Array.set_comm {α : Type u_1} (a b : α) {i j : Nat} (as : Array α) {hi : i < as.size} {hj : j < (as.set i a hi).size} (h : i ≠ j) : (as.set i a hi).set j b hj = (as.set j b ⋯).set i a ⋯

@[simp]

theorem Array.set_set {α : Type u_1} (a b : α) (as : Array α) (i : Nat) (h : i < as.size) : (as.set i a h).set i b ⋯ = as.set i b h

theorem Array.mem_set {α : Type u_1} (as : Array α) (i : Nat) (h : i < as.size) (a : α) : a ∈ as.set i a h

theorem Array.mem_or_eq_of_mem_set {α : Type u_1} {as : Array α} {i : Nat} {a b : α} {w : i < as.size} (h : a ∈ as.set i b w) : a ∈ as ∨ a = b

@[simp]

theorem Array.toList_set {α : Type u_1} (a : Array α) (i : Nat) (x : α) (h : i < a.size) : (a.set i x h).toList = a.toList.set i x

setIfInBounds

@[simp]

theorem Array.set!_eq_setIfInBounds : @Array.set! = @Array.setIfInBounds

@[reducible, inline, deprecated Array.set!_eq_setIfInBounds (since := "2024-12-12")]

abbrev Array.set!_is_setIfInBounds : @Array.set! = @Array.setIfInBounds

Equations

• Array.set!_is_setIfInBounds = Array.set!_eq_setIfInBounds

@[simp]

theorem Array.size_setIfInBounds {α : Type u_1} (as : Array α) (index : Nat) (val : α) : (as.setIfInBounds index val).size = as.size

theorem Array.getElem_setIfInBounds {α : Type u_1} (as : Array α) (i : Nat) (v : α) (j : Nat) (hj : j < (as.setIfInBounds i v).size) : (as.setIfInBounds i v)[j] = if i = j then v else as[j]

@[simp]

theorem Array.getElem_setIfInBounds_self {α : Type u_1} (as : Array α) {i : Nat} (v : α) (h : i < (as.setIfInBounds i v).size) : (as.setIfInBounds i v)[i] = v

@[reducible, inline, deprecated Array.getElem_setIfInBounds_self (since := "2024-12-11")]

abbrev Array.getElem_setIfInBounds_eq {α : Type u_1} (as : Array α) {i : Nat} (v : α) (h : i < (as.setIfInBounds i v).size) : (as.setIfInBounds i v)[i] = v

Equations

• @Array.getElem_setIfInBounds_eq = @Array.getElem_setIfInBounds_self

@[simp]

theorem Array.getElem_setIfInBounds_ne {α : Type u_1} (as : Array α) {i : Nat} (v : α) {j : Nat} (hj : j < (as.setIfInBounds i v).size) (h : i ≠ j) : (as.setIfInBounds i v)[j] = as[j]

theorem Array.getElem?_setIfInBounds {α : Type u_1} {as : Array α} {i j : Nat} {a : α} : (as.setIfInBounds i a)[j]? = if i = j then if i < as.size then some a else none else as[j]?

theorem Array.getElem?_setIfInBounds_self {α : Type u_1} (as : Array α) {i : Nat} (v : α) : (as.setIfInBounds i v)[i]? = if i < as.size then some v else none

@[simp]

theorem Array.getElem?_setIfInBounds_self_of_lt {α : Type u_1} (as : Array α) {i : Nat} (v : α) (h : i < as.size) : (as.setIfInBounds i v)[i]? = some v

@[reducible, inline, deprecated Array.getElem?_setIfInBounds_self (since := "2024-12-11")]

abbrev Array.getElem?_setIfInBounds_eq {α : Type u_1} (as : Array α) {i : Nat} (v : α) : (as.setIfInBounds i v)[i]? = if i < as.size then some v else none

Equations

• @Array.getElem?_setIfInBounds_eq = @Array.getElem?_setIfInBounds_self

@[simp]

theorem Array.getElem?_setIfInBounds_ne {α : Type u_1} {as : Array α} {i j : Nat} (h : i ≠ j) {a : α} : (as.setIfInBounds i a)[j]? = as[j]?

theorem Array.setIfInBounds_eq_of_size_le {α : Type u_1} {l : Array α} {n : Nat} (h : l.size ≤ n) {a : α} : l.setIfInBounds n a = l

@[simp]

theorem Array.setIfInBounds_eq_empty_iff {α : Type u_1} {as : Array α} (n : Nat) (a : α) : as.setIfInBounds n a = #[] ↔ as = #[]

theorem Array.setIfInBounds_comm {α : Type u_1} (a b : α) {i j : Nat} (as : Array α) (h : i ≠ j) : (as.setIfInBounds i a).setIfInBounds j b = (as.setIfInBounds j b).setIfInBounds i a

@[simp]

theorem Array.setIfInBounds_setIfInBounds {α : Type u_1} (a b : α) (as : Array α) (i : Nat) : (as.setIfInBounds i a).setIfInBounds i b = as.setIfInBounds i b

theorem Array.mem_setIfInBounds {α : Type u_1} (as : Array α) (i : Nat) (h : i < as.size) (a : α) : a ∈ as.setIfInBounds i a

theorem Array.mem_or_eq_of_mem_setIfInBounds {α : Type u_1} {as : Array α} {i : Nat} {a b : α} (h : a ∈ as.setIfInBounds i b) : a ∈ as ∨ a = b

@[simp]

theorem Array.getD_get?_setIfInBounds {α : Type u_1} (a : Array α) (i : Nat) (v d : α) : (a.setIfInBounds i v)[i]?.getD d = if i < a.size then v else d

Simplifies a normal form from get!

@[simp]

theorem Array.toList_setIfInBounds {α : Type u_1} (a : Array α) (i : Nat) (x : α) : (a.setIfInBounds i x).toList = a.toList.set i x

BEq

@[simp]

theorem Array.beq_empty_iff {α : Type u_1} [BEq α] {xs : Array α} : (xs == #[]) = xs.isEmpty

@[simp]

theorem Array.empty_beq_iff {α : Type u_1} [BEq α] {xs : Array α} : (#[] == xs) = xs.isEmpty

@[simp]

theorem Array.push_beq_push {α : Type u_1} [BEq α] {a b : α} {v w : Array α} : (v.push a == w.push b) = (v == w && a == b)

theorem Array.size_eq_of_beq {α : Type u_1} [BEq α] {xs ys : Array α} (h : (xs == ys) = true) : xs.size = ys.size

@[simp, irreducible]

theorem Array.mkArray_beq_mkArray {α : Type u_1} [BEq α] {a b : α} {n : Nat} : (mkArray n a == mkArray n b) = (n == 0 || a == b)

@[simp]

theorem Array.reflBEq_iff {α : Type u_1} [BEq α] : ReflBEq (Array α) ↔ ReflBEq α

@[simp]

theorem Array.lawfulBEq_iff {α : Type u_1} [BEq α] : LawfulBEq (Array α) ↔ LawfulBEq α

isEqv

@[simp]

theorem Array.isEqv_eq {α : Type u_1} [DecidableEq α] {l₁ l₂ : Array α} : ((l₁.isEqv l₂ fun (x1 x2 : α) => x1 == x2) = true) = (l₁ = l₂)

Content below this point has not yet been aligned with List.

theorem Array.singleton_eq_toArray_singleton {α : Type u_1} (a : α) : #[a] = #[a]

@[simp]

theorem Array.singleton_def {α : Type u_1} (v : α) : Array.singleton v = #[v]

@[simp]

theorem Array.toArray_toList {α : Type u_1} (a : Array α) : a.toList.toArray = a

@[simp]

theorem Array.length_toList {α : Type u_1} {l : Array α} : l.toList.length = l.size

@[simp]

theorem Array.mkEmpty_eq (α : Type u_1) (n : Nat) : Array.mkEmpty n = #[]

@[deprecated Array.size_toArray (since := "2024-12-11")]

theorem Array.size_mk {α : Type u_1} (as : List α) : { toList := as }.size = as.length

theorem Array.foldrM_push {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [Monad m] (f : α → β → m β) (init : β) (arr : Array α) (a : α) : Array.foldrM f init (arr.push a) = do let init ← f a init Array.foldrM f init arr

@[simp]

theorem Array.foldrM_push' {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [Monad m] (f : α → β → m β) (init : β) (arr : Array α) (a : α) {start : Nat} (h : start = arr.size + 1) : Array.foldrM f init (arr.push a) start = do let init ← f a init Array.foldrM f init arr

Variant of foldrM_push with h : start = arr.size + 1 rather than (arr.push a).size as the argument.

theorem Array.foldr_push {α : Type u_1} {β : Type u_2} (f : α → β → β) (init : β) (arr : Array α) (a : α) : Array.foldr f init (arr.push a) = Array.foldr f (f a init) arr

@[simp]

theorem Array.foldr_push' {α : Type u_1} {β : Type u_2} (f : α → β → β) (init : β) (arr : Array α) (a : α) {start : Nat} (h : start = arr.size + 1) : Array.foldr f init (arr.push a) start = Array.foldr f (f a init) arr

Variant of foldr_push with the h : start = arr.size + 1 rather than (arr.push a).size as the argument.

