mathlib documentation

data.fintype.basic

@[class]

structure fintype :

Type u_4 → Type u_4

elems : finset α
complete : ∀ (x : α), x ∈ fintype.elems α

fintype α means that α is finite, i.e. there are only finitely many distinct elements of type α. The evidence of this is a finset elems (a list up to permutation without duplicates), together with a proof that everything of type α is in the list.

Instances

def finset.univ {α : Type u_1} [fintype α] :

univ is the universal finite set of type finset α implied from the assumption fintype α.

Equations

finset.univ = fintype.elems α

@[simp]

theorem finset.mem_univ {α : Type u_1} [fintype α] (x : α) :

x ∈ finset.univ

@[simp]

theorem finset.mem_univ_val {α : Type u_1} [fintype α] (x : α) :

x ∈ finset.univ.val

@[simp]

theorem finset.coe_univ {α : Type u_1} [fintype α] :

↑finset.univ = set.univ

theorem finset.univ_nonempty_iff {α : Type u_1} [fintype α] :

finset.univ.nonempty ↔ nonempty α

theorem finset.univ_nonempty {α : Type u_1} [fintype α] [nonempty α] :

finset.univ.nonempty

theorem finset.univ_eq_empty {α : Type u_1} [fintype α] :

finset.univ = ∅ ↔ ¬nonempty α

theorem finset.subset_univ {α : Type u_1} [fintype α] (s : finset α) :

s ⊆ finset.univ

@[instance]

def finset.order_top {α : Type u_1} [fintype α] :

order_top (finset α)

Equations

finset.order_top = {top := finset.univ _inst_1, le := partial_order.le finset.partial_order, lt := partial_order.lt finset.partial_order, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_top := _}

@[instance]

def finset.boolean_algebra {α : Type u_1} [fintype α] [decidable_eq α] :

boolean_algebra (finset α)

Equations

finset.boolean_algebra = {sup := distrib_lattice.sup finset.distrib_lattice, le := distrib_lattice.le finset.distrib_lattice, lt := distrib_lattice.lt finset.distrib_lattice, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_sup_left := _, le_sup_right := _, sup_le := _, inf := distrib_lattice.inf finset.distrib_lattice, inf_le_left := _, inf_le_right := _, le_inf := _, le_sup_inf := _, top := order_top.top finset.order_top, le_top := _, bot := semilattice_inf_bot.bot finset.semilattice_inf_bot, bot_le := _, compl := λ (s : finset α), finset.univ \ s, sdiff := has_sdiff.sdiff finset.has_sdiff, inf_compl_le_bot := _, top_le_sup_compl := _, sdiff_eq := _}

theorem finset.compl_eq_univ_sdiff {α : Type u_1} [fintype α] [decidable_eq α] (s : finset α) :

sᶜ = finset.univ \ s

@[simp]

theorem finset.mem_compl {α : Type u_1} [fintype α] [decidable_eq α] {s : finset α} {x : α} :

x ∈ sᶜ ↔ x ∉ s

@[simp]

theorem finset.coe_compl {α : Type u_1} [fintype α] [decidable_eq α] (s : finset α) :

↑sᶜ = (↑s)ᶜ

theorem finset.eq_univ_iff_forall {α : Type u_1} [fintype α] {s : finset α} :

s = finset.univ ↔ ∀ (x : α), x ∈ s

@[simp]

theorem finset.univ_inter {α : Type u_1} [fintype α] [decidable_eq α] (s : finset α) :

finset.univ ∩ s = s

@[simp]

theorem finset.inter_univ {α : Type u_1} [fintype α] [decidable_eq α] (s : finset α) :

s ∩ finset.univ = s

@[simp]

theorem finset.piecewise_univ {α : Type u_1} [fintype α] [Π (i : α), decidable (i ∈ finset.univ)] {δ : α → Sort u_2} (f g : Π (i : α), δ i) :

finset.univ.piecewise f g = f

theorem finset.univ_map_equiv_to_embedding {α : Type u_1} {β : Type u_2} [fintype α] [fintype β] (e : α ≃ β) :

finset.map e.to_embedding finset.univ = finset.univ

@[instance]

def fintype.decidable_pi_fintype {α : Type u_1} {β : α → Type u_2} [Π (a : α), decidable_eq (β a)] [fintype α] :

decidable_eq (Π (a : α), β a)

Equations

fintype.decidable_pi_fintype = λ (f g : Π (a : α), β a), decidable_of_iff (∀ (a : α), a ∈ fintype.elems α → f a = g a) _

@[instance]

def fintype.decidable_forall_fintype {α : Type u_1} {p : α → Prop} [decidable_pred p] [fintype α] :

decidable (∀ (a : α), p a)

Equations

fintype.decidable_forall_fintype = decidable_of_iff (∀ (a : α), a ∈ finset.univ → p a) fintype.decidable_forall_fintype._proof_1

@[instance]

def fintype.decidable_exists_fintype {α : Type u_1} {p : α → Prop} [decidable_pred p] [fintype α] :

decidable (∃ (a : α), p a)

Equations

fintype.decidable_exists_fintype = decidable_of_iff (∃ (a : α) (H : a ∈ finset.univ), p a) fintype.decidable_exists_fintype._proof_1

@[instance]

def fintype.decidable_eq_equiv_fintype {α : Type u_1} {β : Type u_2} [decidable_eq β] [fintype α] :

decidable_eq (α ≃ β)

Equations

fintype.decidable_eq_equiv_fintype = λ (a b : α ≃ β), decidable_of_iff (a.to_fun = b.to_fun) _

@[instance]

def fintype.decidable_injective_fintype {α : Type u_1} {β : Type u_2} [decidable_eq α] [decidable_eq β] [fintype α] :

decidable_pred function.injective

Equations

fintype.decidable_injective_fintype = λ (x : α → β), _.mpr fintype.decidable_forall_fintype

@[instance]

def fintype.decidable_surjective_fintype {α : Type u_1} {β : Type u_2} [decidable_eq β] [fintype α] [fintype β] :

decidable_pred function.surjective

Equations

fintype.decidable_surjective_fintype = λ (x : α → β), _.mpr fintype.decidable_forall_fintype

