Loogle!

Result

Found 276 declarations mentioning Array.map. Of these, only the first 200 are shown.

• Array.map 📋 Init.Data.Array.Basic
  {α : Type u} {β : Type v} (f : α → β) (as : Array α) : Array β
• Array.map_id 📋 Init.Data.Array.Lemmas
  {α : Type u_1} (xs : Array α) : Array.map id xs = xs
• Array.map_id' 📋 Init.Data.Array.Lemmas
  {α : Type u_1} (xs : Array α) : Array.map (fun a => a) xs = xs
• Array.map_id_fun 📋 Init.Data.Array.Lemmas
  {α : Type u_1} : Array.map id = id
• Array.map_id_fun' 📋 Init.Data.Array.Lemmas
  {α : Type u_1} : (Array.map fun a => a) = id
• Array.size_map 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {xs : Array α} : (Array.map f xs).size = xs.size
• Array.map_empty 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} : Array.map f #[] = #[]
• Array.map_const' 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {xs : Array α} {b : β} : Array.map (fun x => b) xs = Array.replicate xs.size b
• Array.map_id'' 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {f : α → α} (h : ∀ (x : α), f x = x) (xs : Array α) : Array.map f xs = xs
• Array.map_const 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {xs : Array α} {b : β} : Array.map (Function.const α b) xs = Array.replicate xs.size b
• Array.map_replicate 📋 Init.Data.Array.Lemmas
  {n : ℕ} {α✝ : Type u_1} {a : α✝} {α✝¹ : Type u_2} {f : α✝ → α✝¹} : Array.map f (Array.replicate n a) = Array.replicate n (f a)
• Array.back?_map 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {xs : Array α} : (Array.map f xs).back? = Option.map f xs.back?
• Array.flatMap_def 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {xs : Array α} {f : α → Array β} : Array.flatMap f xs = (Array.map f xs).flatten
• Array.map_const_fun 📋 Init.Data.Array.Lemmas
  {β : Type u_1} {α : Type u_2} {x : β} : Array.map (Function.const α x) = fun x_1 => Array.replicate x_1.size x
• Array.map_pop 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {xs : Array α} : Array.map f xs.pop = (Array.map f xs).pop
• Array.map_reverse 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {xs : Array α} : Array.map f xs.reverse = (Array.map f xs).reverse
• Array.toList_map 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {xs : Array α} : (Array.map f xs).toList = List.map f xs.toList
• List.map_toArray 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {l : List α} : Array.map f l.toArray = (List.map f l).toArray
• Array.flatten_flatten 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {xss : Array (Array (Array α))} : xss.flatten.flatten = (Array.map Array.flatten xss).flatten
• Array.flatten_map_toArray 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {L : List (List α)} : (Array.map List.toArray L.toArray).flatten = L.flatten.toArray
• Array.flatten_map_toArray_toArray 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {L : List (List α)} : (Array.map List.toArray L.toArray).flatten = L.flatten.toArray
• Array.flatten_reverse 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {xss : Array (Array α)} : xss.reverse.flatten = (Array.map Array.reverse xss).flatten.reverse
• Array.map_inj 📋 Init.Data.Array.Lemmas
  {α✝ : Type u_1} {α✝¹ : Type u_2} {f g : α✝ → α✝¹} : Array.map f = Array.map g ↔ f = g
• Array.reverse_flatten 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {xss : Array (Array α)} : xss.flatten.reverse = (Array.map Array.reverse xss).reverse.flatten
• List.flatten_toArray 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {L : List (List α)} : (Array.map List.toArray L.toArray).flatten = L.flatten.toArray
• Array.size_flatten 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {xss : Array (Array α)} : xss.flatten.size = (Array.map Array.size xss).sum
• Array.eq_empty_of_map_eq_empty 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {xs : Array α} (h : Array.map f xs = #[]) : xs = #[]
• Array.fst_unzip 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {xs : Array (α × β)} : xs.unzip.1 = Array.map Prod.fst xs
• Array.map_eq_empty_iff 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {xs : Array α} : Array.map f xs = #[] ↔ xs = #[]
• Array.map_eq_flatMap 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {xs : Array α} : Array.map f xs = Array.flatMap (fun x => #[f x]) xs
• Array.map_push 📋 Init.Data.Array.Lemmas
  {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)
• Array.map_singleton 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {a : α} : Array.map f #[a] = #[f a]
• Array.snd_unzip 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {xs : Array (α × β)} : xs.unzip.2 = Array.map Prod.snd xs
• Array.mem_map_of_mem 📋 Init.Data.Array.Lemmas
  {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
• Array.map_flatten 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {xss : Array (Array α)} : Array.map f xss.flatten = (Array.map (Array.map f) xss).flatten
• Array.flatMap_map 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β} {g : β → Array γ} {xs : Array α} : Array.flatMap g (Array.map f xs) = Array.flatMap (fun a => g (f a)) xs
• Array.map_setIfInBounds 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {xs : Array α} {i : ℕ} {a : α} : Array.map f (xs.setIfInBounds i a) = (Array.map f xs).setIfInBounds i (f a)
• Array.map_map 📋 Init.Data.Array.Lemmas
  {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
• Array.size_flatMap 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {xs : Array α} {f : α → Array β} : (Array.flatMap f xs).size = (Array.map (fun a => (f a).size) xs).sum
• Array.map_flatMap 📋 Init.Data.Array.Lemmas
  {β : Type u_1} {γ : Type u_2} {α : Type u_3} {f : β → γ} {g : α → Array β} {xs : Array α} : Array.map f (Array.flatMap g xs) = Array.flatMap (fun a => Array.map f (g a)) xs
• Array.exists_of_mem_map 📋 Init.Data.Array.Lemmas
  {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 ∈ l, f a = b
• Array.filterMap_eq_map' 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {stop : ℕ} {as : Array α} {f : α → β} (w : stop = as.size) : Array.filterMap (fun x => some (f x)) as 0 stop = Array.map f as