@[inline]

def Array.toListRev {α : Type u_1} (arr : Array α) : List α

A more efficient version of arr.toList.reverse.

Equations

• arr.toListRev = Array.foldl (fun (l : List α) (t : α) => t :: l) [] arr

@[simp]

theorem Array.toListRev_eq {α : Type u_1} (arr : Array α) : arr.toListRev = arr.toList.reverse

theorem Array.mapM_eq_foldlM {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [Monad m] [LawfulMonad m] (f : α → m β) (arr : Array α) : Array.mapM f arr = Array.foldlM (fun (bs : Array β) (a : α) => bs.push <$> f a) #[] arr

@[irreducible]

theorem Array.mapM_eq_foldlM.aux {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [Monad m] [LawfulMonad m] (f : α → m β) (arr : Array α) (i : Nat) (r : Array β) : Array.mapM.map f arr i r = List.foldlM (fun (bs : Array β) (a : α) => bs.push <$> f a) r (List.drop i arr.toList)

@[simp]

theorem Array.toList_map {α : Type u_1} {β : Type u_2} (f : α → β) (arr : Array α) : (Array.map f arr).toList = List.map f arr.toList

@[simp]

theorem Array.size_map {α : Type u_1} {β : Type u_2} (f : α → β) (arr : Array α) : (Array.map f arr).size = arr.size

@[simp]

theorem Array.mapM_empty {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [Monad m] (f : α → m β) : Array.mapM f #[] = pure #[]

@[simp]

theorem Array.map_empty {α : Type u_1} {β : Type u_2} (f : α → β) : Array.map f #[] = #[]

@[simp]

theorem Array.appendList_nil {α : Type u_1} (arr : Array α) : arr ++ [] = arr

@[simp]

theorem Array.appendList_cons {α : Type u_1} (arr : Array α) (a : α) (l : List α) : arr ++ a :: l = arr.push a ++ l

theorem Array.foldl_toList_eq_flatMap {α : Type u_1} {β : Type u_2} (l : List α) (acc : Array β) (F : Array β → α → Array β) (G : α → List β) (H : ∀ (acc : Array β) (a : α), (F acc a).toList = acc.toList ++ G a) : (List.foldl F acc l).toList = acc.toList ++ l.flatMap G

theorem Array.foldl_toList_eq_map {α : Type u_1} {β : Type u_2} (l : List α) (acc : Array β) (G : α → β) : (List.foldl (fun (acc : Array β) (a : α) => acc.push (G a)) acc l).toList = acc.toList ++ List.map G l

uset

theorem Array.size_uset {α : Type u_1} (a : Array α) (v : α) (i : USize) (h : i.toNat < a.size) : (a.uset i v h).size = a.size

get

@[deprecated Array.getElem?_eq_getElem (since := "2024-12-11")]

theorem Array.getElem?_lt {α : Type u_1} (a : Array α) {i : Nat} (h : i < a.size) : a[i]? = some a[i]

@[deprecated Array.getElem?_eq_none (since := "2024-12-11")]

theorem Array.getElem?_ge {α : Type u_1} (a : Array α) {i : Nat} (h : i ≥ a.size) : a[i]? = none

@[simp]

theorem Array.get?_eq_getElem? {α : Type u_1} (a : Array α) (i : Nat) : a.get? i = a[i]?

@[deprecated Array.getElem?_eq_none (since := "2024-12-11")]

theorem Array.getElem?_len_le {α : Type u_1} (a : Array α) {i : Nat} (h : a.size ≤ i) : a[i]? = none

@[reducible, inline, deprecated Array.getD_getElem? (since := "2024-12-11")]

abbrev Array.getD_get? {α : Type u_1} (a : Array α) (i : Nat) (d : α) : a[i]?.getD d = if p : i < a.size then a[i] else d

Equations

• @Array.getD_get? = @Array.getD_getElem?

@[simp]

theorem Array.getD_eq_get? {α : Type u_1} (a : Array α) (i : Nat) (d : α) : a.getD i d = a[i]?.getD d

theorem Array.get!_eq_getD {α : Type u_1} {n : Nat} [Inhabited α] (a : Array α) : a.get! n = a.getD n default

theorem Array.get!_eq_getElem? {α : Type u_1} [Inhabited α] (a : Array α) (i : Nat) : a.get! i = (a.get? i).getD default

ofFn

@[simp, irreducible]

theorem Array.size_ofFn_go {α : Type u_1} {n : Nat} (f : Fin n → α) (i : Nat) (acc : Array α) : (Array.ofFn.go f i acc).size = acc.size + (n - i)

@[simp]

theorem Array.size_ofFn {n : Nat} {α : Type u_1} (f : Fin n → α) : (Array.ofFn f).size = n

@[irreducible]

theorem Array.getElem_ofFn_go {n : Nat} {α : Type u_1} (f : Fin n → α) (i : Nat) {acc : Array α} {k : Nat} (hki : k < n) (hin : i ≤ n) (hi : i = acc.size) (hacc : ∀ (j : Nat) (hj : j < acc.size), acc[j] = f ⟨j, ⋯⟩) : (Array.ofFn.go f i acc)[k] = f ⟨k, hki⟩

@[simp]

theorem Array.getElem_ofFn {n : Nat} {α : Type u_1} (f : Fin n → α) (i : Nat) (h : i < (Array.ofFn f).size) : (Array.ofFn f)[i] = f ⟨i, ⋯⟩

theorem Array.getElem?_ofFn {n : Nat} {α : Type u_1} (f : Fin n → α) (i : Nat) : (Array.ofFn f)[i]? = if h : i < n then some (f ⟨i, h⟩) else none

@[simp]

theorem Array.ofFn_zero {α : Type u_1} (f : Fin 0 → α) : Array.ofFn f = #[]

theorem Array.ofFn_succ {n : Nat} {α : Type u_1} (f : Fin (n + 1) → α) : Array.ofFn f = (Array.ofFn fun (i : Fin n) => f i.castSucc).push (f ⟨n, ⋯⟩)

mem

@[simp]

theorem Array.mem_toList {α : Type u_1} {a : α} {l : Array α} : a ∈ l.toList ↔ a ∈ l

theorem Array.not_mem_nil {α : Type u_1} (a : α) : ¬a ∈ #[]

get lemmas

theorem Array.lt_of_getElem {α : Type u_1} {x : α} {a : Array α} {idx : Nat} {hidx : idx < a.size} : a[idx] = x → idx < a.size

theorem Array.getElem_fin_eq_getElem_toList {α : Type u_1} (a : Array α) (i : Fin a.size) : a[i] = a.toList[i]

@[simp]

theorem Array.ugetElem_eq_getElem {α : Type u_1} (a : Array α) {i : USize} (h : i.toNat < a.size) : a[i] = a[i.toNat]

theorem Array.getElem?_size_le {α : Type u_1} (a : Array α) (i : Nat) (h : a.size ≤ i) : a[i]? = none

@[reducible, inline, deprecated Array.getElem?_size_le (since := "2024-10-21")]

abbrev Array.get?_len_le {α : Type u_1} (a : Array α) (i : Nat) (h : a.size ≤ i) : a[i]? = none

Equations

• @Array.get?_len_le = @Array.getElem?_size_le

theorem Array.getElem_mem_toList {α : Type u_1} {i : Nat} (a : Array α) (h : i < a.size) : a[i] ∈ a.toList

theorem Array.get?_eq_get?_toList {α : Type u_1} (a : Array α) (i : Nat) : a.get? i = a.toList.get? i

theorem Array.get!_eq_get? {α : Type u_1} {n : Nat} [Inhabited α] (a : Array α) : a.get! n = (a.get? n).getD default

theorem Array.back!_eq_back? {α : Type u_1} [Inhabited α] (a : Array α) : a.back! = a.back?.getD default