@[instance]

def fintype.decidable_bijective_fintype {α : Type u_1} {β : Type u_2} [decidable_eq α] [decidable_eq β] [fintype α] [fintype β] :

decidable_pred function.bijective

Equations

fintype.decidable_bijective_fintype = λ (x : α → β), _.mpr and.decidable

@[instance]

def fintype.decidable_left_inverse_fintype {α : Type u_1} {β : Type u_2} [decidable_eq α] [fintype α] (f : α → β) (g : β → α) :

decidable (function.right_inverse f g)

Equations

fintype.decidable_left_inverse_fintype f g = show decidable (∀ (x : α), g (f x) = x), from fintype.decidable_forall_fintype

@[instance]

def fintype.decidable_right_inverse_fintype {α : Type u_1} {β : Type u_2} [decidable_eq β] [fintype β] (f : α → β) (g : β → α) :

decidable (function.left_inverse f g)

Equations

fintype.decidable_right_inverse_fintype f g = show decidable (∀ (x : β), f (g x) = x), from fintype.decidable_forall_fintype

def fintype.of_multiset {α : Type u_1} [decidable_eq α] (s : multiset α) :

(∀ (x : α), x ∈ s) → fintype α

Construct a proof of fintype α from a universal multiset

Equations

fintype.of_multiset s H = {elems := s.to_finset, complete := _}

def fintype.of_list {α : Type u_1} [decidable_eq α] (l : list α) :

(∀ (x : α), x ∈ l) → fintype α

Construct a proof of fintype α from a universal list

Equations

fintype.of_list l H = {elems := l.to_finset, complete := _}

theorem fintype.exists_univ_list (α : Type u_1) [fintype α] :

∃ (l : list α), l.nodup ∧ ∀ (x : α), x ∈ l

def fintype.card (α : Type u_1) [fintype α] :

card α is the number of elements in α, defined when α is a fintype.

Equations

fintype.card α = finset.univ.card

def fintype.equiv_fin_of_forall_mem_list {α : Type u_1} [decidable_eq α] {l : list α} :

(∀ (x : α), x ∈ l) → l.nodup → α ≃ fin l.length

If l lists all the elements of α without duplicates, then α ≃ fin (l.length).

Equations

fintype.equiv_fin_of_forall_mem_list h nd = {to_fun := λ (a : α), ⟨list.index_of a l, _⟩, inv_fun := λ (i : fin l.length), l.nth_le i.val _, left_inv := _, right_inv := _}

def fintype.equiv_fin (α : Type u_1) [decidable_eq α] [fintype α] :

trunc (α ≃ fin (fintype.card α))

There is (computably) a bijection between α and fin n where n = card α. Since it is not unique, and depends on which permutation of the universe list is used, the bijection is wrapped in trunc to preserve computability.

Equations

fintype.equiv_fin α = _.mpr (quot.rec_on_subsingleton finset.univ.val (λ (l : list α) (h : ∀ (x : α), x ∈ l) (nd : l.nodup), trunc.mk (fintype.equiv_fin_of_forall_mem_list h nd)) finset.mem_univ_val _)

theorem fintype.exists_equiv_fin (α : Type u_1) [fintype α] :

∃ (n : ℕ), nonempty (α ≃ fin n)

@[instance]

def fintype.subsingleton (α : Type u_1) :

subsingleton (fintype α)

def fintype.subtype {α : Type u_1} {p : α → Prop} (s : finset α) :

(∀ (x : α), x ∈ s ↔ p x) → fintype {x // p x}

Given a predicate that can be represented by a finset, the subtype associated to the predicate is a fintype.

Equations

fintype.subtype s H = {elems := {val := multiset.pmap subtype.mk s.val _, nodup := _}, complete := _}

theorem fintype.subtype_card {α : Type u_1} {p : α → Prop} (s : finset α) (H : ∀ (x : α), x ∈ s ↔ p x) :

fintype.card {x // p x} = s.card

theorem fintype.card_of_subtype {α : Type u_1} {p : α → Prop} (s : finset α) (H : ∀ (x : α), x ∈ s ↔ p x) [fintype {x // p x}] :

fintype.card {x // p x} = s.card

def fintype.of_finset {α : Type u_1} {p : set α} (s : finset α) :

(∀ (x : α), x ∈ s ↔ x ∈ p) → fintype ↥p

Construct a fintype from a finset with the same elements.

Equations

fintype.of_finset s H = fintype.subtype s H

@[simp]

theorem fintype.card_of_finset {α : Type u_1} {p : set α} (s : finset α) (H : ∀ (x : α), x ∈ s ↔ x ∈ p) :

fintype.card ↥p = s.card

theorem fintype.card_of_finset' {α : Type u_1} {p : set α} (s : finset α) (H : ∀ (x : α), x ∈ s ↔ x ∈ p) [fintype ↥p] :

fintype.card ↥p = s.card

def fintype.of_bijective {α : Type u_1} {β : Type u_2} [fintype α] (f : α → β) :

function.bijective f → fintype β

If f : α → β is a bijection and α is a fintype, then β is also a fintype.

Equations

fintype.of_bijective f H = {elems := finset.map {to_fun := f, inj' := _} finset.univ, complete := _}

def fintype.of_surjective {α : Type u_1} {β : Type u_2} [decidable_eq β] [fintype α] (f : α → β) :

function.surjective f → fintype β

If f : α → β is a surjection and α is a fintype, then β is also a fintype.

Equations

fintype.of_surjective f H = {elems := finset.image f finset.univ, complete := _}

def fintype.of_injective {α : Type u_1} {β : Type u_2} [fintype β] (f : α → β) :

function.injective f → fintype α

Given an injective function to a fintype, the domain is also a fintype. This is noncomputable because injectivity alone cannot be used to construct preimages.

Equations

fintype.of_injective f H = let _inst : Π (p : Prop), decidable p := classical.dec in dite (nonempty α) (λ (hα : nonempty α), let _inst_2 : inhabited α := classical.inhabited_of_nonempty hα in fintype.of_surjective (function.inv_fun f) _) (λ (hα : ¬nonempty α), {elems := ∅, complete := _})

def fintype.of_equiv {β : Type u_2} (α : Type u_1) [fintype α] :

α ≃ β → fintype β

If f : α ≃ β and α is a fintype, then β is also a fintype.