• Array.forall_mem_map 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {xs : Array α} {P : β → Prop} : (∀ i ∈ Array.map f xs, P i) ↔ ∀ j ∈ xs, P (f j)
• Array.map_congr_left 📋 Init.Data.Array.Lemmas
  {α✝ : Type u_1} {xs : Array α✝} {α✝¹ : Type u_2} {f g : α✝ → α✝¹} (h : ∀ a ∈ xs, f a = g a) : Array.map f xs = Array.map g xs
• Array.map_eq_replicate_iff 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {xs : Array α} {f : α → β} {b : β} : Array.map f xs = Array.replicate xs.size b ↔ ∀ x ∈ xs, f x = b
• Array.map_eq_map_iff 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f g : α → β} {xs : Array α} : Array.map f xs = Array.map g xs ↔ ∀ a ∈ xs, f a = g a
• Array.map_inj_left 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {xs : Array α} {f g : α → β} : Array.map f xs = Array.map g xs ↔ ∀ a ∈ xs, f a = g a
• Array.mem_map 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {b : β} {f : α → β} {xs : Array α} : b ∈ Array.map f xs ↔ ∃ a ∈ xs, f a = b
• Array.contains_map 📋 Init.Data.Array.Lemmas
  {β : Type u_1} {α : Type u_2} [BEq β] {xs : Array α} {x : β} {f : α → β} : (Array.map f xs).contains x = xs.any fun a => x == f a
• Array.filterMap_eq_map 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {stop : ℕ} {as : Array α} {f : α → β} (w : stop = as.size) : Array.filterMap (some ∘ f) as 0 stop = Array.map f as
• Array.map_eq_foldl 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {xs : Array α} : Array.map f xs = Array.foldl (fun bs a => bs.push (f a)) #[] xs
• Array.map_inj_right 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {xs ys : Array α} {f : α → β} (w : ∀ (x y : α), f x = f y → x = y) : Array.map f xs = Array.map f ys ↔ xs = ys
• Array.eq_iff_flatten_eq 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {xss₁ xss₂ : Array (Array α)} : xss₁ = xss₂ ↔ xss₁.flatten = xss₂.flatten ∧ Array.map Array.size xss₁ = Array.map Array.size xss₂
• Array.map_eq_singleton_iff 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {xs : Array α} {b : β} : Array.map f xs = #[b] ↔ ∃ a, xs = #[a] ∧ f a = b
• Array.map_filterMap_of_inv 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → Option β} {g : β → α} (H : ∀ (x : α), Option.map g (f x) = some x) {xs : Array α} : Array.map g (Array.filterMap f xs) = xs
• Array.map_set 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {xs : Array α} {i : ℕ} {h : i < xs.size} {a : α} : Array.map f (xs.set i a h) = (Array.map f xs).set i (f a) ⋯
• Array.all_map 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {xs : Array α} {p : β → Bool} : (Array.map f xs).all p = xs.all (p ∘ f)
• Array.any_map 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {xs : Array α} {p : β → Bool} : (Array.map f xs).any p = xs.any (p ∘ f)
• Array.all_map' 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {stop : ℕ} {f : α → β} {xs : Array α} {p : β → Bool} (w : stop = xs.size) : (Array.map f xs).all p 0 stop = xs.all (p ∘ f)
• Array.any_map' 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {stop : ℕ} {f : α → β} {xs : Array α} {p : β → Bool} (w : stop = xs.size) : (Array.map f xs).any p 0 stop = xs.any (p ∘ f)
• Array.map_filterMap 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → Option β} {g : β → γ} {xs : Array α} : Array.map g (Array.filterMap f xs) = Array.filterMap (fun x => Option.map g (f x)) xs
• Array.map_append 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {xs ys : Array α} : Array.map f (xs ++ ys) = Array.map f xs ++ Array.map f ys
• Array.filter_flatten 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {p : α → Bool} {xss : Array (Array α)} {stop : ℕ} (w : stop = xss.flatten.size) : Array.filter p xss.flatten 0 stop = (Array.map (fun as => Array.filter p as) xss).flatten
• Array.filter_map 📋 Init.Data.Array.Lemmas
  {β : Type u_1} {α : Type u_2} {p : α → Bool} {f : β → α} {xs : Array β} : Array.filter p (Array.map f xs) = Array.map f (Array.filter (p ∘ f) xs)
• Array.filterMap_map 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β} {g : β → Option γ} {xs : Array α} : Array.filterMap g (Array.map f xs) = Array.filterMap (g ∘ f) xs
• Array.foldl_map 📋 Init.Data.Array.Lemmas
  {β₁ : Type u_1} {β₂ : Type u_2} {α : Type u_3} {f : β₁ → β₂} {g : α → β₂ → α} {xs : Array β₁} {init : α} : Array.foldl g init (Array.map f xs) = Array.foldl (fun x y => g x (f y)) init xs
• Array.foldr_map 📋 Init.Data.Array.Lemmas
  {α₁ : Type u_1} {α₂ : Type u_2} {β : Type u_3} {f : α₁ → α₂} {g : α₂ → β → β} {xs : Array α₁} {init : β} : Array.foldr g init (Array.map f xs) = Array.foldr (fun x y => g (f x) y) init xs
• Array.map_eq_push_iff 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {xs : Array α} {ys : Array β} {b : β} : Array.map f xs = ys.push b ↔ ∃ zs a, xs = zs.push a ∧ Array.map f zs = ys ∧ f a = b
• Array.foldl_map' 📋 Init.Data.Array.Lemmas
  {β₁ : Type u_1} {β₂ : Type u_2} {α : Type u_3} {f : β₁ → β₂} {g : α → β₂ → α} {xs : Array β₁} {init : α} {stop : ℕ} (w : stop = xs.size) : Array.foldl g init (Array.map f xs) 0 stop = Array.foldl (fun x y => g x (f y)) init xs
• Array.foldr_map' 📋 Init.Data.Array.Lemmas
  {α₁ : Type u_1} {α₂ : Type u_2} {β : Type u_3} {f : α₁ → α₂} {g : α₂ → β → β} {xs : Array α₁} {init : β} {start : ℕ} (w : start = xs.size) : Array.foldr g init (Array.map f xs) start = Array.foldr (fun x y => g (f x) y) init xs
• Array.filterMap_flatten 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → Option β} {xss : Array (Array α)} {stop : ℕ} (w : stop = xss.flatten.size) : Array.filterMap f xss.flatten 0 stop = (Array.map (fun as => Array.filterMap f as) xss).flatten
• Array.foldl_push_eq_append 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {stop : ℕ} {as : Array α} {bs : Array β} {f : α → β} (w : stop = as.size) : Array.foldl (fun acc a => acc.push (f a)) bs as 0 stop = bs ++ Array.map f as
• Array.getElem?_map 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {xs : Array α} {i : ℕ} : (Array.map f xs)[i]? = Option.map f xs[i]?