@[simp]

theorem Array.back?_push {α : Type u_1} {x : α} (a : Array α) : (a.push x).back? = some x

@[simp]

theorem Array.back!_push {α : Type u_1} {x : α} [Inhabited α] (a : Array α) : (a.push x).back! = x

theorem Array.mem_of_back?_eq_some {α : Type u_1} {xs : Array α} {a : α} (h : xs.back? = some a) : a ∈ xs

theorem Array.getElem?_push_lt {α : Type u_1} (a : Array α) (x : α) (i : Nat) (h : i < a.size) : (a.push x)[i]? = some a[i]

@[reducible, inline, deprecated Array.getElem?_push_lt (since := "2024-10-21")]

abbrev Array.get?_push_lt {α : Type u_1} (a : Array α) (x : α) (i : Nat) (h : i < a.size) : (a.push x)[i]? = some a[i]

Equations

• @Array.get?_push_lt = @Array.getElem?_push_lt

theorem Array.getElem?_push_eq {α : Type u_1} (a : Array α) (x : α) : (a.push x)[a.size]? = some x

@[reducible, inline, deprecated Array.getElem?_push_eq (since := "2024-10-21")]

abbrev Array.get?_push_eq {α : Type u_1} (a : Array α) (x : α) : (a.push x)[a.size]? = some x

Equations

• @Array.get?_push_eq = @Array.getElem?_push_eq

@[reducible, inline, deprecated Array.getElem?_push (since := "2024-10-21")]

abbrev Array.get?_push {α : Type u_1} {i : Nat} {a : Array α} {x : α} : (a.push x)[i]? = if i = a.size then some x else a[i]?

Equations

• @Array.get?_push = @Array.getElem?_push

@[simp]

theorem Array.getElem?_size {α : Type u_1} {a : Array α} : a[a.size]? = none

@[reducible, inline, deprecated Array.getElem?_size (since := "2024-10-21")]

abbrev Array.get?_size {α : Type u_1} {a : Array α} : a[a.size]? = none

Equations

• @Array.get?_size = @Array.getElem?_size

theorem Array.get_set_eq {α : Type u_1} (a : Array α) (i : Nat) (v : α) (h : i < a.size) : (a.set i v h)[i] = v

theorem Array.get?_set_eq {α : Type u_1} (a : Array α) (i : Nat) (v : α) (h : i < a.size) : (a.set i v h)[i]? = some v

@[simp]

theorem Array.get?_set_ne {α : Type u_1} (a : Array α) (i : Nat) (h' : i < a.size) {j : Nat} (v : α) (h : i ≠ j) : (a.set i v h')[j]? = a[j]?

theorem Array.get?_set {α : Type u_1} (a : Array α) (i : Nat) (h : i < a.size) (j : Nat) (v : α) : (a.set i v h)[j]? = if i = j then some v else a[j]?

theorem Array.get_set {α : Type u_1} (a : Array α) (i : Nat) (hi : i < a.size) (j : Nat) (hj : j < a.size) (v : α) : (a.set i v hi)[j] = if i = j then v else a[j]

@[simp]

theorem Array.get_set_ne {α : Type u_1} (a : Array α) (i : Nat) (hi : i < a.size) {j : Nat} (v : α) (hj : j < a.size) (h : i ≠ j) : (a.set i v hi)[j] = a[j]

theorem Array.swap_def {α : Type u_1} (a : Array α) (i j : Nat) (hi : i < a.size) (hj : j < a.size) : a.swap i j hi hj = (a.set i a[j] hi).set j a[i] ⋯

@[simp]

theorem Array.toList_swap {α : Type u_1} (a : Array α) (i j : Nat) (hi : i < a.size) (hj : j < a.size) : (a.swap i j hi hj).toList = (a.toList.set i a[j]).set j a[i]

theorem Array.getElem?_swap {α : Type u_1} (a : Array α) (i j : Nat) (hi : i < a.size) (hj : j < a.size) (k : Nat) : (a.swap i j hi hj)[k]? = if j = k then some a[i] else if i = k then some a[j] else a[k]?

@[simp]

theorem Array.swapAt_def {α : Type u_1} (a : Array α) (i : Nat) (v : α) (hi : i < a.size) : a.swapAt i v hi = (a[i], a.set i v hi)

theorem Array.size_swapAt {α : Type u_1} (a : Array α) (i : Nat) (v : α) (hi : i < a.size) : (a.swapAt i v hi).snd.size = a.size

@[simp]

theorem Array.swapAt!_def {α : Type u_1} (a : Array α) (i : Nat) (v : α) (h : i < a.size) : a.swapAt! i v = (a[i], a.set i v h)

@[simp]

theorem Array.size_swapAt! {α : Type u_1} (a : Array α) (i : Nat) (v : α) : (a.swapAt! i v).snd.size = a.size

@[simp]

theorem Array.toList_pop {α : Type u_1} (a : Array α) : a.pop.toList = a.toList.dropLast

@[simp]

theorem Array.pop_empty {α : Type u_1} : #[].pop = #[]

@[simp]

theorem Array.pop_push {α : Type u_1} {x : α} (a : Array α) : (a.push x).pop = a

@[simp]

theorem Array.getElem_pop {α : Type u_1} (a : Array α) (i : Nat) (hi : i < a.pop.size) : a.pop[i] = a[i]

theorem Array.eq_push_pop_back!_of_size_ne_zero {α : Type u_1} [Inhabited α] {as : Array α} (h : as.size ≠ 0) : as = as.pop.push as.back!

theorem Array.eq_push_of_size_ne_zero {α : Type u_1} {as : Array α} (h : as.size ≠ 0) : ∃ (bs : Array α), ∃ (c : α), as = bs.push c

theorem Array.size_eq_length_toList {α : Type u_1} (as : Array α) : as.size = as.toList.length

@[simp]

theorem Array.size_swapIfInBounds {α : Type u_1} (a : Array α) (i j : Nat) : (a.swapIfInBounds i j).size = a.size

@[reducible, inline, deprecated Array.size_swapIfInBounds (since := "2024-11-24")]

abbrev Array.size_swap! {α : Type u_1} (a : Array α) (i j : Nat) : (a.swapIfInBounds i j).size = a.size

Equations

• @Array.size_swap! = @Array.size_swapIfInBounds

@[simp]

theorem Array.size_reverse {α : Type u_1} (a : Array α) : a.reverse.size = a.size

@[irreducible]

theorem Array.size_reverse.go {α : Type u_1} (as : Array α) (i : Nat) (j : Fin as.size) : (Array.reverse.loop as i j).size = as.size

@[simp]

theorem Array.size_range {n : Nat} : (Array.range n).size = n

@[simp]

theorem Array.toList_range (n : Nat) : (Array.range n).toList = List.range n

@[simp]

theorem Array.getElem_range {n x : Nat} (h : x < (Array.range n).size) : (Array.range n)[x] = x

@[simp]

theorem Array.toList_reverse {α : Type u_1} (a : Array α) : a.reverse.toList = a.toList.reverse

@[irreducible]

theorem Array.toList_reverse.go {α : Type u_1} (a as : Array α) (i j : Nat) (hj : j < as.size) (h : i + j + 1 = a.size) (h₂ : as.size = a.size) (H : ∀ (k : Nat), as.toList[k]? = if i ≤ k ∧ k ≤ j then a.toList[k]? else a.toList.reverse[k]?) (k : Nat) : (Array.reverse.loop as i ⟨j, hj⟩).toList[k]? = a.toList.reverse[k]?

take

@[simp]

theorem Array.size_take_loop {α : Type u_1} (a : Array α) (n : Nat) : (Array.take.loop n a).size = a.size - n

@[simp]

theorem Array.getElem_take_loop {α : Type u_1} (a : Array α) (n i : Nat) (h : i < (Array.take.loop n a).size) : (Array.take.loop n a)[i] = a[i]

@[simp]

theorem Array.size_take {α : Type u_1} (a : Array α) (n : Nat) : (a.take n).size = min n a.size

@[simp]

theorem Array.getElem_take {α : Type u_1} (a : Array α) (n i : Nat) (h : i < (a.take n).size) : (a.take n)[i] = a[i]

@[simp]

theorem Array.toList_take {α : Type u_1} (a : Array α) (n : Nat) : (a.take n).toList = List.take n a.toList

forIn

@[simp]

theorem Array.forIn_toList {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [Monad m] (as : Array α) (b : β) (f : α → β → m (ForInStep β)) : forIn as.toList b f = forIn as b f