Equations

fintype.of_equiv α f = fintype.of_bijective ⇑f _

theorem fintype.of_equiv_card {α : Type u_1} {β : Type u_2} [fintype α] (f : α ≃ β) :

fintype.card β = fintype.card α

theorem fintype.card_congr {α : Type u_1} {β : Type u_2} [fintype α] [fintype β] :

α ≃ β → fintype.card α = fintype.card β

theorem fintype.card_eq {α : Type u_1} {β : Type u_2} [F : fintype α] [G : fintype β] :

fintype.card α = fintype.card β ↔ nonempty (α ≃ β)

def fintype.of_subsingleton {α : Type u_1} (a : α) [subsingleton α] :

Subsingleton types are fintypes (with zero or one terms).

Equations

fintype.of_subsingleton a = {elems := {a}, complete := _}

@[simp]

theorem fintype.univ_of_subsingleton {α : Type u_1} (a : α) [subsingleton α] :

finset.univ = {a}

@[simp]

theorem fintype.card_of_subsingleton {α : Type u_1} (a : α) [subsingleton α] :

fintype.card α = 1

def set.to_finset {α : Type u_1} (s : set α) [fintype ↥s] :

Construct a finset enumerating a set s, given a fintype instance.

Equations

s.to_finset = {val := multiset.map subtype.val finset.univ.val, nodup := _}

@[simp]

theorem set.mem_to_finset {α : Type u_1} {s : set α} [fintype ↥s] {a : α} :

a ∈ s.to_finset ↔ a ∈ s

@[simp]

theorem set.mem_to_finset_val {α : Type u_1} {s : set α} [fintype ↥s] {a : α} :

a ∈ s.to_finset.val ↔ a ∈ s

@[simp]

theorem set.to_finset_card {α : Type u_1} (s : set α) [fintype ↥s] :

s.to_finset.card = fintype.card ↥s

@[simp]

theorem set.coe_to_finset {α : Type u_1} (s : set α) [fintype ↥s] :

↑(s.to_finset) = s

@[simp]

theorem set.to_finset_inj {α : Type u_1} {s t : set α} [fintype ↥s] [fintype ↥t] :

s.to_finset = t.to_finset ↔ s = t

theorem finset.card_univ {α : Type u_1} [fintype α] :

finset.univ.card = fintype.card α

theorem finset.eq_univ_of_card {α : Type u_1} [fintype α] (s : finset α) :

s.card = fintype.card α → s = finset.univ

theorem finset.card_le_univ {α : Type u_1} [fintype α] (s : finset α) :

s.card ≤ fintype.card α

theorem finset.card_univ_diff {α : Type u_1} [decidable_eq α] [fintype α] (s : finset α) :

(finset.univ \ s).card = fintype.card α - s.card

theorem finset.card_compl {α : Type u_1} [decidable_eq α] [fintype α] (s : finset α) :

sᶜ.card = fintype.card α - s.card

@[instance]

def fin.fintype (n : ℕ) :

fintype (fin n)

Equations

fin.fintype n = {elems := finset.fin_range n, complete := _}

theorem fin.univ_def (n : ℕ) :

finset.univ = finset.fin_range n

@[simp]

theorem fintype.card_fin (n : ℕ) :

fintype.card (fin n) = n

@[simp]

theorem finset.card_fin (n : ℕ) :

finset.univ.card = n

theorem fin.equiv_iff_eq {m n : ℕ} :

nonempty (fin m ≃ fin n) ↔ m = n

theorem fin.univ_succ (n : ℕ) :

finset.univ = insert 0 (finset.image fin.succ finset.univ)

Embed fin n into fin (n + 1) by prepending zero to the univ

theorem fin.univ_cast_succ (n : ℕ) :

finset.univ = insert (fin.last n) (finset.image fin.cast_succ finset.univ)

Embed fin n into fin (n + 1) by appending a new fin.last n to the univ

theorem fin.univ_succ_above (n : ℕ) (p : fin (n + 1)) :

finset.univ = insert p (finset.image p.succ_above finset.univ)

Embed fin n into fin (n + 1) by inserting around a specified pivot p : fin (n + 1) into the univ

@[instance]

def unique.fintype {α : Type u_1} [unique α] :

Equations

unique.fintype = fintype.of_subsingleton (default α)

@[simp]

theorem univ_unique {α : Type u_1} [unique α] [f : fintype α] :

finset.univ = {default α}

@[instance]

def empty.fintype :

Equations

empty.fintype = {elems := ∅, complete := empty.fintype._proof_1}

@[simp]

theorem fintype.univ_empty :

finset.univ = ∅

@[simp]

theorem fintype.card_empty :

fintype.card empty = 0

@[instance]

def pempty.fintype :

Equations

pempty.fintype = {elems := ∅, complete := pempty.fintype._proof_1}

@[simp]

theorem fintype.univ_pempty :

finset.univ = ∅

@[simp]

theorem fintype.card_pempty :

fintype.card pempty = 0

@[instance]

def unit.fintype :

Equations

unit.fintype = fintype.of_subsingleton ()

theorem fintype.univ_unit :

finset.univ = {()}

theorem fintype.card_unit :

fintype.card unit = 1

@[instance]

def punit.fintype :

Equations

punit.fintype = fintype.of_subsingleton punit.star

@[simp]

theorem fintype.univ_punit :

finset.univ = {punit.star}

@[simp]

theorem fintype.card_punit :

fintype.card punit = 1

@[instance]

def bool.fintype :

Equations

bool.fintype = {elems := {val := tt ::ₘ ff ::ₘ 0, nodup := bool.fintype._proof_1}, complete := bool.fintype._proof_2}

@[simp]

theorem fintype.univ_bool :

finset.univ = {tt, ff}

@[instance]

def units_int.fintype :

fintype (units ℤ)

Equations

units_int.fintype = {elems := {1, -1}, complete := units_int.fintype._proof_1}

@[instance]

def additive.fintype {α : Type u_1} [fintype α] :

fintype (additive α)

Equations

additive.fintype = id

@[instance]

def multiplicative.fintype {α : Type u_1} [fintype α] :

fintype (multiplicative α)

Equations

multiplicative.fintype = id

@[simp]

theorem fintype.card_units_int :

fintype.card (units ℤ) = 2

@[instance]

def units.fintype {α : Type u_1} [monoid α] [fintype α] :

fintype (units α)

Equations

units.fintype = fintype.of_injective units.val units.ext

@[simp]

theorem fintype.card_bool :

fintype.card bool = 2

def finset.insert_none {α : Type u_1} :

finset α → finset (option α)

Given a finset on α, lift it to being a finset on option α using option.some and then insert option.none.