• Array.foldr_cons_eq_append 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {start : ℕ} {as : Array α} {bs : List β} {f : α → β} (w : start = as.size) : Array.foldr (fun a acc => f a :: acc) bs as start = (Array.map f as).toList ++ bs
• Array.foldr_push_eq_append 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {start : ℕ} {as : Array α} {bs : Array β} {f : α → β} (w : start = as.size) : Array.foldr (fun a xs => xs.push (f a)) bs as start = bs ++ (Array.map f as).reverse
• Array.map_filterMap_some_eq_filterMap_isSome 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → Option β} {xs : Array α} : Array.map some (Array.filterMap f xs) = Array.filter (fun b => b.isSome) (Array.map f xs)
• Array.map_filter_eq_foldl 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {p : α → Bool} {xs : Array α} : Array.map f (Array.filter p xs) = Array.foldl (fun acc x => bif p x then acc.push (f x) else acc) #[] xs
• Array.foldl_cons_eq_append 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {stop : ℕ} {as : Array α} {bs : List β} {f : α → β} (w : stop = as.size) : Array.foldl (fun acc a => f a :: acc) bs as 0 stop = (Array.map f as).reverse.toList ++ bs
• Array.map_eq_iff 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {f : α → β} {xs : Array α} {ys : Array β} : Array.map f xs = ys ↔ ∀ (i : ℕ), ys[i]? = Option.map f xs[i]?
• Array.foldl_append_eq_append 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {xs : Array α} {f : α → Array β} {ys : Array β} : Array.foldl (fun x1 x2 => x1 ++ f x2) ys xs = ys ++ (Array.map f xs).flatten
• Array.foldr_append_eq_append 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {xs : Array α} {f : α → Array β} {ys : Array β} : Array.foldr (fun x1 x2 => f x1 ++ x2) ys xs = (Array.map f xs).flatten ++ ys
• Array.foldl_map_hom 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {g : α → β} {f : α → α → α} {f' : β → β → β} {a : α} {xs : Array α} (h : ∀ (x y : α), f' (g x) (g y) = g (f x y)) : Array.foldl f' (g a) (Array.map g xs) = g (Array.foldl f a xs)
• Array.foldr_map_hom 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {g : α → β} {f : α → α → α} {f' : β → β → β} {a : α} {xs : Array α} (h : ∀ (x y : α), f' (g x) (g y) = g (f x y)) : Array.foldr f' (g a) (Array.map g xs) = g (Array.foldr f a xs)
• Array.foldl_map_hom' 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {g : α → β} {f : α → α → α} {f' : β → β → β} {a : α} {xs : Array α} {stop : ℕ} (h : ∀ (x y : α), f' (g x) (g y) = g (f x y)) (w : stop = xs.size) : Array.foldl f' (g a) (Array.map g xs) 0 stop = g (Array.foldl f a xs)
• Array.foldr_map_hom' 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {g : α → β} {f : α → α → α} {f' : β → β → β} {a : α} {xs : Array α} {start : ℕ} (h : ∀ (x y : α), f' (g x) (g y) = g (f x y)) (w : start = xs.size) : Array.foldr f' (g a) (Array.map g xs) start = g (Array.foldr f a xs)
• Array.foldl_flip_append_eq_append 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {xs : Array α} {f : α → Array β} {ys : Array β} : Array.foldl (fun acc y => f y ++ acc) ys xs = (Array.map f xs).reverse.flatten ++ ys
• Array.foldr_flip_append_eq_append 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {xs : Array α} {f : α → Array β} {ys : Array β} : Array.foldr (fun x acc => acc ++ f x) ys xs = ys ++ (Array.map f xs).reverse.flatten
• Array.getElem_map 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} (f : α → β) {xs : Array α} {i : ℕ} (hi : i < (Array.map f xs).size) : (Array.map f xs)[i] = f xs[i]
• Array.append_eq_map_iff 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {xs ys : Array β} {zs : Array α} {f : α → β} : xs ++ ys = Array.map f zs ↔ ∃ as bs, zs = as ++ bs ∧ Array.map f as = xs ∧ Array.map f bs = ys
• Array.map_eq_append_iff 📋 Init.Data.Array.Lemmas
  {α : Type u_1} {β : Type u_2} {xs : Array α} {ys zs : Array β} {f : α → β} : Array.map f xs = ys ++ zs ↔ ∃ as bs, xs = as ++ bs ∧ Array.map f as = ys ∧ Array.map f bs = zs
• Array.map_extract 📋 Init.Data.Array.Extract
  {α : Type u_1} {α✝ : Type u_2} {f : α → α✝} {as : Array α} {i j : ℕ} : Array.map f (as.extract i j) = (Array.map f as).extract i j
• Array.popWhile_map 📋 Init.Data.Array.Extract
  {α : Type u_1} {β : Type u_2} {f : α → β} {p : β → Bool} {as : Array α} : Array.popWhile p (Array.map f as) = Array.map f (Array.popWhile (p ∘ f) as)
• Array.takeWhile_map 📋 Init.Data.Array.Extract
  {α : Type u_1} {β : Type u_2} {f : α → β} {p : β → Bool} {as : Array α} : Array.takeWhile p (Array.map f as) = Array.map f (Array.takeWhile (p ∘ f) as)
• Array.countP_map 📋 Init.Data.Array.Count
  {α : Type u_2} {β : Type u_1} {p : β → Bool} {f : α → β} {xs : Array α} : Array.countP p (Array.map f xs) = Array.countP (p ∘ f) xs
• Array.countP_flatten 📋 Init.Data.Array.Count
