def List.pmap {α : Type u_1} {β : Type u_2} {P : α → Prop} (f : (a : α) → P a → β) (l : List α) (H : ∀ (a : α), a ∈ l → P a) : List β

O(n). Partial map. If f : Π a, P a → β is a partial function defined on a : α satisfying P, then pmap f l h is essentially the same as map f l but is defined only when all members of l satisfy P, using the proof to apply f.

Equations

• List.pmap f [] x_2 = []
• List.pmap f (a :: l) H = f a ⋯ :: List.pmap f l ⋯

@[implemented_by _private.Init.Data.List.Attach.0.List.attachWithImpl]

def List.attachWith {α : Type u_1} (l : List α) (P : α → Prop) (H : ∀ (x : α), x ∈ l → P x) : List { x : α // P x }

O(1). "Attach" a proof P x that holds for all the elements of l to produce a new list with the same elements but in the type {x // P x}.

Equations

• l.attachWith P H = List.pmap Subtype.mk l H

@[inline]

def List.attach {α : Type u_1} (l : List α) : List { x : α // x ∈ l }

O(1). "Attach" the proof that the elements of l are in l to produce a new list with the same elements but in the type {x // x ∈ l}.

Equations

• l.attach = l.attachWith (Membership.mem l) ⋯

@[simp]

theorem List.pmap_nil {α : Type u_1} {β : Type u_2} {P : α → Prop} (f : (a : α) → P a → β) : List.pmap f [] ⋯ = []

@[simp]

theorem List.pmap_cons {α : Type u_1} {β : Type u_2} {P : α → Prop} (f : (a : α) → P a → β) (a : α) (l : List α) (h : ∀ (b : α), b ∈ a :: l → P b) : List.pmap f (a :: l) h = f a ⋯ :: List.pmap f l ⋯

@[simp]

theorem List.attach_nil {α : Type u_1} : [].attach = []

@[simp]

theorem List.attachWith_nil {α : Type u_1} {P : α → Prop} {H : ∀ (x : α), x ∈ [] → P x} : [].attachWith P H = []

@[simp]

theorem List.pmap_eq_map {α : Type u_1} {β : Type u_2} (p : α → Prop) (f : α → β) (l : List α) (H : ∀ (a : α), a ∈ l → p a) : List.pmap (fun (a : α) (x : p a) => f a) l H = List.map f l

theorem List.pmap_congr_left {α : Type u_1} {β : Type u_2} {p q : α → Prop} {f : (a : α) → p a → β} {g : (a : α) → q a → β} (l : List α) {H₁ : ∀ (a : α), a ∈ l → p a} {H₂ : ∀ (a : α), a ∈ l → q a} (h : ∀ (a : α), a ∈ l → ∀ (h₁ : p a) (h₂ : q a), f a h₁ = g a h₂) : List.pmap f l H₁ = List.pmap g l H₂

@[reducible, inline, deprecated List.pmap_congr_left]

abbrev List.pmap_congr {α : Type u_1} {β : Type u_2} {p q : α → Prop} {f : (a : α) → p a → β} {g : (a : α) → q a → β} (l : List α) {H₁ : ∀ (a : α), a ∈ l → p a} {H₂ : ∀ (a : α), a ∈ l → q a} (h : ∀ (a : α), a ∈ l → ∀ (h₁ : p a) (h₂ : q a), f a h₁ = g a h₂) : List.pmap f l H₁ = List.pmap g l H₂

Equations

• @List.pmap_congr = @List.pmap_congr_left

theorem List.map_pmap {α : Type u_1} {β : Type u_2} {γ : Type u_3} {p : α → Prop} (g : β → γ) (f : (a : α) → p a → β) (l : List α) (H : ∀ (a : α), a ∈ l → p a) : List.map g (List.pmap f l H) = List.pmap (fun (a : α) (h : p a) => g (f a h)) l H

theorem List.pmap_map {β : Type u_1} {γ : Type u_2} {α : Type u_3} {p : β → Prop} (g : (b : β) → p b → γ) (f : α → β) (l : List α) (H : ∀ (a : β), a ∈ List.map f l → p a) : List.pmap g (List.map f l) H = List.pmap (fun (a : α) (h : p (f a)) => g (f a) h) l ⋯

theorem List.attach_congr {α : Type u_1} {l₁ l₂ : List α} (h : l₁ = l₂) : l₁.attach = List.map (fun (x : { x : α // x ∈ l₂ }) => ⟨x.val, ⋯⟩) l₂.attach

theorem List.attachWith_congr {α : Type u_1} {l₁ l₂ : List α} (w : l₁ = l₂) {P : α → Prop} {H : ∀ (x : α), x ∈ l₁ → P x} : l₁.attachWith P H = l₂.attachWith P ⋯

