zip & unzip

This file provides results about List.zipWith, List.zip and List.unzip (definitions are in core Lean). zipWith f l₁ l₂ applies f : α → β → γ pointwise to a list l₁ : List α and l₂ : List β. It applies, until one of the lists is exhausted. For example, zipWith f [0, 1, 2] [6.28, 31] = [f 0 6.28, f 1 31]. zip is zipWith applied to Prod.mk. For example, zip [a₁, a₂] [b₁, b₂, b₃] = [(a₁, b₁), (a₂, b₂)]. unzip undoes zip. For example, unzip [(a₁, b₁), (a₂, b₂)] = ([a₁, a₂], [b₁, b₂]).

@[simp]

theorem List.zip_swap {α : Type u} {β : Type u_1} (l₁ : List α) (l₂ : List β) : List.map Prod.swap (l₁.zip l₂) = l₂.zip l₁

theorem List.forall_zipWith {α : Type u} {β : Type u_1} {γ : Type u_2} {f : α → β → γ} {p : γ → Prop} {l₁ : List α} {l₂ : List β} : l₁.length = l₂.length → (List.Forall p (List.zipWith f l₁ l₂) ↔ List.Forall₂ (fun (x : α) (y : β) => p (f x y)) l₁ l₂)

theorem List.lt_length_left_of_zipWith {α : Type u} {β : Type u_1} {γ : Type u_2} {f : α → β → γ} {i : ℕ} {l : List α} {l' : List β} (h : i < (List.zipWith f l l').length) : i < l.length

theorem List.lt_length_right_of_zipWith {α : Type u} {β : Type u_1} {γ : Type u_2} {f : α → β → γ} {i : ℕ} {l : List α} {l' : List β} (h : i < (List.zipWith f l l').length) : i < l'.length

theorem List.lt_length_left_of_zip {α : Type u} {β : Type u_1} {i : ℕ} {l : List α} {l' : List β} (h : i < (l.zip l').length) : i < l.length

theorem List.lt_length_right_of_zip {α : Type u} {β : Type u_1} {i : ℕ} {l : List α} {l' : List β} (h : i < (l.zip l').length) : i < l'.length

theorem List.mem_zip {α : Type u} {β : Type u_1} {a : α} {b : β} {l₁ : List α} {l₂ : List β} : (a, b) ∈ l₁.zip l₂ → a ∈ l₁ ∧ b ∈ l₂

theorem List.unzip_eq_map {α : Type u} {β : Type u_1} (l : List (α × β)) : l.unzip = (List.map Prod.fst l, List.map Prod.snd l)

theorem List.unzip_left {α : Type u} {β : Type u_1} (l : List (α × β)) : l.unzip.1 = List.map Prod.fst l

theorem List.unzip_right {α : Type u} {β : Type u_1} (l : List (α × β)) : l.unzip.2 = List.map Prod.snd l

theorem List.unzip_swap {α : Type u} {β : Type u_1} (l : List (α × β)) : (List.map Prod.swap l).unzip = l.unzip.swap

theorem List.zip_unzip {α : Type u} {β : Type u_1} (l : List (α × β)) : l.unzip.1.zip l.unzip.2 = l

theorem List.unzip_zip_left {α : Type u} {β : Type u_1} {l₁ : List α} {l₂ : List β} : l₁.length ≤ l₂.length → (l₁.zip l₂).unzip.1 = l₁

theorem List.unzip_zip_right {α : Type u} {β : Type u_1} {l₁ : List α} {l₂ : List β} (h : l₂.length ≤ l₁.length) : (l₁.zip l₂).unzip.2 = l₂

theorem List.unzip_zip {α : Type u} {β : Type u_1} {l₁ : List α} {l₂ : List β} (h : l₁.length = l₂.length) : (l₁.zip l₂).unzip = (l₁, l₂)

theorem List.zip_of_prod {α : Type u} {β : Type u_1} {l : List α} {l' : List β} {lp : List (α × β)} (hl : List.map Prod.fst lp = l) (hr : List.map Prod.snd lp = l') : lp = l.zip l'

theorem List.map_prod_left_eq_zip {α : Type u} {β : Type u_1} {l : List α} (f : α → β) : List.map (fun (x : α) => (x, f x)) l = l.zip (List.map f l)

