Bootstrapping theorems about arrays

This file contains some theorems about Array and List needed for Init.Data.List.Impl.

@[simp]

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

@[simp]

theorem Array.toArray_data {α : Type u_1} (a : Array α) : List.toArray a.data = a

@[simp]

theorem Array.data_length {α : Type u_1} {l : Array α} : l.data.length = l.size

@[simp]

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

@[simp]

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

@[simp]

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

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

@[deprecated Array.getElem_eq_data_getElem]

theorem Array.getElem_eq_data_get {α : Type u_1} {i : Nat} (a : Array α) (h : i < a.size) : a[i] = a.data.get ⟨i, h⟩

@[irreducible]

theorem Array.foldlM_eq_foldlM_data.aux {m : Type u_1 → Type u_2} {β : Type u_1} {α : Type u_3} [Monad m] (f : β → α → m β) (arr : Array α) (i : Nat) (j : Nat) (H : arr.size ≤ i + j) (b : β) : Array.foldlM.loop f arr arr.size ⋯ i j b = List.foldlM f b (List.drop j arr.data)

theorem Array.foldlM_eq_foldlM_data {m : Type u_1 → Type u_2} {β : Type u_1} {α : Type u_3} [Monad m] (f : β → α → m β) (init : β) (arr : Array α) : Array.foldlM f init arr = List.foldlM f init arr.data

theorem Array.foldl_eq_foldl_data {β : Type u_1} {α : Type u_2} (f : β → α → β) (init : β) (arr : Array α) : Array.foldl f init arr = List.foldl f init arr.data

theorem Array.foldrM_eq_reverse_foldlM_data.aux {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [Monad m] (f : α → β → m β) (arr : Array α) (init : β) (i : Nat) (h : i ≤ arr.size) : List.foldlM (fun (x : β) (y : α) => f y x) init (List.take i arr.data).reverse = Array.foldrM.fold f arr 0 i h init

theorem Array.foldrM_eq_reverse_foldlM_data {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) : Array.foldrM f init arr = List.foldlM (fun (x : β) (y : α) => f y x) init arr.data.reverse

theorem Array.foldrM_eq_foldrM_data {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) : Array.foldrM f init arr = List.foldrM f init arr.data

theorem Array.foldr_eq_foldr_data {α : Type u_1} {β : Type u_2} (f : α → β → β) (init : β) (arr : Array α) : Array.foldr f init arr = List.foldr f init arr.data

@[simp]

theorem Array.push_data {α : Type u_1} (arr : Array α) (a : α) : (arr.push a).data = arr.data ++ [a]

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) : Array.foldrM f init (arr.push a) (arr.size + 1) = do let init ← f a init Array.foldrM f init arr

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) : Array.foldr f init (arr.push a) (arr.size + 1) = Array.foldr f (f a init) arr

@[simp]

theorem Array.toListAppend_eq {α : Type u_1} (arr : Array α) (l : List α) : arr.toListAppend l = arr.data ++ l

@[simp]

theorem Array.toList_eq {α : Type u_1} (arr : Array α) : arr.toList = arr.data

@[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.data.reverse

theorem 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]

@[simp]

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

theorem 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

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.data)

@[simp]

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

@[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.pop_data {α : Type u_1} (arr : Array α) : arr.pop.data = arr.data.dropLast

@[simp]

theorem Array.append_eq_append {α : Type u_1} (arr : Array α) (arr' : Array α) : arr.append arr' = arr ++ arr'

@[simp]

theorem Array.append_data {α : Type u_1} (arr : Array α) (arr' : Array α) : (arr ++ arr').data = arr.data ++ arr'.data

@[simp]

theorem Array.appendList_eq_append {α : Type u_1} (arr : Array α) (l : List α) : arr.appendList l = arr ++ l

@[simp]

theorem Array.appendList_data {α : Type u_1} (arr : Array α) (l : List α) : (arr ++ l).data = arr.data ++ l