@[simp]

theorem List.attach_cons {α : Type u_1} {x : α} {xs : List α} : (x :: xs).attach = ⟨x, ⋯⟩ :: List.map (fun (x_1 : { x : α // x ∈ xs }) => match x_1 with | ⟨y, h⟩ => ⟨y, ⋯⟩) xs.attach

@[simp]

theorem List.attachWith_cons {α : Type u_1} {x : α} {xs : List α} {p : α → Prop} (h : ∀ (a : α), a ∈ x :: xs → p a) : (x :: xs).attachWith p h = ⟨x, ⋯⟩ :: xs.attachWith p ⋯

theorem List.pmap_eq_map_attach {α : Type u_1} {β : Type u_2} {p : α → Prop} (f : (a : α) → p a → β) (l : List α) (H : ∀ (a : α), a ∈ l → p a) : List.pmap f l H = List.map (fun (x : { x : α // x ∈ l }) => f x.val ⋯) l.attach

theorem List.attach_map_coe {α : Type u_1} {β : Type u_2} (l : List α) (f : α → β) : List.map (fun (i : { i : α // i ∈ l }) => f i.val) l.attach = List.map f l

theorem List.attach_map_val {α : Type u_1} {β : Type u_2} (l : List α) (f : α → β) : List.map (fun (i : { x : α // x ∈ l }) => f i.val) l.attach = List.map f l

@[simp]

theorem List.attach_map_subtype_val {α : Type u_1} (l : List α) : List.map Subtype.val l.attach = l

theorem List.attachWith_map_coe {α : Type u_1} {β : Type u_2} {p : α → Prop} (f : α → β) (l : List α) (H : ∀ (a : α), a ∈ l → p a) : List.map (fun (i : { i : α // p i }) => f i.val) (l.attachWith p H) = List.map f l

theorem List.attachWith_map_val {α : Type u_1} {β : Type u_2} {p : α → Prop} (f : α → β) (l : List α) (H : ∀ (a : α), a ∈ l → p a) : List.map (fun (i : { x : α // p x }) => f i.val) (l.attachWith p H) = List.map f l

@[simp]

theorem List.attachWith_map_subtype_val {α : Type u_1} {p : α → Prop} (l : List α) (H : ∀ (a : α), a ∈ l → p a) : List.map Subtype.val (l.attachWith p H) = l

@[simp]

theorem List.mem_attach {α : Type u_1} (l : List α) (x : { x : α // x ∈ l }) : x ∈ l.attach

@[simp]

theorem List.mem_pmap {α : Type u_1} {β : Type u_2} {p : α → Prop} {f : (a : α) → p a → β} {l : List α} {H : ∀ (a : α), a ∈ l → p a} {b : β} : b ∈ List.pmap f l H ↔ ∃ (a : α), ∃ (h : a ∈ l), f a ⋯ = b

theorem List.mem_pmap_of_mem {α : Type u_1} {β : Type u_2} {p : α → Prop} {f : (a : α) → p a → β} {l : List α} {H : ∀ (a : α), a ∈ l → p a} {a : α} (h : a ∈ l) : f a ⋯ ∈ List.pmap f l H

@[simp]

theorem List.length_pmap {α : Type u_1} {β : Type u_2} {p : α → Prop} {f : (a : α) → p a → β} {l : List α} {H : ∀ (a : α), a ∈ l → p a} : (List.pmap f l H).length = l.length

@[simp]

theorem List.length_attach {α : Type u_1} {L : List α} : L.attach.length = L.length