@[simp]

theorem Array.forIn'_toList {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [Monad m] (as : Array α) (b : β) (f : (a : α) → a ∈ as.toList → β → m (ForInStep β)) : forIn' as.toList b f = forIn' as b fun (a : α) (m : a ∈ as) (b : β) => f a ⋯ b

foldl / foldr

@[simp]

theorem Array.foldlM_loop_empty {m : Type u_1 → Type u_2} {β : Type u_1} {α : Type u_3} {s : Nat} {h : s ≤ #[].size} [Monad m] (f : β → α → m β) (init : β) (i j : Nat) : Array.foldlM.loop f #[] s h i j init = pure init

@[simp]

theorem Array.foldlM_empty {m : Type u_1 → Type u_2} {β : Type u_1} {α : Type u_3} {start stop : Nat} [Monad m] (f : β → α → m β) (init : β) : Array.foldlM f init #[] start stop = pure init

@[simp]

theorem Array.foldrM_fold_empty {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [Monad m] (f : α → β → m β) (init : β) (i j : Nat) (h : j ≤ #[].size) : Array.foldrM.fold f #[] i j h init = pure init

@[simp]

theorem Array.foldrM_empty {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} {start stop : Nat} [Monad m] (f : α → β → m β) (init : β) : Array.foldrM f init #[] start stop = pure init

theorem Array.foldl_induction {α : Type u_1} {β : Type u_2} {as : Array α} (motive : Nat → β → Prop) {init : β} (h0 : motive 0 init) {f : β → α → β} (hf : ∀ (i : Fin as.size) (b : β), motive (↑i) b → motive (↑i + 1) (f b as[i])) : motive as.size (Array.foldl f init as)

@[irreducible]

theorem Array.foldl_induction.go {α : Type u_1} {β : Type u_2} {as : Array α} (motive : Nat → β → Prop) {f : β → α → β} (hf : ∀ (i : Fin as.size) (b : β), motive (↑i) b → motive (↑i + 1) (f b as[i])) {i j : Nat} {b : β} (h₁ : j ≤ as.size) (h₂ : as.size ≤ i + j) (H : motive j b) : motive as.size (Array.foldlM.loop f as as.size ⋯ i j b)

theorem Array.foldr_induction {α : Type u_1} {β : Type u_2} {as : Array α} (motive : Nat → β → Prop) {init : β} (h0 : motive as.size init) {f : α → β → β} (hf : ∀ (i : Fin as.size) (b : β), motive (↑i + 1) b → motive (↑i) (f as[i] b)) : motive 0 (Array.foldr f init as)

@[irreducible]

theorem Array.foldr_induction.go {α : Type u_1} {β : Type u_2} {as : Array α} (motive : Nat → β → Prop) {f : α → β → β} (hf : ∀ (i : Fin as.size) (b : β), motive (↑i + 1) b → motive (↑i) (f as[i] b)) {i : Nat} {b : β} (hi : i ≤ as.size) (H : motive i b) : motive 0 (Array.foldrM.fold f as 0 i hi b)

theorem Array.foldl_congr {α : Type u_1} {β : Type u_2} {as bs : Array α} (h₀ : as = bs) {f g : β → α → β} (h₁ : f = g) {a b : β} (h₂ : a = b) {start start' stop stop' : Nat} (h₃ : start = start') (h₄ : stop = stop') : Array.foldl f a as start stop = Array.foldl g b bs start' stop'

theorem Array.foldr_congr {α : Type u_1} {β : Type u_2} {as bs : Array α} (h₀ : as = bs) {f g : α → β → β} (h₁ : f = g) {a b : β} (h₂ : a = b) {start start' stop stop' : Nat} (h₃ : start = start') (h₄ : stop = stop') : Array.foldr f a as start stop = Array.foldr g b bs start' stop'

theorem Array.foldl_eq_foldlM {β : Type u_1} {α : Type u_2} (f : β → α → β) (b : β) (l : Array α) : Array.foldl f b l = Array.foldlM f b l

theorem Array.foldr_eq_foldrM {α : Type u_1} {β : Type u_2} (f : α → β → β) (b : β) (l : Array α) : Array.foldr f b l = Array.foldrM f b l

@[simp]

theorem Array.id_run_foldlM {β : Type u_1} {α : Type u_2} (f : β → α → Id β) (b : β) (l : Array α) : (Array.foldlM f b l).run = Array.foldl f b l

@[simp]

theorem Array.id_run_foldrM {α : Type u_1} {β : Type u_2} (f : α → β → Id β) (b : β) (l : Array α) : (Array.foldrM f b l).run = Array.foldr f b l

theorem Array.foldl_hom {α₁ : Type u_1} {α₂ : Type u_2} {β : Type u_3} (f : α₁ → α₂) (g₁ : α₁ → β → α₁) (g₂ : α₂ → β → α₂) (l : Array β) (init : α₁) (H : ∀ (x : α₁) (y : β), g₂ (f x) y = f (g₁ x y)) : Array.foldl g₂ (f init) l = f (Array.foldl g₁ init l)

theorem Array.foldr_hom {β₁ : Type u_1} {β₂ : Type u_2} {α : Type u_3} (f : β₁ → β₂) (g₁ : α → β₁ → β₁) (g₂ : α → β₂ → β₂) (l : Array α) (init : β₁) (H : ∀ (x : α) (y : β₁), g₂ x (f y) = f (g₁ x y)) : Array.foldr g₂ (f init) l = f (Array.foldr g₁ init l)

map

@[simp]

theorem Array.mem_map {α : Type u_1} {β : Type u_2} {b : β} {f : α → β} {l : Array α} : b ∈ Array.map f l ↔ ∃ (a : α), a ∈ l ∧ f a = b

theorem Array.exists_of_mem_map {α✝ : Type u_1} {α✝¹ : Type u_2} {f : α✝ → α✝¹} {l : Array α✝} {b : α✝¹} (h : b ∈ Array.map f l) : ∃ (a : α✝), a ∈ l ∧ f a = b

theorem Array.mem_map_of_mem {α : Type u_1} {β : Type u_2} {l : Array α} {a : α} (f : α → β) (h : a ∈ l) : f a ∈ Array.map f l

theorem Array.mapM_eq_mapM_toList {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [Monad m] [LawfulMonad m] (f : α → m β) (arr : Array α) : Array.mapM f arr = List.toArray <$> List.mapM f arr.toList

@[simp]

theorem Array.toList_mapM {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [Monad m] [LawfulMonad m] (f : α → m β) (arr : Array α) : Array.toList <$> Array.mapM f arr = List.mapM f arr.toList

@[irreducible]

theorem Array.mapM_map_eq_foldl {α : Type u_1} {β : Type u_2} {b : Array β} (as : Array α) (f : α → β) (i : Nat) : Array.mapM.map f as i b = Array.foldl (fun (r : Array β) (a : α) => r.push (f a)) b as i

theorem Array.map_eq_foldl {α : Type u_1} {β : Type u_2} (as : Array α) (f : α → β) : Array.map f as = Array.foldl (fun (r : Array β) (a : α) => r.push (f a)) #[] as

theorem Array.map_induction {α : Type u_1} {β : Type u_2} (as : Array α) (f : α → β) (motive : Nat → Prop) (h0 : motive 0) (p : Fin as.size → β → Prop) (hs : ∀ (i : Fin as.size), motive ↑i → p i (f as[i]) ∧ motive (↑i + 1)) : motive as.size ∧ ∃ (eq : (Array.map f as).size = as.size), ∀ (i : Nat) (h : i < as.size), p ⟨i, h⟩ (Array.map f as)[i]

theorem Array.map_spec {α : Type u_1} {β : Type u_2} (as : Array α) (f : α → β) (p : Fin as.size → β → Prop) (hs : ∀ (i : Fin as.size), p i (f as[i])) : ∃ (eq : (Array.map f as).size = as.size), ∀ (i : Nat) (h : i < as.size), p ⟨i, h⟩ (Array.map f as)[i]

@[simp]

theorem Array.getElem_map {α : Type u_1} {β : Type u_2} (f : α → β) (as : Array α) (i : Nat) (h : i < (Array.map f as).size) : (Array.map f as)[i] = f as[i]

@[simp]

theorem Array.getElem?_map {α : Type u_1} {β : Type u_2} (f : α → β) (as : Array α) (i : Nat) : (Array.map f as)[i]? = Option.map f as[i]?