Equations

s.insert_none = {val := none ::ₘ multiset.map some s.val, nodup := _}

@[simp]

theorem finset.mem_insert_none {α : Type u_1} {s : finset α} {o : option α} :

o ∈ s.insert_none ↔ ∀ (a : α), a ∈ o → a ∈ s

theorem finset.some_mem_insert_none {α : Type u_1} {s : finset α} {a : α} :

some a ∈ s.insert_none ↔ a ∈ s

@[instance]

def option.fintype {α : Type u_1} [fintype α] :

fintype (option α)

Equations

option.fintype = {elems := finset.univ.insert_none, complete := _}

@[simp]

theorem fintype.card_option {α : Type u_1} [fintype α] :

fintype.card (option α) = fintype.card α + 1

@[instance]

def sigma.fintype {α : Type u_1} (β : α → Type u_2) [fintype α] [Π (a : α), fintype (β a)] :

fintype (sigma β)

Equations

sigma.fintype β = {elems := finset.univ.sigma (λ (_x : α), finset.univ), complete := _}

@[simp]

theorem finset.univ_sigma_univ {α : Type u_1} {β : α → Type u_2} [fintype α] [Π (a : α), fintype (β a)] :

finset.univ.sigma (λ (a : α), finset.univ) = finset.univ

@[instance]

def prod.fintype (α : Type u_1) (β : Type u_2) [fintype α] [fintype β] :

fintype (α × β)

Equations

prod.fintype α β = {elems := finset.univ.product finset.univ, complete := _}

@[simp]

theorem finset.univ_product_univ {α : Type u_1} {β : Type u_2} [fintype α] [fintype β] :

finset.univ.product finset.univ = finset.univ

@[simp]

theorem fintype.card_prod (α : Type u_1) (β : Type u_2) [fintype α] [fintype β] :

fintype.card (α × β) = (fintype.card α) * fintype.card β

def fintype.fintype_prod_left {α : Type u_1} {β : Type u_2} [decidable_eq α] [fintype (α × β)] [nonempty β] :

Given that α × β is a fintype, α is also a fintype.

Equations

fintype.fintype_prod_left = {elems := finset.image prod.fst (fintype.elems (α × β)), complete := _}

def fintype.fintype_prod_right {α : Type u_1} {β : Type u_2} [decidable_eq β] [fintype (α × β)] [nonempty α] :

Given that α × β is a fintype, β is also a fintype.

Equations

fintype.fintype_prod_right = {elems := finset.image prod.snd (fintype.elems (α × β)), complete := _}

@[instance]

def ulift.fintype (α : Type u_1) [fintype α] :

fintype (ulift α)

Equations

ulift.fintype α = fintype.of_equiv α equiv.ulift.symm

@[simp]

theorem fintype.card_ulift (α : Type u_1) [fintype α] :

fintype.card (ulift α) = fintype.card α

@[instance]

def sum.fintype (α : Type u) (β : Type v) [fintype α] [fintype β] :

fintype (α ⊕ β)

Equations

sum.fintype α β = fintype.of_equiv (Σ (b : bool), cond b (ulift α) (ulift β)) ((equiv.sum_equiv_sigma_bool (ulift α) (ulift β)).symm.trans (equiv.ulift.sum_congr equiv.ulift))

theorem fintype.card_le_of_injective {α : Type u_1} {β : Type u_2} [fintype α] [fintype β] (f : α → β) :

function.injective f → fintype.card α ≤ fintype.card β

theorem fintype.exists_ne_map_eq_of_card_lt {α : Type u_1} {β : Type u_2} [fintype α] [fintype β] (f : α → β) :

fintype.card β < fintype.card α → (∃ (x y : α), x ≠ y ∧ f x = f y)

The pigeonhole principle for finitely many pigeons and pigeonholes. This is the fintype version of finset.exists_ne_map_eq_of_card_lt_of_maps_to.

theorem fintype.card_eq_one_iff {α : Type u_1} [fintype α] :

fintype.card α = 1 ↔ ∃ (x : α), ∀ (y : α), y = x

theorem fintype.card_eq_zero_iff {α : Type u_1} [fintype α] :

fintype.card α = 0 ↔ α → false

def fintype.card_eq_zero_equiv_equiv_pempty {α : Type u_1} [fintype α] :

fintype.card α = 0 ≃ (α ≃ pempty)

A fintype with cardinality zero is (constructively) equivalent to pempty.

Equations

fintype.card_eq_zero_equiv_equiv_pempty = {to_fun := λ (h : fintype.card α = 0), {to_fun := λ (a : α), _.elim, inv_fun := λ (a : pempty), a.elim, left_inv := _, right_inv := _}, inv_fun := _, left_inv := _, right_inv := _}

theorem fintype.card_pos_iff {α : Type u_1} [fintype α] :

0 < fintype.card α ↔ nonempty α

theorem fintype.card_le_one_iff {α : Type u_1} [fintype α] :

fintype.card α ≤ 1 ↔ ∀ (a b : α), a = b

theorem fintype.card_le_one_iff_subsingleton {α : Type u_1} [fintype α] :

fintype.card α ≤ 1 ↔ subsingleton α

theorem fintype.one_lt_card_iff_nontrivial {α : Type u_1} [fintype α] :

1 < fintype.card α ↔ nontrivial α

theorem fintype.exists_ne_of_one_lt_card {α : Type u_1} [fintype α] (h : 1 < fintype.card α) (a : α) :

∃ (b : α), b ≠ a

theorem fintype.exists_pair_of_one_lt_card {α : Type u_1} [fintype α] :

1 < fintype.card α → (∃ (a b : α), a ≠ b)

theorem fintype.injective_iff_surjective {α : Type u_1} [fintype α] {f : α → α} :

function.injective f ↔ function.surjective f

theorem fintype.injective_iff_bijective {α : Type u_1} [fintype α] {f : α → α} :

function.injective f ↔ function.bijective f

theorem fintype.surjective_iff_bijective {α : Type u_1} [fintype α] {f : α → α} :

function.surjective f ↔ function.bijective f

theorem fintype.injective_iff_surjective_of_equiv {α : Type u_1} [fintype α] {β : Type u_2} {f : α → β} :