@[simp]

theorem List.length_attachWith {α : Type u_1} {p : α → Prop} {l : List α} {H : ∀ (x : α), x ∈ l → p x} : (l.attachWith p H).length = l.length

@[simp]

theorem List.pmap_eq_nil_iff {α : Type u_1} {β : Type u_2} {p : α → Prop} {f : (a : α) → p a → β} {l : List α} {H : ∀ (a : α), a ∈ l → p a} : List.pmap f l H = [] ↔ l = []

theorem List.pmap_ne_nil_iff {α : Type u_1} {β : Type u_2} {P : α → Prop} (f : (a : α) → P a → β) {xs : List α} (H : ∀ (a : α), a ∈ xs → P a) : List.pmap f xs H ≠ [] ↔ xs ≠ []

theorem List.pmap_eq_self {α : Type u_1} {l : List α} {p : α → Prop} (hp : ∀ (a : α), a ∈ l → p a) (f : (a : α) → p a → α) : List.pmap f l hp = l ↔ ∀ (a : α) (h : a ∈ l), f a ⋯ = a

@[simp]

theorem List.attach_eq_nil_iff {α : Type u_1} {l : List α} : l.attach = [] ↔ l = []

theorem List.attach_ne_nil_iff {α : Type u_1} {l : List α} : l.attach ≠ [] ↔ l ≠ []

@[simp]

theorem List.attachWith_eq_nil_iff {α : Type u_1} {l : List α} {P : α → Prop} {H : ∀ (a : α), a ∈ l → P a} : l.attachWith P H = [] ↔ l = []

theorem List.attachWith_ne_nil_iff {α : Type u_1} {l : List α} {P : α → Prop} {H : ∀ (a : α), a ∈ l → P a} : l.attachWith P H ≠ [] ↔ l ≠ []

@[reducible, inline, deprecated List.pmap_eq_nil_iff]

abbrev List.pmap_eq_nil {α : Type u_1} {β : Type u_2} {p : α → Prop} {f : (a : α) → p a → β} {l : List α} {H : ∀ (a : α), a ∈ l → p a} : List.pmap f l H = [] ↔ l = []

Equations

• @List.pmap_eq_nil = @List.pmap_eq_nil_iff

@[reducible, inline, deprecated List.pmap_ne_nil_iff]

abbrev List.pmap_ne_nil {α : Type u_1} {β : Type u_2} {P : α → Prop} (f : (a : α) → P a → β) {xs : List α} (H : ∀ (a : α), a ∈ xs → P a) : List.pmap f xs H ≠ [] ↔ xs ≠ []

Equations

• @List.pmap_ne_nil = @List.pmap_ne_nil_iff

@[reducible, inline, deprecated List.attach_eq_nil_iff]

abbrev List.attach_eq_nil {α : Type u_1} {l : List α} : l.attach = [] ↔ l = []

Equations

• @List.attach_eq_nil = @List.attach_eq_nil_iff

@[reducible, inline, deprecated List.attach_ne_nil_iff]

abbrev List.attach_ne_nil {α : Type u_1} {l : List α} : l.attach ≠ [] ↔ l ≠ []

Equations

• @List.attach_ne_nil = @List.attach_ne_nil_iff

@[simp]

theorem List.getElem?_pmap {α : Type u_1} {β : Type u_2} {p : α → Prop} (f : (a : α) → p a → β) {l : List α} (h : ∀ (a : α), a ∈ l → p a) (n : Nat) : (List.pmap f l h)[n]? = Option.pmap f l[n]? ⋯

theorem List.get?_pmap {α : Type u_1} {β : Type u_2} {p : α → Prop} (f : (a : α) → p a → β) {l : List α} (h : ∀ (a : α), a ∈ l → p a) (n : Nat) : (List.pmap f l h).get? n = Option.pmap f (l.get? n) ⋯

@[simp]

theorem List.getElem_pmap {α : Type u_1} {β : Type u_2} {p : α → Prop} (f : (a : α) → p a → β) {l : List α} (h : ∀ (a : α), a ∈ l → p a) {n : Nat} (hn : n < (List.pmap f l h).length) : (List.pmap f l h)[n] = f l[n] ⋯

theorem List.get_pmap {α : Type u_1} {β : Type u_2} {p : α → Prop} (f : (a : α) → p a → β) {l : List α} (h : ∀ (a : α), a ∈ l → p a) {n : Nat} (hn : n < (List.pmap f l h).length) : (List.pmap f l h).get ⟨n, hn⟩ = f (l.get ⟨n, ⋯⟩) ⋯