  {α : Type u_1} {p : α → Bool} {xss : Array (Array α)} : Array.countP p xss.flatten = (Array.map (Array.countP p) xss).sum
• Array.count_flatten 📋 Init.Data.Array.Count
  {α : Type u_1} [BEq α] {a : α} {xss : Array (Array α)} : Array.count a xss.flatten = (Array.map (Array.count a) xss).sum
• Array.count_le_count_map 📋 Init.Data.Array.Count
  {α : Type u_2} [BEq α] [LawfulBEq α] {β : Type u_1} [BEq β] [LawfulBEq β] {xs : Array α} {f : α → β} {x : α} : Array.count x xs ≤ Array.count (f x) (Array.map f xs)
• Array.countP_flatMap 📋 Init.Data.Array.Count
  {α : Type u_2} {β : Type u_1} {p : β → Bool} {xs : Array α} {f : α → Array β} : Array.countP p (Array.flatMap f xs) = (Array.map (Array.countP p ∘ f) xs).sum
• Array.count_flatMap 📋 Init.Data.Array.Count
  {β : Type u_1} {α : Type u_2} [BEq β] {xs : Array α} {f : α → Array β} {x : β} : Array.count x (Array.flatMap f xs) = (Array.map (Array.count x ∘ f) xs).sum
• Array.map_wfParam 📋 Init.Data.Array.Attach
  {α : Type u_1} {β : Type u_2} {xs : Array α} {f : α → β} : Array.map f (wfParam xs) = Array.map f xs.attach.unattach
• Array.attach_map_subtype_val 📋 Init.Data.Array.Attach
  {α : Type u_1} (xs : Array α) : Array.map Subtype.val xs.attach = xs
• Array.attachWith_map_subtype_val 📋 Init.Data.Array.Attach
  {α : Type u_1} {p : α → Prop} {xs : Array α} (H : ∀ a ∈ xs, p a) : Array.map Subtype.val (xs.attachWith p H) = xs
• Array.pmap_eq_map 📋 Init.Data.Array.Attach
  {α : Type u_1} {β : Type u_2} {p : α → Prop} {f : α → β} {xs : Array α} (H : ∀ a ∈ xs, p a) : Array.pmap (fun a x => f a) xs H = Array.map f xs
• Array.unattach_flatten 📋 Init.Data.Array.Attach
  {α : Type u_1} {p : α → Prop} {xs : Array (Array { x // p x })} : xs.flatten.unattach = (Array.map Array.unattach xs).flatten
• Array.attachWith_map_val 📋 Init.Data.Array.Attach
  {α : Type u_1} {β : Type u_2} {p : α → Prop} {f : α → β} {xs : Array α} (H : ∀ a ∈ xs, p a) : Array.map (fun i => f ↑i) (xs.attachWith p H) = Array.map f xs
• Array.map_pmap 📋 Init.Data.Array.Attach
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {p : α → Prop} {g : β → γ} {f : (a : α) → p a → β} {xs : Array α} (H : ∀ a ∈ xs, p a) : Array.map g (Array.pmap f xs H) = Array.pmap (fun a h => g (f a h)) xs H
• Array.map_subtype 📋 Init.Data.Array.Attach
  {α : Type u_1} {β : Type u_2} {p : α → Prop} {xs : Array { x // p x }} {f : { x // p x } → β} {g : α → β} (hf : ∀ (x : α) (h : p x), f ⟨x, h⟩ = g x) : Array.map f xs = Array.map g xs.unattach
• Array.attach_map_val 📋 Init.Data.Array.Attach
  {α : Type u_1} {β : Type u_2} (xs : Array α) (f : α → β) : Array.map (fun i => f ↑i) xs.attach = Array.map f xs
• Array.map_unattach 📋 Init.Data.Array.Attach
  {α : Type u_1} {β : Type u_2} {P : α → Prop} {xs : Array (Subtype P)} {f : α → β} : Array.map f xs.unattach = Array.map (fun x => match x with | ⟨x, h⟩ => binderNameHint x f (binderNameHint h () (f (wfParam x)))) xs
• Array.pmap_map 📋 Init.Data.Array.Attach
  {β : Type u_1} {γ : Type u_2} {α : Type u_3} {p : β → Prop} {g : (b : β) → p b → γ} {f : α → β} {xs : Array α} (H : ∀ a ∈ Array.map f xs, p a) : Array.pmap g (Array.map f xs) H = Array.pmap (fun a h => g (f a) h) xs ⋯
• Array.map_attach_eq_pmap 📋 Init.Data.Array.Attach
  {α : Type u_1} {β : Type u_2} {xs : Array α} {f : { x // x ∈ xs } → β} : Array.map f xs.attach = Array.pmap (fun a h => f ⟨a, h⟩) xs ⋯
• Array.attach_congr 📋 Init.Data.Array.Attach
  {α : Type u_1} {xs ys : Array α} (h : xs = ys) : xs.attach = Array.map (fun x => ⟨↑x, ⋯⟩) ys.attach
• Array.pmap_eq_map_attach 📋 Init.Data.Array.Attach
  {α : Type u_1} {β : Type u_2} {p : α → Prop} {f : (a : α) → p a → β} {xs : Array α} (H : ∀ a ∈ xs, p a) : Array.pmap f xs H = Array.map (fun x => f ↑x ⋯) xs.attach
• Array.map_attachWith 📋 Init.Data.Array.Attach
  {α : Type u_1} {β : Type u_2} {xs : Array α} {P : α → Prop} {H : ∀ a ∈ xs, P a} {f : { x // P x } → β} : Array.map f (xs.attachWith P H) = Array.map (fun x => match x with | ⟨x, h⟩ => f ⟨x, ⋯⟩) xs.attach
• Array.map_attachWith_eq_pmap 📋 Init.Data.Array.Attach
  {α : Type u_1} {β : Type u_2} {xs : Array α} {P : α → Prop} {H : ∀ a ∈ xs, P a} {f : { x // P x } → β} : Array.map f (xs.attachWith P H) = Array.pmap (fun a h => f ⟨a, ⋯⟩) xs ⋯
• Array.attachWith_map 📋 Init.Data.Array.Attach
  {α : Type u_1} {β : Type u_2} {xs : Array α} {f : α → β} {P : β → Prop} (H : ∀ b ∈ Array.map f xs, P b) : (Array.map f xs).attachWith P H = Array.map (fun x => match x with | ⟨x, h⟩ => ⟨f x, h⟩) (xs.attachWith (P ∘ f) ⋯)
• Array.attach_reverse 📋 Init.Data.Array.Attach
  {α : Type u_1} {xs : Array α} : xs.reverse.attach = Array.map (fun x => match x with | ⟨x, h⟩ => ⟨x, ⋯⟩) xs.attach.reverse
• Array.reverse_attach 📋 Init.Data.Array.Attach
  {α : Type u_1} {xs : Array α} : xs.attach.reverse = Array.map (fun x => match x with | ⟨x, h⟩ => ⟨x, ⋯⟩) xs.reverse.attach