theorem List.map_prod_right_eq_zip {α : Type u} {β : Type u_1} {l : List α} (f : α → β) : List.map (fun (x : α) => (f x, x)) l = (List.map f l).zip l

theorem List.zipWith_comm {α : Type u} {β : Type u_1} {γ : Type u_2} (f : α → β → γ) (la : List α) (lb : List β) : List.zipWith f la lb = List.zipWith (fun (b : β) (a : α) => f a b) lb la

theorem List.zipWith_congr {α : Type u} {β : Type u_1} {γ : Type u_2} (f : α → β → γ) (g : α → β → γ) (la : List α) (lb : List β) (h : List.Forall₂ (fun (a : α) (b : β) => f a b = g a b) la lb) : List.zipWith f la lb = List.zipWith g la lb

theorem List.zipWith_comm_of_comm {α : Type u} {β : Type u_1} (f : α → α → β) (comm : ∀ (x y : α), f x y = f y x) (l : List α) (l' : List α) : List.zipWith f l l' = List.zipWith f l' l

@[simp]

theorem List.zipWith_same {α : Type u} {δ : Type u_3} (f : α → α → δ) (l : List α) : List.zipWith f l l = List.map (fun (a : α) => f a a) l

theorem List.zipWith_zipWith_left {α : Type u} {β : Type u_1} {γ : Type u_2} {δ : Type u_3} {ε : Type u_4} (f : δ → γ → ε) (g : α → β → δ) (la : List α) (lb : List β) (lc : List γ) : List.zipWith f (List.zipWith g la lb) lc = List.zipWith3 (fun (a : α) (b : β) (c : γ) => f (g a b) c) la lb lc

theorem List.zipWith_zipWith_right {α : Type u} {β : Type u_1} {γ : Type u_2} {δ : Type u_3} {ε : Type u_4} (f : α → δ → ε) (g : β → γ → δ) (la : List α) (lb : List β) (lc : List γ) : List.zipWith f la (List.zipWith g lb lc) = List.zipWith3 (fun (a : α) (b : β) (c : γ) => f a (g b c)) la lb lc

@[simp]

theorem List.zipWith3_same_left {α : Type u} {β : Type u_1} {γ : Type u_2} (f : α → α → β → γ) (la : List α) (lb : List β) : List.zipWith3 f la la lb = List.zipWith (fun (a : α) (b : β) => f a a b) la lb

@[simp]

theorem List.zipWith3_same_mid {α : Type u} {β : Type u_1} {γ : Type u_2} (f : α → β → α → γ) (la : List α) (lb : List β) : List.zipWith3 f la lb la = List.zipWith (fun (a : α) (b : β) => f a b a) la lb

@[simp]

theorem List.zipWith3_same_right {α : Type u} {β : Type u_1} {γ : Type u_2} (f : α → β → β → γ) (la : List α) (lb : List β) : List.zipWith3 f la lb lb = List.zipWith (fun (a : α) (b : β) => f a b b) la lb

instance List.instIsSymmOpZipWith {α : Type u} {β : Type u_1} (f : α → α → β) [IsSymmOp α β f] : IsSymmOp (List α) (List β) (List.zipWith f)

Equations

• ⋯ = ⋯

@[simp]

theorem List.length_revzip {α : Type u} (l : List α) : l.revzip.length = l.length

@[simp]

theorem List.unzip_revzip {α : Type u} (l : List α) : l.revzip.unzip = (l, l.reverse)

@[simp]

theorem List.revzip_map_fst {α : Type u} (l : List α) : List.map Prod.fst l.revzip = l

@[simp]

theorem List.revzip_map_snd {α : Type u} (l : List α) : List.map Prod.snd l.revzip = l.reverse

theorem List.reverse_revzip {α : Type u} (l : List α) : l.revzip.reverse = l.reverse.revzip

theorem List.revzip_swap {α : Type u} (l : List α) : List.map Prod.swap l.revzip = l.reverse.revzip

theorem List.getElem?_zip_with {α : Type u} {β : Type u_1} {γ : Type u_2} (f : α → β → γ) (l₁ : List α) (l₂ : List β) (i : ℕ) : (List.zipWith f l₁ l₂)[i]? = (Option.map f l₁[i]?).bind fun (g : β → γ) => Option.map g l₂[i]?