@[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_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).data = acc.data ++ G a) : (List.foldl F acc l).data = acc.data ++ l.bind G

theorem Array.foldl_data_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).data = acc.data ++ List.map G l

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

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 : Nat) (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 : Nat) (stop : Nat) (h : min stop as.size ≤ start) : Array.anyM p as start stop = pure false

theorem Array.mem_def {α : Type u_1} (a : α) (as : Array α) : a ∈ as ↔ a ∈ as.data

get

@[simp]

theorem Array.get_eq_getElem {α : Type u_1} (a : Array α) (i : Fin a.size) : a.get i = a[↑i]

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

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]?

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

theorem 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

@[simp]

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

@[simp]

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

set

@[simp]

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

@[simp]

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

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

@[simp]

theorem Array.getElem?_set_eq {α : Type u_1} (a : Array α) (i : Fin a.size) (v : α) : (a.set i v)[↑i]? = some v

@[simp]

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

setD

@[simp]

theorem Array.set!_is_setD : @Array.set! = @Array.setD

@[simp]

theorem Array.size_setD {α : Type u_1} (a : Array α) (index : Nat) (val : α) : (a.setD index val).size = a.size

@[simp]

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

@[simp]

theorem Array.getElem?_setD_eq {α : Type u_1} (a : Array α) {i : Nat} (p : i < a.size) (v : α) : (a.setD i v)[i]? = some v

@[simp]

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

Simplifies a normal form from get!

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, ⋯⟩

@[simp]

theorem Array.mkArray_data {α : Type u_1} (n : Nat) (v : α) : (mkArray n v).data = List.replicate n v

mkArray

@[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.mem_data {α : Type u_1} {a : α} {l : Array α} : a ∈ l.data ↔ a ∈ l

mem

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

theorem Array.getElem?_mem {α : Type u_1} {l : Array α} {i : Fin l.size} : l[i] ∈ l

get lemmas

theorem Array.getElem_fin_eq_data_get {α : Type u_1} (a : Array α) (i : Fin a.data.length) : a[i] = a.data.get 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?_eq_getElem {α : Type u_1} (a : Array α) (i : Nat) (h : i < a.size) : a[i]? = some a[i]

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

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

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

theorem Array.get?_eq_data_get? {α : Type u_1} (a : Array α) (i : Nat) : a.get? i = a.data.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

@[simp]

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

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

theorem 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]

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

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

@[simp]

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

@[simp]

theorem Array.data_set {α : Type u_1} (a : Array α) (i : Fin a.size) (v : α) : (a.set i v).data = a.data.set (↑i) v

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

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

@[simp]

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

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

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

@[simp]

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

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

theorem Array.set_set {α : Type u_1} (a : Array α) (i : Fin a.size) (v : α) (v' : α) : (a.set i v).set ⟨↑i, ⋯⟩ v' = a.set i v'

theorem Array.swap_def {α : Type u_1} (a : Array α) (i : Fin a.size) (j : Fin a.size) : a.swap i j = (a.set i (a.get j)).set ⟨↑j, ⋯⟩ (a.get i)

theorem Array.data_swap {α : Type u_1} (a : Array α) (i : Fin a.size) (j : Fin a.size) : (a.swap i j).data = (a.data.set (↑i) (a.get j)).set (↑j) (a.get i)

theorem Array.get?_swap {α : Type u_1} (a : Array α) (i : Fin a.size) (j : Fin a.size) (k : Nat) : (a.swap i j)[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 : Fin a.size) (v : α) : a.swapAt i v = (a[↑i], a.set i v)

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, h⟩ v)

@[simp]

theorem Array.data_pop {α : Type u_1} (a : Array α) : a.pop.data = a.data.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_empty_of_size_eq_zero {α : Type u_1} {as : Array α} (h : as.size = 0) : as = #[]

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_data {α : Type u_1} (as : Array α) : as.size = as.data.length

@[simp]