• Array.attach_map 📋 Init.Data.Array.Attach
  {α : Type u_1} {β : Type u_2} {xs : Array α} {f : α → β} : (Array.map f xs).attach = Array.map (fun x => match x with | ⟨x, h⟩ => ⟨f x, ⋯⟩) xs.attach
• Array.attach_push 📋 Init.Data.Array.Attach
  {α : Type u_1} {a : α} {xs : Array α} : (xs.push a).attach = (Array.map (fun x => match x with | ⟨x, h⟩ => ⟨x, ⋯⟩) xs.attach).push ⟨a, ⋯⟩
• Array.filter_attachWith 📋 Init.Data.Array.Attach
  {α : Type u_1} {stop : ℕ} {q : α → Prop} {xs : Array α} {p : { x // q x } → Bool} (H : ∀ x ∈ xs, q x) (w : stop = (xs.attachWith q H).size) : Array.filter p (xs.attachWith q H) 0 stop = Array.map (fun x => match x with | ⟨x, h⟩ => ⟨x, ⋯⟩) (Array.filter (fun x => match x with | ⟨x, h⟩ => p ⟨x, ⋯⟩) xs.attach)
• Array.attach_append 📋 Init.Data.Array.Attach
  {α : Type u_1} {xs ys : Array α} : (xs ++ ys).attach = Array.map (fun x => match x with | ⟨x, h⟩ => ⟨x, ⋯⟩) xs.attach ++ Array.map (fun x => match x with | ⟨x, h⟩ => ⟨x, ⋯⟩) ys.attach
• Option.toArray_map 📋 Init.Data.Option.Array
  {α : Type u_1} {β : Type u_2} {o : Option α} {f : α → β} : (Option.map f o).toArray = Array.map f o.toArray
• Option.toArray_attach 📋 Init.Data.Option.Attach
  {α : Type u_1} (o : Option α) : o.attach.toArray = Array.map (fun x => ⟨↑x, ⋯⟩) o.toArray.attach
• Option.toArray_pmap 📋 Init.Data.Option.Attach
  {α : Type u_1} {β : Type u_2} {p : α → Prop} {o : Option α} {f : (a : α) → p a → β} (h : ∀ (a : α), o = some a → p a) : (Option.pmap f o h).toArray = Array.map (fun x => f ↑x ⋯) o.attach.toArray
• Option.toArray_attachWith 📋 Init.Data.Option.Attach
  {α : Type u_1} {p : α → Prop} {o : Option α} {h : ∀ (x : α), o = some x → p x} : (o.attachWith p h).toArray = Array.map (fun x => ⟨↑x, ⋯⟩) o.toArray.attach
• Array.idRun_mapM 📋 Init.Data.Array.Monadic
  {α : Type u_1} {β : Type u_2} {xs : Array α} {f : α → Id β} : (Array.mapM f xs).run = Array.map (fun x => (f x).run) xs
• Array.mapM_map 📋 Init.Data.Array.Monadic
  {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_4} {γ : Type u_1} [Monad m] [LawfulMonad m] {f : α → β} {g : β → m γ} {xs : Array α} : Array.mapM g (Array.map f xs) = Array.mapM (g ∘ f) xs
• Array.forM_map 📋 Init.Data.Array.Monadic
  {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_4} [Monad m] [LawfulMonad m] {xs : Array α} {g : α → β} {f : β → m PUnit.{u_1 + 1}} : forM (Array.map g xs) f = forM xs fun a => f (g a)
• Array.mapM_pure 📋 Init.Data.Array.Monadic
  {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [Monad m] [LawfulMonad m] {xs : Array α} {f : α → β} : Array.mapM (fun x => pure (f x)) xs = pure (Array.map f xs)
• Array.foldlM_map 📋 Init.Data.Array.Monadic
  {m : Type u_1 → Type u_2} {β₁ : Type u_3} {β₂ : Type u_4} {α : Type u_1} {stop : ℕ} [Monad m] {f : β₁ → β₂} {g : α → β₂ → m α} {xs : Array β₁} {init : α} {w : stop = xs.size} : Array.foldlM g init (Array.map f xs) 0 stop = Array.foldlM (fun x y => g x (f y)) init xs 0 stop
• Array.foldrM_map 📋 Init.Data.Array.Monadic
  {m : Type u_1 → Type u_2} {β₁ : Type u_3} {β₂ : Type u_4} {α : Type u_1} {start : ℕ} [Monad m] [LawfulMonad m] {f : β₁ → β₂} {g : β₂ → α → m α} {xs : Array β₁} {init : α} {w : start = xs.size} : Array.foldrM g init (Array.map f xs) start = Array.foldrM (fun x y => g (f x) y) init xs start
• Array.forIn_map 📋 Init.Data.Array.Monadic
  {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_4} {γ : Type u_1} {init : γ} [Monad m] [LawfulMonad m] {xs : Array α} {g : α → β} {f : β → γ → m (ForInStep γ)} : forIn (Array.map g xs) init f = forIn xs init fun a y => f (g a) y
• Array.forIn'_map 📋 Init.Data.Array.Monadic
  {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_4} {γ : Type u_1} {init : γ} [Monad m] [LawfulMonad m] {xs : Array α} (g : α → β) (f : (b : β) → b ∈ Array.map g xs → γ → m (ForInStep γ)) : forIn' (Array.map g xs) init f = forIn' xs init fun a h y => f (g a) ⋯ y
• Array.map_ofFn 📋 Init.Data.Array.OfFn
  {n : ℕ} {α : Type u_1} {β : Type u_2} {f : Fin n → α} {g : α → β} : Array.map g (Array.ofFn f) = Array.ofFn (g ∘ f)
• Array.mapIdx_eq_zipIdx_map 📋 Init.Data.Array.MapIdx
  {α : Type u_1} {β : Type u_2} {xs : Array α} {f : ℕ → α → β} : Array.mapIdx f xs = Array.map (fun x => match x with | (a, i) => f i a) xs.zipIdx
• Array.mapFinIdx_eq_zipIdx_map 📋 Init.Data.Array.MapIdx
  {α : Type u_1} {β : Type u_2} {xs : Array α} {f : (i : ℕ) → α → i < xs.size → β} : xs.mapFinIdx f = Array.map (fun x => match x with | ⟨(x, i), m⟩ => f i x ⋯) xs.zipIdx.attach
• Std.IterM.toArray_map 📋 Init.Data.Iterators.Lemmas.Combinators.Monadic.FilterMap
  {α β γ : Type w} {m : Type w → Type w'} [Monad m] [LawfulMonad m] [Std.Iterator α m β] [Std.Iterators.Finite α m] {f : β → γ} (it : Std.IterM m β) : (Std.IterM.map f it).toArray = (fun x => Array.map f x) <$> it.toArray