α ≃ β → (function.injective f ↔ function.surjective f)

theorem fintype.nonempty_equiv_of_card_eq {α : Type u_1} {β : Type u_2} [fintype α] [fintype β] :

fintype.card α = fintype.card β → nonempty (α ≃ β)

theorem fintype.bijective_iff_injective_and_card {α : Type u_1} {β : Type u_2} [fintype α] [fintype β] (f : α → β) :

function.bijective f ↔ function.injective f ∧ fintype.card α = fintype.card β

theorem fintype.bijective_iff_surjective_and_card {α : Type u_1} {β : Type u_2} [fintype α] [fintype β] (f : α → β) :

function.bijective f ↔ function.surjective f ∧ fintype.card α = fintype.card β

theorem fintype.coe_image_univ {α : Type u_1} {β : Type u_2} [fintype α] [decidable_eq β] {f : α → β} :

↑(finset.image f finset.univ) = set.range f

@[instance]

def list.subtype.fintype {α : Type u_1} [decidable_eq α] (l : list α) :

fintype {x // x ∈ l}

Equations

list.subtype.fintype l = fintype.of_list l.attach _

@[instance]

def multiset.subtype.fintype {α : Type u_1} [decidable_eq α] (s : multiset α) :

fintype {x // x ∈ s}

Equations

multiset.subtype.fintype s = fintype.of_multiset s.attach _

@[instance]

def finset.subtype.fintype {α : Type u_1} (s : finset α) :

fintype {x // x ∈ s}

Equations

finset.subtype.fintype s = {elems := s.attach, complete := _}

@[instance]

def finset_coe.fintype {α : Type u_1} (s : finset α) :

fintype ↥↑s

Equations

finset_coe.fintype s = finset.subtype.fintype s

@[simp]

theorem fintype.card_coe {α : Type u_1} (s : finset α) :

fintype.card ↥↑s = s.card

theorem finset.attach_eq_univ {α : Type u_1} {s : finset α} :

s.attach = finset.univ

theorem finset.card_le_one_iff {α : Type u_1} {s : finset α} :

s.card ≤ 1 ↔ ∀ {x y : α}, x ∈ s → y ∈ s → x = y

theorem finset.card_le_one_of_subsingleton {α : Type u_1} [subsingleton α] (s : finset α) :

A finset of a subsingleton type has cardinality at most one.

theorem finset.one_lt_card_iff {α : Type u_1} {s : finset α} :

1 < s.card ↔ ∃ (x y : α), x ∈ s ∧ y ∈ s ∧ x ≠ y

@[instance]

def plift.fintype (p : Prop) [decidable p] :

fintype (plift p)

Equations

plift.fintype p = {elems := dite p (λ (h : p), {{down := h}}) (λ (h : ¬p), ∅), complete := _}

@[instance]

def Prop.fintype :

Equations

Prop.fintype = {elems := {val := true ::ₘ false ::ₘ 0, nodup := Prop.fintype._proof_1}, complete := Prop.fintype._proof_2}

@[instance]

def subtype.fintype {α : Type u_1} (p : α → Prop) [decidable_pred p] [fintype α] :

fintype {x // p x}

Equations

subtype.fintype p = fintype.subtype (finset.filter p finset.univ) _

def set_fintype {α : Type u_1} [fintype α] (s : set α) [decidable_pred s] :

A set on a fintype, when coerced to a type, is a fintype.

Equations

set_fintype s = subtype.fintype (λ (x : α), x ∈ s)

def function.embedding.equiv_of_fintype_self_embedding {α : Type u_1} [fintype α] :

(α ↪ α) → α ≃ α

An embedding from a fintype to itself can be promoted to an equivalence.

Equations

e.equiv_of_fintype_self_embedding = equiv.of_bijective ⇑e _

@[simp]

theorem function.embedding.equiv_of_fintype_self_embedding_to_embedding {α : Type u_1} [fintype α] (e : α ↪ α) :

e.equiv_of_fintype_self_embedding.to_embedding = e

@[simp]

theorem finset.univ_map_embedding {α : Type u_1} [fintype α] (e : α ↪ α) :

finset.map e finset.univ = finset.univ

def fintype.pi_finset {α : Type u_1} [decidable_eq α] [fintype α] {δ : α → Type u_4} :

(Π (a : α), finset (δ a)) → finset (Π (a : α), δ a)

Given for all a : α a finset t a of δ a, then one can define the finset fintype.pi_finset t of all functions taking values in t a for all a. This is the analogue of finset.pi where the base finset is univ (but formally they are not the same, as there is an additional condition i ∈ finset.univ in the finset.pi definition).

Equations

fintype.pi_finset t = finset.map {to_fun := λ (f : Π (a : α), a ∈ finset.univ → δ a) (a : α), f a _, inj' := _} (finset.univ.pi t)

@[simp]

theorem fintype.mem_pi_finset {α : Type u_1} [decidable_eq α] [fintype α] {δ : α → Type u_4} {t : Π (a : α), finset (δ a)} {f : Π (a : α), δ a} :

f ∈ fintype.pi_finset t ↔ ∀ (a : α), f a ∈ t a

theorem fintype.pi_finset_subset {α : Type u_1} [decidable_eq α] [fintype α] {δ : α → Type u_4} (t₁ t₂ : Π (a : α), finset (δ a)) :

(∀ (a : α), t₁ a ⊆ t₂ a) → fintype.pi_finset t₁ ⊆ fintype.pi_finset t₂

theorem fintype.pi_finset_disjoint_of_disjoint {α : Type u_1} [decidable_eq α] [fintype α] {δ : α → Type u_4} [Π (a : α), decidable_eq (δ a)] (t₁ t₂ : Π (a : α), finset (δ a)) {a : α} :

disjoint (t₁ a) (t₂ a) → disjoint (fintype.pi_finset t₁) (fintype.pi_finset t₂)

pi

@[instance]

def pi.fintype {α : Type u_1} {β : α → Type u_2} [decidable_eq α] [fintype α] [Π (a : α), fintype (β a)] :

fintype (Π (a : α), β a)

A dependent product of fintypes, indexed by a fintype, is a fintype.