@[simp]

theorem List.getElem?_attachWith {α : Type u_1} {xs : List α} {i : Nat} {P : α → Prop} {H : ∀ (a : α), a ∈ xs → P a} : (xs.attachWith P H)[i]? = Option.pmap Subtype.mk xs[i]? ⋯

@[simp]

theorem List.getElem?_attach {α : Type u_1} {xs : List α} {i : Nat} : xs.attach[i]? = Option.pmap Subtype.mk xs[i]? ⋯

@[simp]

theorem List.getElem_attachWith {α : Type u_1} {xs : List α} {P : α → Prop} {H : ∀ (a : α), a ∈ xs → P a} {i : Nat} (h : i < (xs.attachWith P H).length) : (xs.attachWith P H)[i] = ⟨xs[i], ⋯⟩

@[simp]

theorem List.getElem_attach {α : Type u_1} {xs : List α} {i : Nat} (h : i < xs.attach.length) : xs.attach[i] = ⟨xs[i], ⋯⟩

@[simp]

theorem List.head?_pmap {α : Type u_1} {β : Type u_2} {P : α → Prop} (f : (a : α) → P a → β) (xs : List α) (H : ∀ (a : α), a ∈ xs → P a) : (List.pmap f xs H).head? = Option.map (fun (x : { x : α // x ∈ xs }) => match x with | ⟨a, m⟩ => f a ⋯) xs.attach.head?

@[simp]

theorem List.head_pmap {α : Type u_1} {β : Type u_2} {P : α → Prop} (f : (a : α) → P a → β) (xs : List α) (H : ∀ (a : α), a ∈ xs → P a) (h : List.pmap f xs H ≠ []) : (List.pmap f xs H).head h = f (xs.head ⋯) ⋯

@[simp]

theorem List.head?_attachWith {α : Type u_1} {P : α → Prop} {xs : List α} (H : ∀ (a : α), a ∈ xs → P a) : (xs.attachWith P H).head? = xs.head?.pbind fun (a : α) (h : a ∈ xs.head?) => some ⟨a, ⋯⟩

@[simp]

theorem List.head_attachWith {α : Type u_1} {P : α → Prop} {xs : List α} {H : ∀ (a : α), a ∈ xs → P a} (h : xs.attachWith P H ≠ []) : (xs.attachWith P H).head h = ⟨xs.head ⋯, ⋯⟩

@[simp]

theorem List.head?_attach {α : Type u_1} (xs : List α) : xs.attach.head? = xs.head?.pbind fun (a : α) (h : a ∈ xs.head?) => some ⟨a, ⋯⟩

@[simp]

theorem List.head_attach {α : Type u_1} {xs : List α} (h : xs.attach ≠ []) : xs.attach.head h = ⟨xs.head ⋯, ⋯⟩

@[simp]

theorem List.tail_pmap {α : Type u_1} {β : Type u_2} {P : α → Prop} (f : (a : α) → P a → β) (xs : List α) (H : ∀ (a : α), a ∈ xs → P a) : (List.pmap f xs H).tail = List.pmap f xs.tail ⋯

@[simp]

theorem List.tail_attachWith {α : Type u_1} {P : α → Prop} {xs : List α} {H : ∀ (a : α), a ∈ xs → P a} : (xs.attachWith P H).tail = xs.tail.attachWith P ⋯