• Std.Iter.toArray_map 📋 Init.Data.Iterators.Lemmas.Combinators.FilterMap
  {α β γ : Type w} [Std.Iterator α Id β] {it : Std.Iter β} [Std.Iterators.Finite α Id] {f : β → γ} : (Std.Iter.map f it).toArray = Array.map f it.toArray
• Array.findIdx?_map 📋 Init.Data.Array.Find
  {β : Type u_1} {α : Type u_2} {f : β → α} {xs : Array β} {p : α → Bool} : Array.findIdx? p (Array.map f xs) = Array.findIdx? (p ∘ f) xs
• Array.find?_map 📋 Init.Data.Array.Find
  {β : Type u_1} {α : Type u_2} {p : α → Bool} {f : β → α} {xs : Array β} : Array.find? p (Array.map f xs) = Option.map f (Array.find? (p ∘ f) xs)
• Array.findSome?_map 📋 Init.Data.Array.Find
  {β : Type u_1} {γ : Type u_2} {α✝ : Type u_3} {p : γ → Option α✝} {f : β → γ} {xs : Array β} : Array.findSome? p (Array.map f xs) = Array.findSome? (p ∘ f) xs
• Array.findIdx?_flatten 📋 Init.Data.Array.Find
  {α : Type u_1} {xss : Array (Array α)} {p : α → Bool} : Array.findIdx? p xss.flatten = Option.map (fun i => (Array.map Array.size (xss.take i)).sum + (Option.map (fun xs => Array.findIdx p xs) xss[i]?).getD 0) (Array.findIdx? (fun x => x.any p) xss)
• Vector.toArray_map 📋 Init.Data.Vector.Lemmas
  {α : Type u_1} {β : Type u_2} {n : ℕ} {f : α → β} {xs : Vector α n} : (Vector.map f xs).toArray = Array.map f xs.toArray
• Vector.map_mk 📋 Init.Data.Vector.Lemmas
  {α : Type u_1} {β : Type u_2} {n : ℕ} {f : α → β} {xs : Array α} (h : xs.size = n) : Vector.map f (Vector.mk xs h) = Vector.mk (Array.map f xs) ⋯
• Vector.flatten_mk 📋 Init.Data.Vector.Lemmas
  {α : Type u_1} {n m : ℕ} {xss : Array (Vector α n)} (h : xss.size = m) : (Vector.mk xss h).flatten = Vector.mk (Array.map Vector.toArray xss).flatten ⋯
• Vector.vector₂_induction 📋 Init.Data.Vector.Lemmas
  {α : Type u_1} {n m : ℕ} (P : Vector (Vector α n) m → Prop) (of : ∀ (xss : Array (Array α)) (h₁ : xss.size = m) (h₂ : ∀ xs ∈ xss, xs.size = n), P (Vector.mk (Array.map (fun x => match x with | ⟨xs, m⟩ => Vector.mk xs ⋯) xss.attach) ⋯)) (xss : Vector (Vector α n) m) : P xss
• Vector.vector₃_induction 📋 Init.Data.Vector.Lemmas
  {α : Type u_1} {n m k : ℕ} (P : Vector (Vector (Vector α n) m) k → Prop) (of : ∀ (xss : Array (Array (Array α))) (h₁ : xss.size = k) (h₂ : ∀ xs ∈ xss, xs.size = m) (h₃ : ∀ xs ∈ xss, ∀ as ∈ xs, as.size = n), P (Vector.mk (Array.map (fun x => match x with | ⟨xs, m_1⟩ => Vector.mk (Array.map (fun x => match x with | ⟨as, m'⟩ => Vector.mk as ⋯) xs.attach) ⋯) xss.attach) ⋯)) (xss : Vector (Vector (Vector α n) m) k) : P xss
• Array.finRange_reverse 📋 Init.Data.Array.FinRange
  {n : ℕ} : (Array.finRange n).reverse = Array.map Fin.rev (Array.finRange n)
• Array.finRange_succ_last 📋 Init.Data.Array.FinRange
  {n : ℕ} : Array.finRange (n + 1) = Array.map Fin.castSucc (Array.finRange n) ++ #[Fin.last n]
• Array.finRange_succ 📋 Init.Data.Array.FinRange
  {n : ℕ} : Array.finRange (n + 1) = #[0] ++ Array.map Fin.succ (Array.finRange n)
• Std.Rci.toArray_succ_succ_eq_map 📋 Init.Data.Range.Polymorphic.Lemmas
  {α : Type u} [LE α] [DecidableLE α] [Std.PRange.UpwardEnumerable α] [Std.PRange.LinearlyUpwardEnumerable α] [Std.PRange.LawfulUpwardEnumerableLE α] [Std.PRange.InfinitelyUpwardEnumerable α] [Std.Rxi.IsAlwaysFinite α] [Std.PRange.LawfulUpwardEnumerable α] {lo : α} : (Std.PRange.succ lo)...*.toArray = Array.map Std.PRange.succ lo...*.toArray
• Std.Roi.toArray_succ_succ_eq_map 📋 Init.Data.Range.Polymorphic.Lemmas
  {α : Type u} [LT α] [DecidableLT α] [Std.PRange.UpwardEnumerable α] [Std.PRange.LinearlyUpwardEnumerable α] [Std.PRange.InfinitelyUpwardEnumerable α] [Std.Rxi.IsAlwaysFinite α] [Std.PRange.LawfulUpwardEnumerable α] [Std.PRange.LawfulUpwardEnumerableLT α] {lo : α} : (Std.PRange.succ lo)<...*.toArray = Array.map Std.PRange.succ lo<...*.toArray
• Std.Rcc.toArray_succ_succ_eq_map 📋 Init.Data.Range.Polymorphic.Lemmas
  {α : Type u} [LE α] [DecidableLE α] [Std.PRange.UpwardEnumerable α] [Std.PRange.LinearlyUpwardEnumerable α] [Std.PRange.LawfulUpwardEnumerableLE α] [Std.PRange.InfinitelyUpwardEnumerable α] [Std.Rxc.IsAlwaysFinite α] [Std.PRange.LawfulUpwardEnumerable α] {lo hi : α} : ((Std.PRange.succ lo)...=Std.PRange.succ hi).toArray = Array.map Std.PRange.succ (lo...=hi).toArray
• Std.Roo.toArray_succ_succ_eq_map 📋 Init.Data.Range.Polymorphic.Lemmas
  {α : Type u} [LT α] [DecidableLT α] [Std.PRange.UpwardEnumerable α] [Std.PRange.LinearlyUpwardEnumerable α] [Std.PRange.LawfulUpwardEnumerableLT α] [Std.PRange.InfinitelyUpwardEnumerable α] [Std.Rxo.IsAlwaysFinite α] [Std.PRange.LawfulUpwardEnumerable α] {lo hi : α} : ((Std.PRange.succ lo)<...Std.PRange.succ hi).toArray = Array.map Std.PRange.succ (lo<...hi).toArray
• Std.Rco.toArray_succ_succ_eq_map 📋 Init.Data.Range.Polymorphic.Lemmas