@[simp]

theorem Array.map_push {α : Type u_1} {β : Type u_2} {f : α → β} {as : Array α} {x : α} : Array.map f (as.push x) = (Array.map f as).push (f x)

@[simp]

theorem Array.map_pop {α : Type u_1} {β : Type u_2} {f : α → β} {as : Array α} : Array.map f as.pop = (Array.map f as).pop

modify

@[simp]

theorem Array.size_modify {α : Type u_1} (a : Array α) (i : Nat) (f : α → α) : (a.modify i f).size = a.size

theorem Array.getElem_modify {α : Type u_1} {f : α → α} {as : Array α} {x i : Nat} (h : i < (as.modify x f).size) : (as.modify x f)[i] = if x = i then f as[i] else as[i]

@[simp]

theorem Array.toList_modify {α : Type u_1} {x : Nat} (as : Array α) (f : α → α) : (as.modify x f).toList = List.modify f x as.toList

theorem Array.getElem_modify_self {α : Type u_1} {as : Array α} {i : Nat} (f : α → α) (h : i < (as.modify i f).size) : (as.modify i f)[i] = f as[i]

theorem Array.getElem_modify_of_ne {α : Type u_1} {j : Nat} {as : Array α} {i : Nat} (h : i ≠ j) (f : α → α) (hj : j < (as.modify i f).size) : (as.modify i f)[j] = as[j]

theorem Array.getElem?_modify {α : Type u_1} {as : Array α} {i : Nat} {f : α → α} {j : Nat} : (as.modify i f)[j]? = if i = j then Option.map f as[j]? else as[j]?

filter

@[simp]

theorem Array.toList_filter {α : Type u_1} (p : α → Bool) (l : Array α) : (Array.filter p l).toList = List.filter p l.toList

@[simp]

theorem Array.filter_filter {α : Type u_1} {p : α → Bool} (q : α → Bool) (l : Array α) : Array.filter p (Array.filter q l) = Array.filter (fun (a : α) => p a && q a) l

@[simp]

theorem Array.mem_filter {α✝ : Type u_1} {p : α✝ → Bool} {as : Array α✝} {x : α✝} : x ∈ Array.filter p as ↔ x ∈ as ∧ p x = true

theorem Array.mem_of_mem_filter {α : Type u_1} {p : α → Bool} {a : α} {l : Array α} (h : a ∈ Array.filter p l) : a ∈ l

theorem Array.filter_congr {α : Type u_1} {as bs : Array α} (h : as = bs) {f g : α → Bool} (h' : f = g) {start stop start' stop' : Nat} (h₁ : start = start') (h₂ : stop = stop') : Array.filter f as start stop = Array.filter g bs start' stop'

filterMap

@[simp]

theorem Array.toList_filterMap {α : Type u_1} {β : Type u_2} (f : α → Option β) (l : Array α) : (Array.filterMap f l).toList = List.filterMap f l.toList

@[simp]

theorem Array.mem_filterMap {α : Type u_1} {β : Type u_2} {f : α → Option β} {l : Array α} {b : β} : b ∈ Array.filterMap f l ↔ ∃ (a : α), a ∈ l ∧ f a = some b

theorem Array.filterMap_congr {α : Type u_1} {β : Type u_2} {as bs : Array α} (h : as = bs) {f g : α → Option β} (h' : f = g) {start stop start' stop' : Nat} (h₁ : start = start') (h₂ : stop = stop') : Array.filterMap f as start stop = Array.filterMap g bs start' stop'

empty

theorem Array.size_empty {α : Type u_1} : #[].size = 0

append

theorem Array.push_eq_append_singleton {α : Type u_1} (as : Array α) (x : α) : as.push x = as ++ #[x]

@[simp]

theorem Array.mem_append {α : Type u_1} {a : α} {s t : Array α} : a ∈ s ++ t ↔ a ∈ s ∨ a ∈ t

theorem Array.mem_append_left {α : Type u_1} {a : α} {l₁ : Array α} (l₂ : Array α) (h : a ∈ l₁) : a ∈ l₁ ++ l₂

theorem Array.mem_append_right {α : Type u_1} {a : α} (l₁ : Array α) {l₂ : Array α} (h : a ∈ l₂) : a ∈ l₁ ++ l₂

@[simp]

theorem Array.size_append {α : Type u_1} (as bs : Array α) : (as ++ bs).size = as.size + bs.size

theorem Array.empty_append {α : Type u_1} (as : Array α) : #[] ++ as = as

theorem Array.append_empty {α : Type u_1} (as : Array α) : as ++ #[] = as

theorem Array.getElem_append {α : Type u_1} {i : Nat} {as bs : Array α} (h : i < (as ++ bs).size) : (as ++ bs)[i] = if h' : i < as.size then as[i] else bs[i - as.size]

theorem Array.getElem_append_left {α : Type u_1} {i : Nat} {as bs : Array α} {h : i < (as ++ bs).size} (hlt : i < as.size) : (as ++ bs)[i] = as[i]

theorem Array.getElem_append_right {α : Type u_1} {i : Nat} {as bs : Array α} {h : i < (as ++ bs).size} (hle : as.size ≤ i) : (as ++ bs)[i] = bs[i - as.size]

theorem Array.getElem?_append_left {α : Type u_1} {as bs : Array α} {i : Nat} (hn : i < as.size) : (as ++ bs)[i]? = as[i]?

theorem Array.getElem?_append_right {α : Type u_1} {as bs : Array α} {i : Nat} (h : as.size ≤ i) : (as ++ bs)[i]? = bs[i - as.size]?

theorem Array.getElem?_append {α : Type u_1} {as bs : Array α} {i : Nat} : (as ++ bs)[i]? = if i < as.size then as[i]? else bs[i - as.size]?

@[simp]

theorem Array.toArray_eq_append_iff {α : Type u_1} {xs : List α} {as bs : Array α} : xs.toArray = as ++ bs ↔ xs = as.toList ++ bs.toList

@[simp]

theorem Array.append_eq_toArray_iff {α : Type u_1} {as bs : Array α} {xs : List α} : as ++ bs = xs.toArray ↔ as.toList ++ bs.toList = xs

flatten

@[simp]

theorem Array.toList_flatten {α : Type u_1} {l : Array (Array α)} : l.flatten.toList = (List.map Array.toList l.toList).flatten

theorem Array.mem_flatten {α : Type u_1} {a : α} {L : Array (Array α)} : a ∈ L.flatten ↔ ∃ (l : Array α), l ∈ L ∧ a ∈ l

extract

theorem Array.extract_loop_zero {α : Type u_1} (as bs : Array α) (start : Nat) : Array.extract.loop as 0 start bs = bs

theorem Array.extract_loop_succ {α : Type u_1} (as bs : Array α) (size start : Nat) (h : start < as.size) : Array.extract.loop as (size + 1) start bs = Array.extract.loop as size (start + 1) (bs.push as[start])

theorem Array.extract_loop_of_ge {α : Type u_1} (as bs : Array α) (size start : Nat) (h : start ≥ as.size) : Array.extract.loop as size start bs = bs

theorem Array.extract_loop_eq_aux {α : Type u_1} (as bs : Array α) (size start : Nat) : Array.extract.loop as size start bs = bs ++ Array.extract.loop as size start #[]

theorem Array.extract_loop_eq {α : Type u_1} (as bs : Array α) (size start : Nat) (h : start + size ≤ as.size) : Array.extract.loop as size start bs = bs ++ as.extract start (start + size)

theorem Array.size_extract_loop {α : Type u_1} (as bs : Array α) (size start : Nat) : (Array.extract.loop as size start bs).size = bs.size + min size (as.size - start)

@[simp]

theorem Array.size_extract {α : Type u_1} (as : Array α) (start stop : Nat) : (as.extract start stop).size = min stop as.size - start

theorem Array.getElem_extract_loop_lt_aux {α : Type u_1} {i : Nat} (as bs : Array α) (size start : Nat) (hlt : i < bs.size) : i < (Array.extract.loop as size start bs).size

theorem Array.getElem_extract_loop_lt {α : Type u_1} {i : Nat} (as bs : Array α) (size start : Nat) (hlt : i < bs.size) (h : i < (Array.extract.loop as size start bs).size := ⋯) : (Array.extract.loop as size start bs)[i] = bs[i]

theorem Array.getElem_extract_loop_ge_aux {α : Type u_1} {i : Nat} (as bs : Array α) (size start : Nat) (hge : i ≥ bs.size) (h : i < (Array.extract.loop as size start bs).size) : start + i - bs.size < as.size

theorem Array.getElem_extract_loop_ge {α : Type u_1} {i : Nat} (as bs : Array α) (size start : Nat) (hge : i ≥ bs.size) (h : i < (Array.extract.loop as size start bs).size) (h' : start + i - bs.size < as.size := ⋯) : (Array.extract.loop as size start bs)[i] = as[start + i - bs.size]

theorem Array.getElem_extract_aux {α : Type u_1} {i : Nat} {as : Array α} {start stop : Nat} (h : i < (as.extract start stop).size) : start + i < as.size