@[simp]

theorem List.tail_attach {α : Type u_1} (xs : List α) : xs.attach.tail = List.map (fun (x : { x : α // x ∈ xs.tail }) => match x with | ⟨x, h⟩ => ⟨x, ⋯⟩) xs.tail.attach

theorem List.foldl_pmap {α : Type u_1} {β : Type u_2} {γ : Type u_3} (l : List α) {P : α → Prop} (f : (a : α) → P a → β) (H : ∀ (a : α), a ∈ l → P a) (g : γ → β → γ) (x : γ) : List.foldl g x (List.pmap f l H) = List.foldl (fun (acc : γ) (a : { x : α // x ∈ l }) => g acc (f a.val ⋯)) x l.attach

theorem List.foldr_pmap {α : Type u_1} {β : Type u_2} {γ : Type u_3} (l : List α) {P : α → Prop} (f : (a : α) → P a → β) (H : ∀ (a : α), a ∈ l → P a) (g : β → γ → γ) (x : γ) : List.foldr g x (List.pmap f l H) = List.foldr (fun (a : { x : α // x ∈ l }) (acc : γ) => g (f a.val ⋯) acc) x l.attach

theorem List.foldl_attach {α : Type u_1} {β : Type u_2} (l : List α) (f : β → α → β) (b : β) : List.foldl (fun (acc : β) (t : { x : α // x ∈ l }) => f acc t.val) b l.attach = List.foldl f b l

If we fold over l.attach with a function that ignores the membership predicate, we get the same results as folding over l directly.

This is useful when we need to use attach to show termination.

Unfortunately this can't be applied by simp because of the higher order unification problem, and even when rewriting we need to specify the function explicitly. See however foldl_subtype below.

theorem List.foldr_attach {α : Type u_1} {β : Type u_2} (l : List α) (f : α → β → β) (b : β) : List.foldr (fun (t : { x : α // x ∈ l }) (acc : β) => f t.val acc) b l.attach = List.foldr f b l

If we fold over l.attach with a function that ignores the membership predicate, we get the same results as folding over l directly.

This is useful when we need to use attach to show termination.

Unfortunately this can't be applied by simp because of the higher order unification problem, and even when rewriting we need to specify the function explicitly. See however foldr_subtype below.

theorem List.attach_map {α : Type u_1} {β : Type u_2} {l : List α} (f : α → β) : (List.map f l).attach = List.map (fun (x : { x : α // x ∈ l }) => match x with | ⟨x, h⟩ => ⟨f x, ⋯⟩) l.attach

theorem List.attachWith_map {α : Type u_1} {β : Type u_2} {l : List α} (f : α → β) {P : β → Prop} {H : ∀ (b : β), b ∈ List.map f l → P b} : (List.map f l).attachWith P H = List.map (fun (x : { x : α // (P ∘ f) x }) => match x with | ⟨x, h⟩ => ⟨f x, h⟩) (l.attachWith (P ∘ f) ⋯)

theorem List.map_attachWith {α : Type u_1} {β : Type u_2} {l : List α} {P : α → Prop} {H : ∀ (a : α), a ∈ l → P a} (f : { x : α // P x } → β) : List.map f (l.attachWith P H) = List.pmap (fun (a : α) (h : a ∈ l ∧ P a) => f ⟨a, ⋯⟩) l ⋯

theorem List.map_attach {α : Type u_1} {β : Type u_2} {l : List α} (f : { x : α // x ∈ l } → β) : List.map f l.attach = List.pmap (fun (a : α) (h : a ∈ l) => f ⟨a, h⟩) l ⋯

See also pmap_eq_map_attach for writing pmap in terms of map and attach.

theorem List.attach_filterMap {α : Type u_1} {β : Type u_2} {l : List α} {f : α → Option β} : (List.filterMap f l).attach = List.filterMap (fun (x : { x : α // x ∈ l }) => match x with | ⟨x, h⟩ => (f x).pbind fun (b : β) (m : b ∈ f x) => some ⟨b, ⋯⟩) l.attach

theorem List.attach_filter {α : Type u_1} {l : List α} (p : α → Bool) : (List.filter p l).attach = List.filterMap (fun (x : { x : α // x ∈ l }) => if w : p x.val = true then some ⟨x.val, ⋯⟩ else none) l.attach

theorem List.pmap_pmap {α : Type u_1} {β : Type u_2} {γ : Type u_3} {p : α → Prop} {q : β → Prop} (g : (a : α) → p a → β) (f : (b : β) → q b → γ) (l : List α) (H₁ : ∀ (a : α), a ∈ l → p a) (H₂ : ∀ (a : β), a ∈ List.pmap g l H₁ → q a) : List.pmap f (List.pmap g l H₁) H₂ = List.pmap (fun (a : { x : α // x ∈ l }) (h : p a.val) => f (g a.val h) ⋯) l.attach ⋯