Equations

pi.fintype = {elems := fintype.pi_finset (λ (_x : α), finset.univ), complete := _}

@[simp]

theorem fintype.pi_finset_univ {α : Type u_1} {β : α → Type u_2} [decidable_eq α] [fintype α] [Π (a : α), fintype (β a)] :

fintype.pi_finset (λ (a : α), finset.univ) = finset.univ

@[instance]

def d_array.fintype {n : ℕ} {α : fin n → Type u_1} [Π (n : fin n), fintype (α n)] :

fintype (d_array n α)

Equations

d_array.fintype = fintype.of_equiv (Π (i : fin n), α i) (equiv.d_array_equiv_fin α).symm

@[instance]

def array.fintype {n : ℕ} {α : Type u_1} [fintype α] :

fintype (array n α)

Equations

array.fintype = d_array.fintype

@[instance]

def vector.fintype {α : Type u_1} [fintype α] {n : ℕ} :

fintype (vector α n)

Equations

vector.fintype = fintype.of_equiv (fin n → α) (equiv.vector_equiv_fin α n).symm

@[instance]

def quotient.fintype {α : Type u_1} [fintype α] (s : setoid α) [decidable_rel has_equiv.equiv] :

fintype (quotient s)

Equations

quotient.fintype s = fintype.of_surjective quotient.mk _

@[instance]

def finset.fintype {α : Type u_1} [fintype α] :

fintype (finset α)

Equations

finset.fintype = {elems := finset.univ.powerset, complete := _}

@[simp]

theorem fintype.card_finset {α : Type u_1} [fintype α] :

fintype.card (finset α) = 2 ^ fintype.card α

@[simp]

theorem set.to_finset_univ {α : Type u_1} [fintype α] :

set.univ.to_finset = finset.univ

@[simp]

theorem set.to_finset_empty {α : Type u_1} [fintype α] :

∅.to_finset = ∅

theorem fintype.card_subtype_le {α : Type u_1} [fintype α] (p : α → Prop) [decidable_pred p] :

fintype.card {x // p x} ≤ fintype.card α

theorem fintype.card_subtype_lt {α : Type u_1} [fintype α] {p : α → Prop} [decidable_pred p] {x : α} :

¬p x → fintype.card {x // p x} < fintype.card α

@[instance]

def psigma.fintype {α : Type u_1} {β : α → Type u_2} [fintype α] [Π (a : α), fintype (β a)] :

fintype (Σ' (a : α), β a)

Equations

psigma.fintype = fintype.of_equiv (Σ (i : α), β i) (equiv.psigma_equiv_sigma (λ (a : α), β a)).symm

@[instance]

def psigma.fintype_prop_left {α : Prop} {β : α → Type u_1} [decidable α] [Π (a : α), fintype (β a)] :

fintype (Σ' (a : α), β a)

Equations

psigma.fintype_prop_left = dite α (λ (h : α), fintype.of_equiv (β h) {to_fun := λ (x : β h), ⟨h, x⟩, inv_fun := psigma.snd (λ (a : α), β a), left_inv := _, right_inv := _}) (λ (h : ¬α), {elems := ∅, complete := _})

@[instance]

def psigma.fintype_prop_right {α : Type u_1} {β : α → Prop} [Π (a : α), decidable (β a)] [fintype α] :

fintype (Σ' (a : α), β a)

Equations

psigma.fintype_prop_right = fintype.of_equiv {a // β a} {to_fun := λ (_x : {a // β a}), psigma.fintype_prop_right._match_1 _x, inv_fun := λ (_x : Σ' (a : α), β a), psigma.fintype_prop_right._match_2 _x, left_inv := _, right_inv := _}
psigma.fintype_prop_right._match_1 ⟨x, y⟩ = ⟨x, y⟩
psigma.fintype_prop_right._match_2 ⟨x, y⟩ = ⟨x, y⟩

@[instance]

def psigma.fintype_prop_prop {α : Prop} {β : α → Prop} [decidable α] [Π (a : α), decidable (β a)] :

fintype (Σ' (a : α), β a)

Equations

psigma.fintype_prop_prop = dite (∃ (a : α), β a) (λ (h : ∃ (a : α), β a), {elems := {⟨_, _⟩}, complete := _}) (λ (h : ¬∃ (a : α), β a), {elems := ∅, complete := _})

@[instance]

def set.fintype {α : Type u_1} [fintype α] :

fintype (set α)

Equations

set.fintype = {elems := finset.map {to_fun := coe lift_base, inj' := _} finset.univ.powerset, complete := _}

@[instance]

def pfun_fintype (p : Prop) [decidable p] (α : p → Type u_1) [Π (hp : p), fintype (α hp)] :

fintype (Π (hp : p), α hp)

Equations

pfun_fintype p α = dite p (λ (hp : p), fintype.of_equiv (α hp) {to_fun := λ (a : α hp) (_x : p), a, inv_fun := λ (f : Π (hp : p), α hp), f hp, left_inv := _, right_inv := _}) (λ (hp : ¬p), {elems := {λ (h : p), _.elim}, complete := _})

@[simp]

theorem finset.univ_pi_univ {α : Type u_1} {β : α → Type u_2} [decidable_eq α] [fintype α] [Π (a : α), fintype (β a)] :

finset.univ.pi (λ (a : α), finset.univ) = finset.univ

theorem mem_image_univ_iff_mem_range {α : Type u_1} {β : Type u_2} [fintype α] [decidable_eq β] {f : α → β} {b : β} :

b ∈ finset.image f finset.univ ↔ b ∈ set.range f

theorem card_lt_card_of_injective_of_not_mem {α : Type u_1} {β : Type u_2} [fintype α] [fintype β] (f : α → β) (h : function.injective f) {b : β} :

b ∉ set.range f → fintype.card α < fintype.card β

def quotient.fin_choice_aux {ι : Type u_1} [decidable_eq ι] {α : ι → Type u_2} [S : Π (i : ι), setoid (α i)] (l : list ι) :

(Π (i : ι), i ∈ l → quotient (S i)) → quotient pi_setoid

An auxiliary function for quotient.fin_choice. Given a collection of setoids indexed by a type ι, a (finite) list l of indices, and a function that for each i ∈ l gives a term of the corresponding quotient type, then there is a corresponding term in the quotient of the product of the setoids indexed by l.