@[simp]

theorem Array.getElem_extract {α : Type u_1} {i : Nat} {as : Array α} {start stop : Nat} (h : i < (as.extract start stop).size) : (as.extract start stop)[i] = as[start + i]

theorem Array.getElem?_extract {α : Type u_1} {i : Nat} {as : Array α} {start stop : Nat} : (as.extract start stop)[i]? = if i < min stop as.size - start then as[start + i]? else none

@[simp]

theorem Array.toList_extract {α : Type u_1} (as : Array α) (start stop : Nat) : (as.extract start stop).toList = List.take (stop - start) (List.drop start as.toList)

@[simp]

theorem Array.extract_all {α : Type u_1} (as : Array α) : as.extract 0 as.size = as

theorem Array.extract_empty_of_stop_le_start {α : Type u_1} (as : Array α) {start stop : Nat} (h : stop ≤ start) : as.extract start stop = #[]

theorem Array.extract_empty_of_size_le_start {α : Type u_1} (as : Array α) {start stop : Nat} (h : as.size ≤ start) : as.extract start stop = #[]

@[simp]

theorem Array.extract_empty {α : Type u_1} (start stop : Nat) : #[].extract start stop = #[]

contains

theorem Array.contains_def {α : Type u_1} [DecidableEq α] {a : α} {as : Array α} : as.contains a = true ↔ a ∈ as

instance Array.instDecidableMemOfDecidableEq {α : Type u_1} [DecidableEq α] (a : α) (as : Array α) : Decidable (a ∈ as)

Equations

• Array.instDecidableMemOfDecidableEq a as = decidable_of_iff (as.contains a = true) ⋯

swap

@[simp]

theorem Array.getElem_swap_right {α : Type u_1} (a : Array α) {i j : Nat} {hi : i < a.size} {hj : j < a.size} : (a.swap i j hi hj)[j] = a[i]

@[simp]

theorem Array.getElem_swap_left {α : Type u_1} (a : Array α) {i j : Nat} {hi : i < a.size} {hj : j < a.size} : (a.swap i j hi hj)[i] = a[j]

@[simp]

theorem Array.getElem_swap_of_ne {α : Type u_1} {p : Nat} (a : Array α) {i j : Nat} {hi : i < a.size} {hj : j < a.size} (hp : p < a.size) (hi' : p ≠ i) (hj' : p ≠ j) : (a.swap i j hi hj)[p] = a[p]

theorem Array.getElem_swap' {α : Type u_1} (a : Array α) (i j : Nat) {hi : i < a.size} {hj : j < a.size} (k : Nat) (hk : k < a.size) : (a.swap i j hi hj)[k] = if k = i then a[j] else if k = j then a[i] else a[k]

theorem Array.getElem_swap {α : Type u_1} (a : Array α) (i j : Nat) {hi : i < a.size} {hj : j < a.size} (k : Nat) (hk : k < (a.swap i j hi hj).size) : (a.swap i j hi hj)[k] = if k = i then a[j] else if k = j then a[i] else a[k]

@[simp]

theorem Array.swap_swap {α : Type u_1} (a : Array α) {i j : Nat} (hi : i < a.size) (hj : j < a.size) : (a.swap i j hi hj).swap i j ⋯ ⋯ = a

theorem Array.swap_comm {α : Type u_1} (a : Array α) {i j : Nat} {hi : i < a.size} {hj : j < a.size} : a.swap i j hi hj = a.swap j i hj hi

eraseIdx

theorem Array.eraseIdx_eq_eraseIdxIfInBounds {α : Type u_1} {a : Array α} {i : Nat} (h : i < a.size) : a.eraseIdx i h = a.eraseIdxIfInBounds i

isPrefixOf

@[simp]

theorem Array.isPrefixOf_toList {α : Type u_1} [BEq α] {as bs : Array α} : as.toList.isPrefixOf bs.toList = as.isPrefixOf bs

zipWith

@[simp]

theorem Array.toList_zipWith {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β → γ) (as : Array α) (bs : Array β) : (as.zipWith bs f).toList = List.zipWith f as.toList bs.toList

@[simp]

theorem Array.toList_zip {α : Type u_1} {β : Type u_2} (as : Array α) (bs : Array β) : (as.zip bs).toList = as.toList.zip bs.toList

@[simp]

theorem Array.toList_zipWithAll {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : Option α → Option β → γ) (as : Array α) (bs : Array β) : (as.zipWithAll bs f).toList = List.zipWithAll f as.toList bs.toList

@[simp]

theorem Array.size_zipWith {α : Type u_1} {β : Type u_2} {γ : Type u_3} (as : Array α) (bs : Array β) (f : α → β → γ) : (as.zipWith bs f).size = min as.size bs.size

@[simp]

theorem Array.size_zip {α : Type u_1} {β : Type u_2} (as : Array α) (bs : Array β) : (as.zip bs).size = min as.size bs.size

@[simp]

theorem Array.getElem_zipWith {α : Type u_1} {β : Type u_2} {γ : Type u_3} (as : Array α) (bs : Array β) (f : α → β → γ) (i : Nat) (hi : i < (as.zipWith bs f).size) : (as.zipWith bs f)[i] = f as[i] bs[i]

findSomeM?, findM?, findSome?, find?

@[simp]

theorem Array.findSomeM?_toList {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [Monad m] [LawfulMonad m] (p : α → m (Option β)) (as : Array α) : List.findSomeM? p as.toList = Array.findSomeM? p as

@[simp]

theorem Array.findM?_toList {m : Type → Type} {α : Type} [Monad m] [LawfulMonad m] (p : α → m Bool) (as : Array α) : List.findM? p as.toList = Array.findM? p as

@[simp]

theorem Array.findSome?_toList {α : Type u_1} {β : Type u_2} (p : α → Option β) (as : Array α) : List.findSome? p as.toList = Array.findSome? p as

@[simp]

theorem Array.find?_toList {α : Type u_1} (p : α → Bool) (as : Array α) : List.find? p as.toList = Array.find? p as

More theorems about List.toArray, followed by an Array operation.

Our goal is to have simp "pull List.toArray outwards" as much as possible.

theorem List.toListRev_toArray {α : Type u_1} (l : List α) : l.toArray.toListRev = l.reverse

@[simp]

theorem List.take_toArray {α : Type u_1} (l : List α) (n : Nat) : l.toArray.take n = (List.take n l).toArray

@[simp]

theorem List.mapM_toArray {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [Monad m] [LawfulMonad m] (f : α → m β) (l : List α) : Array.mapM f l.toArray = List.toArray <$> List.mapM f l

@[simp]

theorem List.map_toArray {α : Type u_1} {β : Type u_2} (f : α → β) (l : List α) : Array.map f l.toArray = (List.map f l).toArray

theorem List.uset_toArray {α : Type u_1} (l : List α) (i : USize) (a : α) (h : i.toNat < l.toArray.size) : l.toArray.uset i a h = (l.set i.toNat a).toArray

@[simp]

theorem List.swap_toArray {α : Type u_1} (l : List α) (i j : Nat) {hi : i < l.toArray.size} {hj : j < l.toArray.size} : l.toArray.swap i j hi hj = ((l.set i l[j]).set j l[i]).toArray

@[simp]

theorem List.reverse_toArray {α : Type u_1} (l : List α) : l.toArray.reverse = l.reverse.toArray

@[simp]

theorem List.modify_toArray {α : Type u_1} {i : Nat} (f : α → α) (l : List α) : l.toArray.modify i f = (List.modify f i l).toArray

@[simp]

theorem List.filter_toArray' {α : Type u_1} {stop : Nat} (p : α → Bool) (l : List α) (h : stop = l.toArray.size) : Array.filter p l.toArray 0 stop = (List.filter p l).toArray

@[simp]

theorem List.filterMap_toArray' {α : Type u_1} {β : Type u_2} {stop : Nat} (f : α → Option β) (l : List α) (h : stop = l.toArray.size) : Array.filterMap f l.toArray 0 stop = (List.filterMap f l).toArray

theorem List.filter_toArray {α : Type u_1} (p : α → Bool) (l : List α) : Array.filter p l.toArray = (List.filter p l).toArray

theorem List.filterMap_toArray {α : Type u_1} {β : Type u_2} (f : α → Option β) (l : List α) : Array.filterMap f l.toArray = (List.filterMap f l).toArray