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

@[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.reverse_data {α : Type u_1} (a : Array α) : a.reverse.data = a.data.reverse

@[irreducible]

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

foldl / foldr

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 : Nat} {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)

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.mapM_eq_mapM_data {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 = do let __do_lift ← List.mapM f arr.data pure { data := __do_lift }

@[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]

mapIdx

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

theorem Array.mapIdx_induction.go {α : Type u_1} {β : Type u_2} (as : Array α) (f : Fin as.size → α → β) (motive : Nat → Prop) (p : Fin as.size → β → Prop) (hs : ∀ (i : Fin as.size), motive ↑i → p i (f i as[i]) ∧ motive (↑i + 1)) {bs : Array β} {i : Nat} {j : Nat} {h : i + j = as.size} (h₁ : j = bs.size) (h₂ : ∀ (i : Nat) (h : i < as.size) (h' : i < bs.size), p ⟨i, h⟩ bs[i]) (hm : motive j) : let arr := Array.mapIdxM.map as f i j h bs; motive as.size ∧ ∃ (eq : arr.size = as.size), ∀ (i_1 : Nat) (h : i_1 < as.size), p ⟨i_1, h⟩ arr[i_1]

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

@[simp]

theorem Array.size_mapIdx {α : Type u_1} {β : Type u_2} (a : Array α) (f : Fin a.size → α → β) : (a.mapIdx f).size = a.size

@[simp]

theorem Array.size_zipWithIndex {α : Type u_1} (as : Array α) : as.zipWithIndex.size = as.size

@[simp]

theorem Array.getElem_mapIdx {α : Type u_1} {β : Type u_2} (a : Array α) (f : Fin a.size → α → β) (i : Nat) (h : i < (a.mapIdx f).size) : (a.mapIdx f)[i] = f ⟨i, ⋯⟩ a[i]

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.get_modify {α : Type u_1} {f : α → α} {arr : Array α} {x : Nat} {i : Nat} (h : i < arr.size) : (arr.modify x f).get ⟨i, ⋯⟩ = if x = i then f (arr.get ⟨i, h⟩) else arr.get ⟨i, h⟩

filter

@[simp]

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

@[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) => decide (p a = true ∧ q a = true)) l

@[simp]

theorem Array.mem_filter : ∀ {α : Type u_1} {x : α} {p : α → Bool} {as : Array 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

filterMap

@[simp]

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

@[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

empty

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

theorem Array.empty_data {α : Type u_1} : #[].data = []

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 : Array α} {t : Array α} : a ∈ s ++ t ↔ a ∈ s ∨ a ∈ t

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

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

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

@[simp]

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

@[simp]

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

theorem Array.append_assoc {α : Type u_1} (as : Array α) (bs : Array α) (cs : Array α) : as ++ bs ++ cs = as ++ (bs ++ cs)

extract

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

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

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

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

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

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

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

@[simp]

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

@[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 : Nat} {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 : Nat} {stop : Nat} (h : as.size ≤ start) : as.extract start stop = #[]

@[simp]

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

any

@[irreducible]

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

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

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

all

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

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

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

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.get_swap_right {α : Type u_1} (a : Array α) {i : Fin a.size} {j : Fin a.size} : (a.swap i j)[↑j] = a[i]

@[simp]

theorem Array.get_swap_left {α : Type u_1} (a : Array α) {i : Fin a.size} {j : Fin a.size} : (a.swap i j)[↑i] = a[j]

@[simp]

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

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

theorem Array.get_swap' {α : Type u_1} (a : Array α) (i : Fin a.size) (j : Fin a.size) (k : Nat) (hk' : k < (a.swap i j).size) : (a.swap i j)[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 : Fin a.size} {j : Fin a.size} : (a.swap i j).swap ⟨↑i, ⋯⟩ ⟨↑j, ⋯⟩ = a

theorem Array.swap_comm {α : Type u_1} (a : Array α) {i : Fin a.size} {j : Fin a.size} : a.swap i j = a.swap j i