@[simp]

theorem List.pmap_append {ι : Type u_1} {α : Type u_2} {p : ι → Prop} (f : (a : ι) → p a → α) (l₁ l₂ : List ι) (h : ∀ (a : ι), a ∈ l₁ ++ l₂ → p a) : List.pmap f (l₁ ++ l₂) h = List.pmap f l₁ ⋯ ++ List.pmap f l₂ ⋯

theorem List.pmap_append' {α : Type u_1} {β : Type u_2} {p : α → Prop} (f : (a : α) → p a → β) (l₁ l₂ : List α) (h₁ : ∀ (a : α), a ∈ l₁ → p a) (h₂ : ∀ (a : α), a ∈ l₂ → p a) : List.pmap f (l₁ ++ l₂) ⋯ = List.pmap f l₁ h₁ ++ List.pmap f l₂ h₂

@[simp]

theorem List.attach_append {α : Type u_1} (xs ys : List α) : (xs ++ ys).attach = List.map (fun (x : { x : α // x ∈ xs }) => match x with | ⟨x, h⟩ => ⟨x, ⋯⟩) xs.attach ++ List.map (fun (x : { x : α // x ∈ ys }) => match x with | ⟨x, h⟩ => ⟨x, ⋯⟩) ys.attach

@[simp]

theorem List.attachWith_append {α : Type u_1} {P : α → Prop} {xs ys : List α} {H : ∀ (a : α), a ∈ xs ++ ys → P a} : (xs ++ ys).attachWith P H = xs.attachWith P ⋯ ++ ys.attachWith P ⋯

@[simp]

theorem List.pmap_reverse {α : Type u_1} {β : Type u_2} {P : α → Prop} (f : (a : α) → P a → β) (xs : List α) (H : ∀ (a : α), a ∈ xs.reverse → P a) : List.pmap f xs.reverse H = (List.pmap f xs ⋯).reverse

theorem List.reverse_pmap {α : Type u_1} {β : Type u_2} {P : α → Prop} (f : (a : α) → P a → β) (xs : List α) (H : ∀ (a : α), a ∈ xs → P a) : (List.pmap f xs H).reverse = List.pmap f xs.reverse ⋯

@[simp]

theorem List.attachWith_reverse {α : Type u_1} {P : α → Prop} {xs : List α} {H : ∀ (a : α), a ∈ xs.reverse → P a} : xs.reverse.attachWith P H = (xs.attachWith P ⋯).reverse

theorem List.reverse_attachWith {α : Type u_1} {P : α → Prop} {xs : List α} {H : ∀ (a : α), a ∈ xs → P a} : (xs.attachWith P H).reverse = xs.reverse.attachWith P ⋯

@[simp]

theorem List.attach_reverse {α : Type u_1} (xs : List α) : xs.reverse.attach = List.map (fun (x : { x : α // x ∈ xs }) => match x with | ⟨x, h⟩ => ⟨x, ⋯⟩) xs.attach.reverse

theorem List.reverse_attach {α : Type u_1} (xs : List α) : xs.attach.reverse = List.map (fun (x : { x : α // x ∈ xs.reverse }) => match x with | ⟨x, h⟩ => ⟨x, ⋯⟩) xs.reverse.attach

@[simp]

theorem List.getLast?_pmap {α : Type u_1} {β : Type u_2} {P : α → Prop} (f : (a : α) → P a → β) (xs : List α) (H : ∀ (a : α), a ∈ xs → P a) : (List.pmap f xs H).getLast? = Option.map (fun (x : { x : α // x ∈ xs }) => match x with | ⟨a, m⟩ => f a ⋯) xs.attach.getLast?