@[simp]

theorem List.flatten_toArray {α : Type u_1} (l : List (List α)) : (Array.map List.toArray l.toArray).flatten = l.flatten.toArray

@[simp]

theorem List.toArray_range (n : Nat) : (List.range n).toArray = Array.range n

@[simp]

theorem List.extract_toArray {α : Type u_1} (l : List α) (start stop : Nat) : l.toArray.extract start stop = (List.take (stop - start) (List.drop start l)).toArray

@[simp]

theorem List.toArray_ofFn {n : Nat} {α : Type u_1} (f : Fin n → α) : (List.ofFn f).toArray = Array.ofFn f

@[simp, irreducible]

theorem List.eraseIdx_toArray {α : Type u_1} (l : List α) (i : Nat) (h : i < l.toArray.size) : l.toArray.eraseIdx i h = (l.eraseIdx i).toArray

@[simp]

theorem List.eraseIdxIfInBounds_toArray {α : Type u_1} (l : List α) (i : Nat) : l.toArray.eraseIdxIfInBounds i = (l.eraseIdx i).toArray

@[simp]

theorem Array.mapM_id {α : Type u_1} {β : Type u_2} {l : Array α} {f : α → Id β} : Array.mapM f l = Array.map f l

@[simp]

theorem Array.toList_ofFn {n : Nat} {α : Type u_1} (f : Fin n → α) : (Array.ofFn f).toList = List.ofFn f

@[simp]

theorem Array.toList_takeWhile {α : Type u_1} (p : α → Bool) (as : Array α) : (Array.takeWhile p as).toList = List.takeWhile p as.toList

@[simp]

theorem Array.toList_eraseIdx {α : Type u_1} (as : Array α) (i : Nat) (h : i < as.size) : (as.eraseIdx i h).toList = as.toList.eraseIdx i

@[simp]

theorem Array.toList_eraseIdxIfInBounds {α : Type u_1} (as : Array α) (i : Nat) : (as.eraseIdxIfInBounds i).toList = as.toList.eraseIdx i

map

@[simp]

theorem Array.map_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β} {g : β → γ} {as : Array α} : Array.map g (Array.map f as) = Array.map (g ∘ f) as

@[simp]

theorem Array.map_id_fun {α : Type u_1} : Array.map id = id

@[simp]

theorem Array.map_id_fun' {α : Type u_1} : (Array.map fun (a : α) => a) = id

map_id_fun' differs from map_id_fun by representing the identity function as a lambda, rather than id.

theorem Array.map_id {α : Type u_1} (as : Array α) : Array.map id as = as

theorem Array.map_id' {α : Type u_1} (as : Array α) : Array.map (fun (a : α) => a) as = as

map_id' differs from map_id by representing the identity function as a lambda, rather than id.

theorem Array.map_id'' {α : Type u_1} {f : α → α} (h : ∀ (x : α), f x = x) (as : Array α) : Array.map f as = as

Variant of map_id, with a side condition that the function is pointwise the identity.

theorem Array.array_array_induction {α : Type u_1} (P : Array (Array α) → Prop) (h : ∀ (xss : List (List α)), P (List.map List.toArray xss).toArray) (ass : Array (Array α)) : P ass

theorem Array.foldl_map {β₁ : Type u_1} {β₂ : Type u_2} {α : Type u_3} (f : β₁ → β₂) (g : α → β₂ → α) (l : Array β₁) (init : α) : Array.foldl g init (Array.map f l) = Array.foldl (fun (x : α) (y : β₁) => g x (f y)) init l

theorem Array.foldr_map {α₁ : Type u_1} {α₂ : Type u_2} {β : Type u_3} (f : α₁ → α₂) (g : α₂ → β → β) (l : Array α₁) (init : β) : Array.foldr g init (Array.map f l) = Array.foldr (fun (x : α₁) (y : β) => g (f x) y) init l

theorem Array.foldl_filterMap {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → Option β) (g : γ → β → γ) (l : Array α) (init : γ) : Array.foldl g init (Array.filterMap f l) = Array.foldl (fun (x : γ) (y : α) => match f y with | some b => g x b | none => x) init l

theorem Array.foldr_filterMap {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → Option β) (g : β → γ → γ) (l : Array α) (init : γ) : Array.foldr g init (Array.filterMap f l) = Array.foldr (fun (x : α) (y : γ) => match f x with | some b => g b y | none => y) init l

theorem Array.foldl_map' {α : Type u_1} {β : Type u_2} (g : α → β) (f : α → α → α) (f' : β → β → β) (a : α) (l : Array α) (h : ∀ (x y : α), f' (g x) (g y) = g (f x y)) : Array.foldl f' (g a) (Array.map g l) = g (Array.foldl f a l)

theorem Array.foldr_map' {α : Type u_1} {β : Type u_2} (g : α → β) (f : α → α → α) (f' : β → β → β) (a : α) (l : List α) (h : ∀ (x y : α), f' (g x) (g y) = g (f x y)) : List.foldr f' (g a) (List.map g l) = g (List.foldr f a l)

flatten

@[simp]

theorem Array.flatten_empty {α : Type u_1} : #[].flatten = #[]

@[simp]

theorem Array.flatten_toArray_map_toArray {α : Type u_1} (xss : List (List α)) : (List.map List.toArray xss).toArray.flatten = xss.flatten.toArray

reverse

@[simp]

theorem Array.mem_reverse {α : Type u_1} {x : α} {as : Array α} : x ∈ as.reverse ↔ x ∈ as

@[simp]

theorem Array.getElem_reverse {α : Type u_1} (as : Array α) (i : Nat) (hi : i < as.reverse.size) : as.reverse[i] = as[as.size - 1 - i]

findSomeRevM?, findRevM?, findSomeRev?, findRev?

@[simp]

theorem Array.findSomeRevM?_eq_findSomeM?_reverse {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [Monad m] [LawfulMonad m] (f : α → m (Option β)) (as : Array α) : Array.findSomeRevM? f as = Array.findSomeM? f as.reverse

@[simp]

theorem Array.findRevM?_eq_findM?_reverse {m : Type → Type} {α : Type} [Monad m] [LawfulMonad m] (f : α → m Bool) (as : Array α) : Array.findRevM? f as = Array.findM? f as.reverse

@[simp]

theorem Array.findSomeRev?_eq_findSome?_reverse {α : Type u_1} {β : Type u_2} (f : α → Option β) (as : Array α) : Array.findSomeRev? f as = Array.findSome? f as.reverse

@[simp]

theorem Array.findRev?_eq_find?_reverse {α : Type} (f : α → Bool) (as : Array α) : Array.findRev? f as = Array.find? f as.reverse

unzip

@[simp]

theorem Array.fst_unzip {α : Type u_1} {β : Type u_2} (as : Array (α × β)) : as.unzip.fst = Array.map Prod.fst as

@[simp]

theorem Array.snd_unzip {α : Type u_1} {β : Type u_2} (as : Array (α × β)) : as.unzip.snd = Array.map Prod.snd as

take

@[simp]

theorem Array.take_size {α : Type u_1} (a : Array α) : a.take a.size = a

@[simp]

theorem List.unzip_toArray {α : Type u_1} {β : Type u_2} (as : List (α × β)) : as.toArray.unzip = Prod.map List.toArray List.toArray as.unzip

theorem Array.toList_fst_unzip {α : Type u_1} {β : Type u_2} (as : Array (α × β)) : as.unzip.fst.toList = as.toList.unzip.fst

theorem Array.toList_snd_unzip {α : Type u_1} {β : Type u_2} (as : Array (α × β)) : as.unzip.snd.toList = as.toList.unzip.snd

@[simp]

theorem Array.flatMap_empty {α : Type u_1} {β : Type u_2} (f : α → Array β) : Array.flatMap f #[] = #[]

theorem Array.flatMap_toArray_cons {α : Type u_1} {β : Type u_2} (f : α → Array β) (a : α) (as : List α) : Array.flatMap f (a :: as).toArray = f a ++ Array.flatMap f as.toArray

@[simp]

theorem Array.flatMap_toArray {α : Type u_1} {β : Type u_2} (f : α → Array β) (as : List α) : Array.flatMap f as.toArray = (as.flatMap fun (a : α) => (f a).toList).toArray

Deprecations

@[reducible, inline, deprecated List.back!_toArray (since := "2024-10-31")]

abbrev List.back_toArray {α : Type u_1} [Inhabited α] (l : List α) : l.toArray.back! = l.getLast!