  {α : Type u} [LE α] [DecidableLE α] [LT α] [DecidableLT α] [Std.PRange.UpwardEnumerable α] [Std.PRange.LinearlyUpwardEnumerable α] [Std.PRange.LawfulUpwardEnumerableLE α] [Std.PRange.LawfulUpwardEnumerableLT α] [Std.PRange.InfinitelyUpwardEnumerable α] [Std.Rxo.IsAlwaysFinite α] [Std.PRange.LawfulUpwardEnumerable α] {lo hi : α} : ((Std.PRange.succ lo)...Std.PRange.succ hi).toArray = Array.map Std.PRange.succ (lo...hi).toArray
• Std.Roc.toArray_succ_succ_eq_map 📋 Init.Data.Range.Polymorphic.Lemmas
  {α : Type u} [LE α] [DecidableLE α] [LT α] [DecidableLT α] [Std.PRange.UpwardEnumerable α] [Std.PRange.LinearlyUpwardEnumerable α] [Std.PRange.LawfulUpwardEnumerableLE α] [Std.PRange.LawfulUpwardEnumerableLT α] [Std.PRange.InfinitelyUpwardEnumerable α] [Std.Rxc.IsAlwaysFinite α] [Std.PRange.LawfulUpwardEnumerable α] {lo hi : α} : ((Std.PRange.succ lo)<...=Std.PRange.succ hi).toArray = Array.map Std.PRange.succ (lo<...=hi).toArray
• Std.Ric.toArray_succ_eq_map 📋 Init.Data.Range.Polymorphic.Lemmas
  {α : Type u} {r : Std.Ric α} [LE α] [DecidableLE α] [Std.PRange.Least? α] [Std.PRange.UpwardEnumerable α] [Std.PRange.LinearlyUpwardEnumerable α] [Std.PRange.LawfulUpwardEnumerableLE α] [Std.PRange.InfinitelyUpwardEnumerable α] [Std.Rxc.IsAlwaysFinite α] [Std.PRange.LawfulUpwardEnumerable α] [Std.PRange.LawfulUpwardEnumerableLeast? α] {hi : α} : (*...=Std.PRange.succ hi).toArray = #[Std.PRange.UpwardEnumerable.least] ++ Array.map Std.PRange.succ (*...=hi).toArray
• Std.Rio.toArray_succ_eq_map 📋 Init.Data.Range.Polymorphic.Lemmas
  {α : Type u} {r : Std.Rio α} [LT α] [DecidableLT α] [Std.PRange.Least? α] [Std.PRange.UpwardEnumerable α] [Std.PRange.LinearlyUpwardEnumerable α] [Std.PRange.LawfulUpwardEnumerableLT α] [Std.PRange.InfinitelyUpwardEnumerable α] [Std.Rxo.IsAlwaysFinite α] [Std.PRange.LawfulUpwardEnumerable α] [Std.PRange.LawfulUpwardEnumerableLeast? α] {hi : α} : (*...Std.PRange.succ hi).toArray = #[Std.PRange.UpwardEnumerable.least] ++ Array.map Std.PRange.succ (*...hi).toArray
• Nat.map_add_toArray_ric 📋 Init.Data.Range.Polymorphic.NatLemmas
  {n k : ℕ} : Array.map (fun x => x + k) (*...=n).toArray = (k...=n + k).toArray
• Nat.map_add_toArray_ric' 📋 Init.Data.Range.Polymorphic.NatLemmas
  {n k : ℕ} : Array.map (fun x => k + x) (*...=n).toArray = (k...=k + n).toArray
• Nat.map_add_toArray_rio 📋 Init.Data.Range.Polymorphic.NatLemmas
  {n k : ℕ} : Array.map (fun x => x + k) (*...n).toArray = (k...n + k).toArray
• Nat.map_add_toArray_rio' 📋 Init.Data.Range.Polymorphic.NatLemmas
  {n k : ℕ} : Array.map (fun x => k + x) (*...n).toArray = (k...k + n).toArray
• Nat.toArray_rcc_add_right_eq_map 📋 Init.Data.Range.Polymorphic.NatLemmas
  {m n : ℕ} : (m...=m + n).toArray = Array.map (fun x => x + m) (0...=n).toArray
• Nat.toArray_rco_add_right_eq_map 📋 Init.Data.Range.Polymorphic.NatLemmas
  {m n : ℕ} : (m...m + n).toArray = Array.map (fun x => x + m) (0...n).toArray
• Nat.map_add_toArray_rcc 📋 Init.Data.Range.Polymorphic.NatLemmas
  {m n k : ℕ} : Array.map (fun x => x + k) (m...=n).toArray = ((m + k)...=n + k).toArray
• Nat.map_add_toArray_rcc' 📋 Init.Data.Range.Polymorphic.NatLemmas
  {m n k : ℕ} : Array.map (fun x => k + x) (m...=n).toArray = ((k + m)...=k + n).toArray
• Nat.map_add_toArray_rco 📋 Init.Data.Range.Polymorphic.NatLemmas
  {m n k : ℕ} : Array.map (fun x => x + k) (m...n).toArray = ((m + k)...n + k).toArray
• Nat.map_add_toArray_rco' 📋 Init.Data.Range.Polymorphic.NatLemmas
  {m n k : ℕ} : Array.map (fun x => k + x) (m...n).toArray = ((k + m)...k + n).toArray
• Nat.map_add_toArray_roc 📋 Init.Data.Range.Polymorphic.NatLemmas
  {m n k : ℕ} : Array.map (fun x => x + k) (m<...=n).toArray = ((m + k)<...=n + k).toArray
• Nat.map_add_toArray_roc' 📋 Init.Data.Range.Polymorphic.NatLemmas
  {m n k : ℕ} : Array.map (fun x => k + x) (m<...=n).toArray = ((k + m)<...=k + n).toArray
• Nat.map_add_toArray_roo 📋 Init.Data.Range.Polymorphic.NatLemmas
  {m n k : ℕ} : Array.map (fun x => x + k) (m<...n).toArray = ((m + k)<...n + k).toArray
• Nat.map_add_toArray_roo' 📋 Init.Data.Range.Polymorphic.NatLemmas
  {m n k : ℕ} : Array.map (fun x => k + x) (m<...n).toArray = ((k + m)<...k + n).toArray
• Nat.toArray_roc_add_right_eq_map 📋 Init.Data.Range.Polymorphic.NatLemmas
  {m n : ℕ} : (m<...=m + n).toArray = Array.map (fun x => x + m + 1) (0...n).toArray
• Nat.toArray_rcc_succ_succ 📋 Init.Data.Range.Polymorphic.NatLemmas
  {m n : ℕ} : ((m + 1)...=n + 1).toArray = Array.map (fun x => x + 1) (m...=n).toArray
• Nat.toArray_rco_succ_succ 📋 Init.Data.Range.Polymorphic.NatLemmas
  {m n : ℕ} : ((m + 1)...n + 1).toArray = Array.map (fun x => x + 1) (m...n).toArray
• Nat.toArray_roc_succ_succ 📋 Init.Data.Range.Polymorphic.NatLemmas
  {m n : ℕ} : ((m + 1)<...=n + 1).toArray = Array.map (fun x => x + 1) (m<...=n).toArray