Equations

quotient.fin_choice_aux (i :: l) f = (f i _).lift_on₂ (quotient.fin_choice_aux l (λ (j : ι) (h : j ∈ l), f j _)) (λ (a : α i) (l_1 : Π (i : ι), i ∈ l → α i), ⟦(λ (j : ι) (h : j ∈ i :: l), dite (j = i) (λ (e : j = i), _.mpr a) (λ (e : ¬j = i), l_1 j _))⟧) _
quotient.fin_choice_aux list.nil f = ⟦(λ (i : ι), false.elim)⟧

theorem quotient.fin_choice_aux_eq {ι : Type u_1} [decidable_eq ι] {α : ι → Type u_2} [S : Π (i : ι), setoid (α i)] (l : list ι) (f : Π (i : ι), i ∈ l → α i) :

quotient.fin_choice_aux l (λ (i : ι) (h : i ∈ l), ⟦f i h⟧) = ⟦f⟧

def quotient.fin_choice {ι : Type u_1} [decidable_eq ι] [fintype ι] {α : ι → Type u_2} [S : Π (i : ι), setoid (α i)] :

(Π (i : ι), quotient (S i)) → quotient pi_setoid

Given a collection of setoids indexed by a fintype ι and a function that for each i : ι gives a term of the corresponding quotient type, then there is corresponding term in the quotient of the product of the setoids.

Equations

quotient.fin_choice f = quotient.lift_on (quotient.rec_on finset.univ.val (λ (l : list ι), quotient.fin_choice_aux l (λ (i : ι) (_x : i ∈ l), f i)) _) (λ (f : Π (i : ι), i ∈ finset.univ.val → α i), ⟦(λ (i : ι), f i _)⟧) quotient.fin_choice._proof_2

theorem quotient.fin_choice_eq {ι : Type u_1} [decidable_eq ι] [fintype ι] {α : ι → Type u_2} [Π (i : ι), setoid (α i)] (f : Π (i : ι), α i) :

quotient.fin_choice (λ (i : ι), ⟦f i⟧) = ⟦f⟧

def perms_of_list {α : Type u_1} [decidable_eq α] :

list α → list (equiv.perm α)

Given a list, produce a list of all permutations of its elements.

Equations

perms_of_list (a :: l) = perms_of_list l ++ l.bind (λ (b : α), list.map (λ (f : equiv.perm α), (equiv.swap a b) * f) (perms_of_list l))
perms_of_list list.nil = [1]

theorem length_perms_of_list {α : Type u_1} [decidable_eq α] (l : list α) :

(perms_of_list l).length = (l.length)!

theorem mem_perms_of_list_of_mem {α : Type u_1} [decidable_eq α] {l : list α} {f : equiv.perm α} :

(∀ (x : α), ⇑f x ≠ x → x ∈ l) → f ∈ perms_of_list l

theorem mem_of_mem_perms_of_list {α : Type u_1} [decidable_eq α] {l : list α} {f : equiv.perm α} (ᾰ : f ∈ perms_of_list l) {x : α} :

⇑f x ≠ x → x ∈ l

theorem mem_perms_of_list_iff {α : Type u_1} [decidable_eq α] {l : list α} {f : equiv.perm α} :

f ∈ perms_of_list l ↔ ∀ {x : α}, ⇑f x ≠ x → x ∈ l

theorem nodup_perms_of_list {α : Type u_1} [decidable_eq α] {l : list α} :

l.nodup → (perms_of_list l).nodup

def perms_of_finset {α : Type u_1} [decidable_eq α] :

finset α → finset (equiv.perm α)

Given a finset, produce the finset of all permutations of its elements.

Equations

perms_of_finset s = quotient.hrec_on s.val (λ (l : list α) (hl : multiset.nodup ⟦l⟧), {val := ↑(perms_of_list l), nodup := _}) perms_of_finset._proof_2 _

theorem mem_perms_of_finset_iff {α : Type u_1} [decidable_eq α] {s : finset α} {f : equiv.perm α} :

f ∈ perms_of_finset s ↔ ∀ {x : α}, ⇑f x ≠ x → x ∈ s

theorem card_perms_of_finset {α : Type u_1} [decidable_eq α] (s : finset α) :

(perms_of_finset s).card = (s.card)!

def fintype_perm {α : Type u_1} [decidable_eq α] [fintype α] :

fintype (equiv.perm α)

The collection of permutations of a fintype is a fintype.

Equations

fintype_perm = {elems := perms_of_finset finset.univ, complete := _}

@[instance]

def equiv.fintype {α : Type u_1} {β : Type u_2} [decidable_eq α] [decidable_eq β] [fintype α] [fintype β] :

fintype (α ≃ β)

Equations

equiv.fintype = dite (fintype.card β = fintype.card α) (λ (h : fintype.card β = fintype.card α), (fintype.equiv_fin α).rec_on_subsingleton (λ (eα : α ≃ fin (fintype.card α)), (fintype.equiv_fin β).rec_on_subsingleton (λ (eβ : β ≃ fin (fintype.card β)), fintype.of_equiv (equiv.perm α) ((equiv.refl α).equiv_congr (eα.trans (h.rec_on eβ.symm)))))) (λ (h : ¬fintype.card β = fintype.card α), {elems := ∅, complete := _})

theorem fintype.card_perm {α : Type u_1} [decidable_eq α] [fintype α] :

fintype.card (equiv.perm α) = (fintype.card α)!

theorem fintype.card_equiv {α : Type u_1} {β : Type u_2} [decidable_eq α] [decidable_eq β] [fintype α] [fintype β] :

α ≃ β → fintype.card (α ≃ β) = (fintype.card α)!

theorem univ_eq_singleton_of_card_one {α : Type u_1} [fintype α] (x : α) :

fintype.card α = 1 → finset.univ = {x}

def fintype.choose_x {α : Type u_1} [fintype α] (p : α → Prop) [decidable_pred p] :

(∃! (a : α), p a) → {a // p a}

Given a fintype α and a predicate p, associate to a proof that there is a unique element of α satisfying p this unique element, as an element of the corresponding subtype.

Equations

fintype.choose_x p hp = ⟨finset.choose p finset.univ _, _⟩

def fintype.choose {α : Type u_1} [fintype α] (p : α → Prop) [decidable_pred p] :

(∃! (a : α), p a) → α

Given a fintype α and a predicate p, associate to a proof that there is a unique element of α satisfying p this unique element, as an element of α.