theorem List.get?_zip_with {α : Type u} {β : Type u_1} {γ : Type u_2} (f : α → β → γ) (l₁ : List α) (l₂ : List β) (i : ℕ) : (List.zipWith f l₁ l₂).get? i = (Option.map f (l₁.get? i)).bind fun (g : β → γ) => Option.map g (l₂.get? i)

theorem List.getElem?_zip_with_eq_some {α : Type u} {β : Type u_1} {γ : Type u_2} (f : α → β → γ) (l₁ : List α) (l₂ : List β) (z : γ) (i : ℕ) : (List.zipWith f l₁ l₂)[i]? = some z ↔ ∃ (x : α), ∃ (y : β), l₁[i]? = some x ∧ l₂[i]? = some y ∧ f x y = z

theorem List.get?_zip_with_eq_some {α : Type u} {β : Type u_1} {γ : Type u_2} (f : α → β → γ) (l₁ : List α) (l₂ : List β) (z : γ) (i : ℕ) : (List.zipWith f l₁ l₂).get? i = some z ↔ ∃ (x : α), ∃ (y : β), l₁.get? i = some x ∧ l₂.get? i = some y ∧ f x y = z

theorem List.getElem?_zip_eq_some {α : Type u} {β : Type u_1} (l₁ : List α) (l₂ : List β) (z : α × β) (i : ℕ) : (l₁.zip l₂)[i]? = some z ↔ l₁[i]? = some z.1 ∧ l₂[i]? = some z.2

theorem List.get?_zip_eq_some {α : Type u} {β : Type u_1} (l₁ : List α) (l₂ : List β) (z : α × β) (i : ℕ) : (l₁.zip l₂).get? i = some z ↔ l₁.get? i = some z.1 ∧ l₂.get? i = some z.2

@[simp]

theorem List.getElem_zipWith {α : Type u} {β : Type u_1} {γ : Type u_2} {f : α → β → γ} {l : List α} {l' : List β} {i : ℕ} {h : i < (List.zipWith f l l').length} : (List.zipWith f l l')[i] = f l[i] l'[i]

@[deprecated List.getElem_zipWith]

theorem List.get_zipWith {α : Type u} {β : Type u_1} {γ : Type u_2} {f : α → β → γ} {l : List α} {l' : List β} {i : Fin (List.zipWith f l l').length} : (List.zipWith f l l').get i = f (l.get ⟨↑i, ⋯⟩) (l'.get ⟨↑i, ⋯⟩)

@[simp, deprecated List.get_zipWith]

theorem List.nthLe_zipWith {α : Type u} {β : Type u_1} {γ : Type u_2} {f : α → β → γ} {l : List α} {l' : List β} {i : ℕ} {h : i < (List.zipWith f l l').length} : (List.zipWith f l l').nthLe i h = f (l.nthLe i ⋯) (l'.nthLe i ⋯)

@[simp]

theorem List.getElem_zip {α : Type u} {β : Type u_1} {l : List α} {l' : List β} {i : ℕ} {h : i < (l.zip l').length} : (l.zip l')[i] = (l[i], l'[i])

@[deprecated List.getElem_zip]

theorem List.get_zip {α : Type u} {β : Type u_1} {l : List α} {l' : List β} {i : Fin (l.zip l').length} : (l.zip l').get i = (l.get ⟨↑i, ⋯⟩, l'.get ⟨↑i, ⋯⟩)

@[simp, deprecated List.get_zip]

theorem List.nthLe_zip {α : Type u} {β : Type u_1} {l : List α} {l' : List β} {i : ℕ} {h : i < (l.zip l').length} : (l.zip l').nthLe i h = (l.nthLe i ⋯, l'.nthLe i ⋯)

theorem List.mem_zip_inits_tails {α : Type u} {l : List α} {init : List α} {tail : List α} : (init, tail) ∈ l.inits.zip l.tails ↔ init ++ tail = l

theorem List.map_uncurry_zip_eq_zipWith {α : Type u} {β : Type u_1} {γ : Type u_2} (f : α → β → γ) (l : List α) (l' : List β) : List.map (Function.uncurry f) (l.zip l') = List.zipWith f l l'

Operations that can be applied before or after a zip_with

theorem List.zipWith_distrib_reverse {α : Type u} {β : Type u_1} {γ : Type u_2} (f : α → β → γ) (l : List α) (l' : List β) (h : l.length = l'.length) : (List.zipWith f l l').reverse = List.zipWith f l.reverse l'.reverse