@[simp]

theorem List.getLast_pmap {α : Type u_1} {β : Type u_2} {P : α → Prop} (f : (a : α) → P a → β) (xs : List α) (H : ∀ (a : α), a ∈ xs → P a) (h : List.pmap f xs H ≠ []) : (List.pmap f xs H).getLast h = f (xs.getLast ⋯) ⋯

@[simp]

theorem List.getLast?_attachWith {α : Type u_1} {P : α → Prop} {xs : List α} {H : ∀ (a : α), a ∈ xs → P a} : (xs.attachWith P H).getLast? = xs.getLast?.pbind fun (a : α) (h : a ∈ xs.getLast?) => some ⟨a, ⋯⟩

@[simp]

theorem List.getLast_attachWith {α : Type u_1} {P : α → Prop} {xs : List α} {H : ∀ (a : α), a ∈ xs → P a} (h : xs.attachWith P H ≠ []) : (xs.attachWith P H).getLast h = ⟨xs.getLast ⋯, ⋯⟩

@[simp]

theorem List.getLast?_attach {α : Type u_1} {xs : List α} : xs.attach.getLast? = xs.getLast?.pbind fun (a : α) (h : a ∈ xs.getLast?) => some ⟨a, ⋯⟩

@[simp]

theorem List.getLast_attach {α : Type u_1} {xs : List α} (h : xs.attach ≠ []) : xs.attach.getLast h = ⟨xs.getLast ⋯, ⋯⟩

@[simp]

theorem List.countP_attach {α : Type u_1} (l : List α) (p : α → Bool) : List.countP (fun (a : { x : α // x ∈ l }) => p a.val) l.attach = List.countP p l

@[simp]

theorem List.countP_attachWith {α : Type u_1} {p : α → Prop} (l : List α) (H : ∀ (a : α), a ∈ l → p a) (q : α → Bool) : List.countP (fun (a : { x : α // p x }) => q a.val) (l.attachWith p H) = List.countP q l

@[simp]

theorem List.count_attach {α : Type u_1} [DecidableEq α] (l : List α) (a : { x : α // x ∈ l }) : List.count a l.attach = List.count a.val l

@[simp]

theorem List.count_attachWith {α : Type u_1} [DecidableEq α] {p : α → Prop} (l : List α) (H : ∀ (a : α), a ∈ l → p a) (a : { x : α // p x }) : List.count a (l.attachWith p H) = List.count a.val l

unattach

List.unattach is the (one-sided) inverse of List.attach. It is a synonym for List.map Subtype.val.

We use it by providing a simp lemma l.attach.unattach = l, and simp lemmas which recognize higher order functions applied to l : List { x // p x } which only depend on the value, not the predicate, and rewrite these in terms of a simpler function applied to l.unattach.

Further, we provide simp lemmas that push unattach inwards.

def List.unattach {α : Type u_1} {p : α → Prop} (l : List { x : α // p x }) : List α

A synonym for l.map (·.val). Mostly this should not be needed by users. It is introduced as an intermediate step by lemmas such as map_subtype, and is ideally subsequently simplified away by unattach_attach.

If not, usually the right approach is simp [List.unattach, -List.map_subtype] to unfold.

Equations

• l.unattach = List.map (fun (x : { x : α // p x }) => x.val) l

@[simp]

theorem List.unattach_nil {α : Type u_1} {p : α → Prop} : [].unattach = []

@[simp]

theorem List.unattach_cons {α : Type u_1} {p : α → Prop} {a : { x : α // p x }} {l : List { x : α // p x }} : (a :: l).unattach = a.val :: l.unattach

@[simp]

theorem List.length_unattach {α : Type u_1} {p : α → Prop} {l : List { x : α // p x }} : l.unattach.length = l.length

@[simp]

theorem List.unattach_attach {α : Type u_1} {l : List α} : l.attach.unattach = l

@[simp]

theorem List.unattach_attachWith {α : Type u_1} {p : α → Prop} {l : List α} {H : ∀ (a : α), a ∈ l → p a} : (l.attachWith p H).unattach = l

@[simp]

theorem List.getElem?_unattach {α : Type u_1} {p : α → Prop} {l : List { x : α // p x }} (i : Nat) : l.unattach[i]? = Option.map Subtype.val l[i]?

@[simp]

theorem List.getElem_unattach {α : Type u_1} {p : α → Prop} {l : List { x : α // p x }} (i : Nat) (h : i < l.unattach.length) : l.unattach[i] = l[i].val

Recognizing higher order functions on subtypes using a function that only depends on the value.