Equations

fintype.choose p hp = ↑(fintype.choose_x p hp)

theorem fintype.choose_spec {α : Type u_1} [fintype α] (p : α → Prop) [decidable_pred p] (hp : ∃! (a : α), p a) :

p (fintype.choose p hp)

def fintype.bij_inv {α : Type u_1} {β : Type u_2} [fintype α] [decidable_eq β] {f : α → β} :

function.bijective f → β → α

bij_inv fis the unique inverse to a bijectionf. This acts as a computable alternative tofunction.inv_fun`.

Equations

fintype.bij_inv f_bij b = fintype.choose (λ (a : α), f a = b) _

theorem fintype.left_inverse_bij_inv {α : Type u_1} {β : Type u_2} [fintype α] [decidable_eq β] {f : α → β} (f_bij : function.bijective f) :

function.left_inverse (fintype.bij_inv f_bij) f

theorem fintype.right_inverse_bij_inv {α : Type u_1} {β : Type u_2} [fintype α] [decidable_eq β] {f : α → β} (f_bij : function.bijective f) :

function.right_inverse (fintype.bij_inv f_bij) f

theorem fintype.bijective_bij_inv {α : Type u_1} {β : Type u_2} [fintype α] [decidable_eq β] {f : α → β} (f_bij : function.bijective f) :

function.bijective (fintype.bij_inv f_bij)

theorem fintype.well_founded_of_trans_of_irrefl {α : Type u_1} [fintype α] (r : α → α → Prop) [is_trans α r] [is_irrefl α r] :

theorem fintype.preorder.well_founded {α : Type u_1} [fintype α] [preorder α] :

well_founded has_lt.lt

@[instance]

theorem fintype.linear_order.is_well_order {α : Type u_1} [fintype α] [linear_order α] :

is_well_order α has_lt.lt

@[class]

structure infinite :

Type u_4 → Prop

not_fintype : fintype α → false

A type is said to be infinite if it has no fintype instance.

Instances

@[simp]

theorem not_nonempty_fintype {α : Type u_1} :

¬nonempty (fintype α) ↔ infinite α

theorem finset.exists_minimal {α : Type u_1} [preorder α] (s : finset α) :

s.nonempty → (∃ (m : α) (H : m ∈ s), ∀ (x : α), x ∈ s → ¬x < m)

theorem finset.exists_maximal {α : Type u_1} [preorder α] (s : finset α) :

s.nonempty → (∃ (m : α) (H : m ∈ s), ∀ (x : α), x ∈ s → ¬m < x)

theorem infinite.exists_not_mem_finset {α : Type u_1} [infinite α] (s : finset α) :

∃ (x : α), x ∉ s

@[instance]

def infinite.nontrivial (α : Type u_1) [H : infinite α] :

theorem infinite.nonempty (α : Type u_1) [infinite α] :

theorem infinite.of_injective {α : Type u_1} {β : Type u_2} [infinite β] (f : β → α) :

function.injective f → infinite α

theorem infinite.of_surjective {α : Type u_1} {β : Type u_2} [infinite β] (f : α → β) :

function.surjective f → infinite α

def infinite.nat_embedding (α : Type u_1) [infinite α] :

Embedding of ℕ into an infinite type.

Equations

infinite.nat_embedding α = {to_fun := nat_embedding_aux α _inst_1, inj' := _}

theorem infinite.exists_subset_card_eq (α : Type u_1) [infinite α] (n : ℕ) :

∃ (s : finset α), s.card = n

theorem not_injective_infinite_fintype {α : Type u_1} {β : Type u_2} [infinite α] [fintype β] (f : α → β) :

¬function.injective f

theorem fintype.exists_ne_map_eq_of_infinite {α : Type u_1} {β : Type u_2} [infinite α] [fintype β] (f : α → β) :

∃ (x y : α), x ≠ y ∧ f x = f y

The pigeonhole principle for infinitely many pigeons in finitely many pigeonholes. If there are infinitely many pigeons in finitely many pigeonholes, then there are at least two pigeons in the same pigeonhole.

See also: fintype.exists_ne_map_eq_of_card_lt, fintype.exists_infinite_fiber.

theorem fintype.exists_infinite_fiber {α : Type u_1} {β : Type u_2} [infinite α] [fintype β] (f : α → β) :

∃ (y : β), infinite ↥(f ⁻¹' {y})

The strong pigeonhole principle for infinitely many pigeons in finitely many pigeonholes. If there are infinitely many pigeons in finitely many pigeonholes, then there is a pigeonhole with infinitely many pigeons.

See also: fintype.exists_ne_map_eq_of_infinite

theorem not_surjective_fintype_infinite {α : Type u_1} {β : Type u_2} [fintype α] [infinite β] (f : α → β) :

¬function.surjective f

@[instance]

def nat.infinite :

@[instance]

def int.infinite :

def trunc_of_multiset_exists_mem {α : Type u_1} (s : multiset α) :

(∃ (x : α), x ∈ s) → trunc α

For s : multiset α, we can lift the existential statement that ∃ x, x ∈ s to a trunc α.

Equations

trunc_of_multiset_exists_mem s = quotient.rec_on_subsingleton s (λ (l : list α) (h : ∃ (x : α), x ∈ ⟦l⟧), trunc_of_multiset_exists_mem._match_1 l h l h)
trunc_of_multiset_exists_mem._match_1 l h (a :: _x) _x_1 = trunc.mk a
trunc_of_multiset_exists_mem._match_1 l h list.nil _x = false.elim _

def trunc_of_nonempty_fintype (α : Type u_1) [nonempty α] [fintype α] :

A nonempty fintype constructively contains an element.

Equations

trunc_of_nonempty_fintype α = trunc_of_multiset_exists_mem finset.univ.val _

def trunc_of_card_pos {α : Type u_1} [fintype α] :

0 < fintype.card α → trunc α

A fintype with positive cardinality constructively contains an element.

Equations

trunc_of_card_pos h = let _inst : nonempty α := _ in trunc_of_nonempty_fintype α

def trunc_sigma_of_exists {α : Type u_1} [fintype α] {P : α → Prop} [decidable_pred P] :

(∃ (a : α), P a) → trunc (Σ' (a : α), P a)

By iterating over the elements of a fintype, we can lift an existential statement ∃ a, P a to trunc (Σ' a, P a), containing data.

Equations

trunc_sigma_of_exists h = trunc_of_nonempty_fintype (Σ' (a : α), P a)