Equations

• @List.back_toArray = @List.back!_toArray

@[reducible, inline, deprecated List.setIfInBounds_toArray (since := "2024-11-24")]

abbrev List.setD_toArray {α : Type u_1} (l : List α) (i : Nat) (a : α) : l.toArray.setIfInBounds i a = (l.set i a).toArray

Equations

• @List.setD_toArray = @List.setIfInBounds_toArray

@[reducible, inline, deprecated Array.foldl_toList_eq_flatMap (since := "2024-10-16")]

abbrev Array.foldl_toList_eq_bind {α : Type u_1} {β : Type u_2} (l : List α) (acc : Array β) (F : Array β → α → Array β) (G : α → List β) (H : ∀ (acc : Array β) (a : α), (F acc a).toList = acc.toList ++ G a) : (List.foldl F acc l).toList = acc.toList ++ l.flatMap G

Equations

• @Array.foldl_toList_eq_bind = @Array.foldl_toList_eq_flatMap

@[reducible, inline, deprecated Array.foldl_toList_eq_flatMap (since := "2024-10-16")]

abbrev Array.foldl_data_eq_bind {α : Type u_1} {β : Type u_2} (l : List α) (acc : Array β) (F : Array β → α → Array β) (G : α → List β) (H : ∀ (acc : Array β) (a : α), (F acc a).toList = acc.toList ++ G a) : (List.foldl F acc l).toList = acc.toList ++ l.flatMap G

Equations

• @Array.foldl_data_eq_bind = @Array.foldl_toList_eq_flatMap

@[reducible, inline, deprecated Array.getElem_mem (since := "2024-10-17")]

abbrev Array.getElem?_mem {α : Type u_1} {l : Array α} {i : Nat} (h : i < l.size) : l[i] ∈ l

Equations

• @Array.getElem?_mem = @Array.getElem_mem

@[reducible, inline, deprecated Array.getElem_fin_eq_getElem_toList (since := "2024-10-17")]

abbrev Array.getElem_fin_eq_toList_get {α : Type u_1} (a : Array α) (i : Fin a.size) : a[i] = a.toList[i]

Equations

• @Array.getElem_fin_eq_toList_get = @Array.getElem_fin_eq_getElem_toList

@[reducible, inline, deprecated "Use reverse direction of `getElem?_toList`" (since := "2024-10-17")]

abbrev Array.getElem?_eq_toList_getElem? {α : Type u_1} {a : Array α} {i : Nat} : a.toList[i]? = a[i]?

Equations

• @Array.getElem?_eq_toList_getElem? = @Array.getElem?_toList

@[reducible, inline, deprecated Array.get?_eq_get?_toList (since := "2024-10-17")]

abbrev Array.get?_eq_toList_get? {α : Type u_1} (a : Array α) (i : Nat) : a.get? i = a.toList.get? i

Equations

• @Array.get?_eq_toList_get? = @Array.get?_eq_get?_toList

@[reducible, inline, deprecated Array.getElem?_swap (since := "2024-10-17")]

abbrev Array.get?_swap {α : Type u_1} (a : Array α) (i j : Nat) (hi : i < a.size) (hj : j < a.size) (k : Nat) : (a.swap i j hi hj)[k]? = if j = k then some a[i] else if i = k then some a[j] else a[k]?

Equations

• @Array.get?_swap = @Array.getElem?_swap

@[reducible, inline, deprecated Array.getElem_push (since := "2024-10-21")]

abbrev Array.get_push {α : Type u_1} (a : Array α) (x : α) (i : Nat) (h : i < (a.push x).size) : (a.push x)[i] = if h : i < a.size then a[i] else x

Equations

• @Array.get_push = @Array.getElem_push

@[reducible, inline, deprecated Array.getElem_push_lt (since := "2024-10-21")]

abbrev Array.get_push_lt {α : Type u_1} (a : Array α) (x : α) (i : Nat) (h : i < a.size) : let_fun this := ⋯; (a.push x)[i] = a[i]

Equations

• @Array.get_push_lt = @Array.getElem_push_lt

@[reducible, inline, deprecated Array.getElem_push_eq (since := "2024-10-21")]

abbrev Array.get_push_eq {α : Type u_1} (a : Array α) (x : α) : (a.push x)[a.size] = x

Equations

• @Array.get_push_eq = @Array.getElem_push_eq

@[reducible, inline, deprecated Array.back!_eq_back? (since := "2024-10-31")]

abbrev Array.back_eq_back? {α : Type u_1} [Inhabited α] (a : Array α) : a.back! = a.back?.getD default

Equations

• @Array.back_eq_back? = @Array.back!_eq_back?

@[reducible, inline, deprecated Array.back!_push (since := "2024-10-31")]

abbrev Array.back_push {α : Type u_1} {x : α} [Inhabited α] (a : Array α) : (a.push x).back! = x

Equations

• @Array.back_push = @Array.back!_push

@[reducible, inline, deprecated Array.eq_push_pop_back!_of_size_ne_zero (since := "2024-10-31")]

abbrev Array.eq_push_pop_back_of_size_ne_zero {α : Type u_1} [Inhabited α] {as : Array α} (h : as.size ≠ 0) : as = as.pop.push as.back!

Equations

• @Array.eq_push_pop_back_of_size_ne_zero = @Array.eq_push_pop_back!_of_size_ne_zero

@[reducible, inline, deprecated Array.set!_is_setIfInBounds (since := "2024-11-24")]

abbrev Array.set_is_setIfInBounds : @Array.set! = @Array.setIfInBounds

Equations

• Array.set_is_setIfInBounds = Array.set!_eq_setIfInBounds

@[reducible, inline, deprecated Array.size_setIfInBounds (since := "2024-11-24")]

abbrev Array.size_setD {α : Type u_1} (as : Array α) (index : Nat) (val : α) : (as.setIfInBounds index val).size = as.size

Equations

• @Array.size_setD = @Array.size_setIfInBounds

@[reducible, inline, deprecated Array.getElem_setIfInBounds_eq (since := "2024-11-24")]

abbrev Array.getElem_setD_eq {α : Type u_1} (as : Array α) {i : Nat} (v : α) (h : i < (as.setIfInBounds i v).size) : (as.setIfInBounds i v)[i] = v

Equations

• @Array.getElem_setD_eq = @Array.getElem_setIfInBounds_self

@[reducible, inline, deprecated Array.getElem?_setIfInBounds_eq (since := "2024-11-24")]

abbrev Array.get?_setD_eq {α : Type u_1} (as : Array α) {i : Nat} (v : α) : (as.setIfInBounds i v)[i]? = if i < as.size then some v else none

Equations

• @Array.get?_setD_eq = @Array.getElem?_setIfInBounds_self

@[reducible, inline, deprecated Array.getD_get?_setIfInBounds (since := "2024-11-24")]

abbrev Array.getD_setD {α : Type u_1} (a : Array α) (i : Nat) (v d : α) : (a.setIfInBounds i v)[i]?.getD d = if i < a.size then v else d

Equations

• @Array.getD_setD = @Array.getD_get?_setIfInBounds

@[reducible, inline, deprecated Array.getElem_setIfInBounds (since := "2024-11-24")]

abbrev Array.getElem_setD {α : Type u_1} (as : Array α) (i : Nat) (v : α) (j : Nat) (hj : j < (as.setIfInBounds i v).size) : (as.setIfInBounds i v)[j] = if i = j then v else as[j]

Equations

• @Array.getElem_setD = @Array.getElem_setIfInBounds

@[deprecated List.getElem_toArray (since := "2024-11-29")]

theorem Array.getElem_mk {α : Type u_1} {xs : List α} {i : Nat} (h : i < xs.length) : { toList := xs }[i] = xs[i]

@[deprecated Array.getElem_toList (since := "2024-12-08")]

theorem Array.getElem_eq_getElem_toList {α : Type u_1} {i : Nat} {a : Array α} (h : i < a.size) : a[i] = a.toList[i]

@[deprecated Array.getElem?_toList (since := "2024-12-08")]

theorem Array.getElem?_eq_getElem?_toList {α : Type u_1} (a : Array α) (i : Nat) : a[i]? = a.toList[i]?

@[deprecated LawfulGetElem.getElem?_def (since := "2024-12-08")]

theorem Array.getElem?_eq {α : Type u_1} {a : Array α} {i : Nat} : a[i]? = if h : i < a.size then some a[i] else none