• Nat.toArray_roo_succ_succ 📋 Init.Data.Range.Polymorphic.NatLemmas
  {m n : ℕ} : ((m + 1)<...n + 1).toArray = Array.map (fun x => x + 1) (m<...n).toArray
• Nat.toArray_roo_add_right_eq_map 📋 Init.Data.Range.Polymorphic.NatLemmas
  {m n : ℕ} : (m<...m + 1 + n).toArray = Array.map (fun x => x + m + 1) (0...n).toArray
• Nat.toArray_rco_succ_right_eq_append_map 📋 Init.Data.Range.Polymorphic.NatLemmas
  {m n : ℕ} (h : m ≤ n) : (m...n + 1).toArray = #[m] ++ Array.map (fun x => x + 1) (m...n).toArray
• Nat.toArray_rcc_succ_right_eq_append_map 📋 Init.Data.Range.Polymorphic.NatLemmas
  {m n : ℕ} (h : m ≤ n + 1) : (m...=n + 1).toArray = #[m] ++ Array.map (fun x => x + 1) (m...=n).toArray
• Nat.toArray_roc_succ_right_eq_append_map 📋 Init.Data.Range.Polymorphic.NatLemmas
  {m n : ℕ} (h : m ≤ n) : (m<...=n + 1).toArray = #[m + 1] ++ Array.map (fun x => x + 1) (m<...=n).toArray
• Nat.toArray_roo_succ_right_eq_append_map 📋 Init.Data.Range.Polymorphic.NatLemmas
  {m n : ℕ} (h : m < n) : (m<...n + 1).toArray = #[m + 1] ++ Array.map (fun x => x + 1) (m<...n).toArray
• Array.map_lt 📋 Init.Data.Array.Lex.Lemmas
  {α : Type u_1} {β : Type u_2} [LT α] [LT β] {xs ys : Array α} {f : α → β} (w : ∀ (x y : α), x < y → f x < f y) (h : xs < ys) : Array.map f xs < Array.map f ys
• Array.map_le 📋 Init.Data.Array.Lex.Lemmas
  {α : Type u_1} {β : Type u_2} [LT α] [LT β] [Std.Asymm fun x1 x2 => x1 < x2] [Std.Trichotomous fun x1 x2 => x1 < x2] [Std.Asymm fun x1 x2 => x1 < x2] [Std.Trichotomous fun x1 x2 => x1 < x2] {xs ys : Array α} {f : α → β} (w : ∀ (x y : α), x < y → f x < f y) (h : xs ≤ ys) : Array.map f xs ≤ Array.map f ys
• Array.zipWith_self 📋 Init.Data.Array.Zip
  {α : Type u_1} {δ : Type u_2} {f : α → α → δ} {xs : Array α} : Array.zipWith f xs xs = Array.map (fun a => f a a) xs
• Array.map_fst_zip 📋 Init.Data.Array.Zip
  {α : Type u_1} {β : Type u_2} {as : Array α} {bs : Array β} (h : as.size ≤ bs.size) : Array.map Prod.fst (as.zip bs) = as
• Array.map_snd_zip 📋 Init.Data.Array.Zip
  {α : Type u_1} {β : Type u_2} {as : Array α} {bs : Array β} (h : bs.size ≤ as.size) : Array.map Prod.snd (as.zip bs) = bs
• Array.map_prod_left_eq_zip 📋 Init.Data.Array.Zip
  {α : Type u_1} {β : Type u_2} {xs : Array α} {f : α → β} : Array.map (fun x => (x, f x)) xs = xs.zip (Array.map f xs)
• Array.map_prod_right_eq_zip 📋 Init.Data.Array.Zip
  {α : Type u_1} {β : Type u_2} {xs : Array α} {f : α → β} : Array.map (fun x => (f x, x)) xs = (Array.map f xs).zip xs
• Array.map_uncurry_zip_eq_zipWith 📋 Init.Data.Array.Zip
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β → γ} {as : Array α} {bs : Array β} : Array.map (Function.uncurry f) (as.zip bs) = Array.zipWith f as bs
• Array.map_zip_eq_zipWith 📋 Init.Data.Array.Zip
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α × β → γ} {as : Array α} {bs : Array β} : Array.map f (as.zip bs) = Array.zipWith (Function.curry f) as bs
• Array.map_zipWith 📋 Init.Data.Array.Zip
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f : α → β} {g : γ → δ → α} {cs : Array γ} {ds : Array δ} : Array.map f (Array.zipWith g cs ds) = Array.zipWith (fun x y => f (g x y)) cs ds
• Array.unzip_eq_map 📋 Init.Data.Array.Zip
  {α : Type u_1} {β : Type u_2} {xs : Array (α × β)} : xs.unzip = (Array.map Prod.fst xs, Array.map Prod.snd xs)
• Array.zipWith_map_left 📋 Init.Data.Array.Zip
  {α : Type u_1} {β : Type u_2} {α' : Type u_3} {γ : Type u_4} {as : Array α} {bs : Array β} {f : α → α'} {g : α' → β → γ} : Array.zipWith g (Array.map f as) bs = Array.zipWith (fun a b => g (f a) b) as bs
• Array.zipWith_map_right 📋 Init.Data.Array.Zip
  {α : Type u_1} {β : Type u_2} {β' : Type u_3} {γ : Type u_4} {as : Array α} {bs : Array β} {f : β → β'} {g : α → β' → γ} : Array.zipWith g as (Array.map f bs) = Array.zipWith (fun a b => g a (f b)) as bs
• Array.zip_map' 📋 Init.Data.Array.Zip
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β} {g : α → γ} {xs : Array α} : (Array.map f xs).zip (Array.map g xs) = Array.map (fun a => (f a, g a)) xs
• Array.map_zipWithAll 📋 Init.Data.Array.Zip
  {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {f : α → β} {g : Option γ → Option δ → α} {cs : Array γ} {ds : Array δ} : Array.map f (Array.zipWithAll g cs ds) = Array.zipWithAll (fun x y => f (g x y)) cs ds
• Array.zip_map_left 📋 Init.Data.Array.Zip
  {α : Type u_1} {γ : Type u_2} {β : Type u_3} {f : α → γ} {as : Array α} {bs : Array β} : (Array.map f as).zip bs = Array.map (Prod.map f id) (as.zip bs)
• Array.zip_map_right 📋 Init.Data.Array.Zip
  {β : Type u_1} {γ : Type u_2} {α : Type u_3} {f : β → γ} {as : Array α} {bs : Array β} : as.zip (Array.map f bs) = Array.map (Prod.map id f) (as.zip bs)

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision a114d38 serving mathlib revision 476fb97