@[simp]

theorem List.foldl_subtype {α : Type u_1} {β : Type u_2} {p : α → Prop} {l : List { x : α // p x }} {f : β → { x : α // p x } → β} {g : β → α → β} {x : β} {hf : ∀ (b : β) (x : α) (h : p x), f b ⟨x, h⟩ = g b x} : List.foldl f x l = List.foldl g x l.unattach

This lemma identifies folds over lists of subtypes, where the function only depends on the value, not the proposition, and simplifies these to the function directly taking the value.

@[simp]

theorem List.foldr_subtype {α : Type u_1} {β : Type u_2} {p : α → Prop} {l : List { x : α // p x }} {f : { x : α // p x } → β → β} {g : α → β → β} {x : β} {hf : ∀ (x : α) (h : p x) (b : β), f ⟨x, h⟩ b = g x b} : List.foldr f x l = List.foldr g x l.unattach

This lemma identifies folds over lists of subtypes, where the function only depends on the value, not the proposition, and simplifies these to the function directly taking the value.

@[simp]

theorem List.map_subtype {α : Type u_1} {β : Type u_2} {p : α → Prop} {l : List { x : α // p x }} {f : { x : α // p x } → β} {g : α → β} {hf : ∀ (x : α) (h : p x), f ⟨x, h⟩ = g x} : List.map f l = List.map g l.unattach

This lemma identifies maps over lists of subtypes, where the function only depends on the value, not the proposition, and simplifies these to the function directly taking the value.

@[simp]

theorem List.filterMap_subtype {α : Type u_1} {β : Type u_2} {p : α → Prop} {l : List { x : α // p x }} {f : { x : α // p x } → Option β} {g : α → Option β} {hf : ∀ (x : α) (h : p x), f ⟨x, h⟩ = g x} : List.filterMap f l = List.filterMap g l.unattach

@[simp]

theorem List.flatMap_subtype {α : Type u_1} {β : Type u_2} {p : α → Prop} {l : List { x : α // p x }} {f : { x : α // p x } → List β} {g : α → List β} {hf : ∀ (x : α) (h : p x), f ⟨x, h⟩ = g x} : l.flatMap f = l.unattach.flatMap g

@[reducible, inline, deprecated List.flatMap_subtype]

abbrev List.bind_subtype {α : Type u_1} {β : Type u_2} {p : α → Prop} {l : List { x : α // p x }} {f : { x : α // p x } → List β} {g : α → List β} {hf : ∀ (x : α) (h : p x), f ⟨x, h⟩ = g x} : l.flatMap f = l.unattach.flatMap g

Equations

• @List.bind_subtype = @List.flatMap_subtype

@[simp]

theorem List.unattach_filter {α : Type u_1} {p : α → Prop} {l : List { x : α // p x }} {f : { x : α // p x } → Bool} {g : α → Bool} {hf : ∀ (x : α) (h : p x), f ⟨x, h⟩ = g x} : (List.filter f l).unattach = List.filter g l.unattach

Simp lemmas pushing unattach inwards.

@[simp]

theorem List.unattach_reverse {α : Type u_1} {p : α → Prop} {l : List { x : α // p x }} : l.reverse.unattach = l.unattach.reverse

@[simp]

theorem List.unattach_append {α : Type u_1} {p : α → Prop} {l₁ l₂ : List { x : α // p x }} : (l₁ ++ l₂).unattach = l₁.unattach ++ l₂.unattach

@[simp]

theorem List.unattach_flatten {α : Type u_1} {p : α → Prop} {l : List (List { x : α // p x })} : l.flatten.unattach = (List.map List.unattach l).flatten

@[reducible, inline, deprecated List.unattach_flatten]

abbrev List.unattach_join {α : Type u_1} {p : α → Prop} {l : List (List { x : α // p x })} : l.flatten.unattach = (List.map List.unattach l).flatten

Equations

• @List.unattach_join = @List.unattach_flatten

@[simp]

theorem List.unattach_replicate {α : Type u_1} {p : α → Prop} {n : Nat} {x : { x : α // p x }} : (List.replicate n x).unattach = List.replicate n x.val