Cardinality of a finite set #
This defines the cardinality of a finset and provides induction principles for finsets.
Main declarations #
finset.card:s.card : ℕreturns the cardinality ofs : finset α.
Induction principles #
finset.strong_induction: Strong inductionfinset.strong_induction_onfinset.strong_downward_inductionfinset.strong_downward_induction_onfinset.case_strong_induction_on
TODO #
Should we add a noncomputable version?
s.card is the number of elements of s, aka its cardinality.
Equations
- s.card = ⇑multiset.card s.val
Alias of the reverse direction of finset.card_pos.
@[simp]
theorem
finset.card_cons
{α : Type u_1}
{s : finset α}
{a : α}
(h : a ∉ s) :
(finset.cons a s h).card = s.card + 1
@[simp]
theorem
finset.card_insert_of_not_mem
{α : Type u_1}
{s : finset α}
{a : α}
[decidable_eq α]
(h : a ∉ s) :
(has_insert.insert a s).card = s.card + 1
theorem
finset.card_insert_of_mem
{α : Type u_1}
{s : finset α}
{a : α}
[decidable_eq α]
(h : a ∈ s) :
(has_insert.insert a s).card = s.card
theorem
finset.card_insert_le
{α : Type u_1}
[decidable_eq α]
(a : α)
(s : finset α) :
(has_insert.insert a s).card ≤ s.card + 1
If a ∈ s is known, see also finset.card_insert_of_mem and finset.card_insert_of_not_mem.
@[simp]
@[simp]
@[simp]
If a ∈ s is known, see also finset.card_erase_of_mem and finset.erase_eq_of_not_mem.
theorem
multiset.to_finset_card_le
{α : Type u_1}
[decidable_eq α]
(m : multiset α) :
m.to_finset.card ≤ ⇑multiset.card m
theorem
multiset.to_finset_card_of_nodup
{α : Type u_1}
[decidable_eq α]
{m : multiset α}
(h : m.nodup) :
m.to_finset.card = ⇑multiset.card m
theorem
finset.card_image_le
{α : Type u_1}
{β : Type u_2}
{s : finset α}
{f : α → β}
[decidable_eq β] :
(finset.image f s).card ≤ s.card
theorem
finset.card_image_of_inj_on
{α : Type u_1}
{β : Type u_2}
{s : finset α}
{f : α → β}
[decidable_eq β]
(H : set.inj_on f ↑s) :
(finset.image f s).card = s.card
theorem
finset.inj_on_of_card_image_eq
{α : Type u_1}
{β : Type u_2}
{s : finset α}
{f : α → β}
[decidable_eq β]
(H : (finset.image f s).card = s.card) :
set.inj_on f ↑s
theorem
finset.card_image_iff
{α : Type u_1}
{β : Type u_2}
{s : finset α}
{f : α → β}
[decidable_eq β] :
(finset.image f s).card = s.card ↔ set.inj_on f ↑s
theorem
finset.card_image_of_injective
{α : Type u_1}
{β : Type u_2}
{f : α → β}
[decidable_eq β]
(s : finset α)
(H : function.injective f) :
(finset.image f s).card = s.card
theorem
finset.fiber_card_ne_zero_iff_mem_image
{α : Type u_1}
{β : Type u_2}
(s : finset α)
(f : α → β)
[decidable_eq β]
(y : β) :
(finset.filter (λ (x : α), f x = y) s).card ≠ 0 ↔ y ∈ finset.image f s
@[simp]
theorem
finset.card_map
{α : Type u_1}
{β : Type u_2}
{s : finset α}
(f : α ↪ β) :
(finset.map f s).card = s.card
@[simp]
theorem
finset.card_subtype
{α : Type u_1}
(p : α → Prop)
[decidable_pred p]
(s : finset α) :
(finset.subtype p s).card = (finset.filter p s).card
theorem
finset.card_filter_le
{α : Type u_1}
(s : finset α)
(p : α → Prop)
[decidable_pred p] :
(finset.filter p s).card ≤ s.card
theorem
finset.map_eq_of_subset
{α : Type u_1}
{s : finset α}
{f : α ↪ α}
(hs : finset.map f s ⊆ s) :
finset.map f s = s
theorem
finset.filter_card_eq
{α : Type u_1}
{s : finset α}
{p : α → Prop}
[decidable_pred p]
(h : (finset.filter p s).card = s.card)
(x : α)
(hx : x ∈ s) :
p x
theorem
finset.surj_on_of_inj_on_of_card_le
{α : Type u_1}
{β : Type u_2}
{s : finset α}
{t : finset β}
(f : Π (a : α), a ∈ s → β)
(hf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t)
(hinj : ∀ (a₁ a₂ : α) (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s), f a₁ ha₁ = f a₂ ha₂ → a₁ = a₂)
(hst : t.card ≤ s.card)
(b : β)
(H : b ∈ t) :
theorem
finset.inj_on_of_surj_on_of_card_le
{α : Type u_1}
{β : Type u_2}
{s : finset α}
{t : finset β}
(f : Π (a : α), a ∈ s → β)
(hf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t)
(hsurj : ∀ (b : β), b ∈ t → (∃ (a : α) (ha : a ∈ s), b = f a ha))
(hst : s.card ≤ t.card)
⦃a₁ a₂ : α⦄
(ha₁ : a₁ ∈ s)
(ha₂ : a₂ ∈ s)
(ha₁a₂ : f a₁ ha₁ = f a₂ ha₂) :
a₁ = a₂
Lattice structure #
theorem
finset.filter_card_add_filter_neg_card_eq_card
{α : Type u_1}
{s : finset α}
(p : α → Prop)
[decidable_pred p] :
(finset.filter p s).card + (finset.filter (not ∘ p) s).card = s.card
Explicit description of a finset from its card #
A finset of a subsingleton type has cardinality at most one.
Inductions #
def
finset.strong_induction
{α : Type u_1}
{p : finset α → Sort u_2}
(H : Π (s : finset α), (Π (t : finset α), t ⊂ s → p t) → p s)
(s : finset α) :
p s
Suppose that, given objects defined on all strict subsets of any finset s, one knows how to
define an object on s. Then one can inductively define an object on all finsets, starting from
the empty set and iterating. This can be used either to define data, or to prove properties.
Equations
- finset.strong_induction H s = H s (λ (t : finset α) (h : t ⊂ s), finset.strong_induction H t)
theorem
finset.strong_induction_eq
{α : Type u_1}
{p : finset α → Sort u_2}
(H : Π (s : finset α), (Π (t : finset α), t ⊂ s → p t) → p s)
(s : finset α) :
finset.strong_induction H s = H s (λ (t : finset α) (h : t ⊂ s), finset.strong_induction H t)
Analogue of strong_induction with order of arguments swapped.
Equations
- s.strong_induction_on = λ (H : Π (s : finset α), (Π (t : finset α), t ⊂ s → p t) → p s), finset.strong_induction H s
theorem
finset.strong_induction_on_eq
{α : Type u_1}
{p : finset α → Sort u_2}
(s : finset α)
(H : Π (s : finset α), (Π (t : finset α), t ⊂ s → p t) → p s) :
s.strong_induction_on H = H s (λ (t : finset α) (h : t ⊂ s), t.strong_induction_on H)
theorem
finset.case_strong_induction_on
{α : Type u_1}
[decidable_eq α]
{p : finset α → Prop}
(s : finset α)
(h₀ : p ∅)
(h₁ : ∀ (a : α) (s : finset α), a ∉ s → (∀ (t : finset α), t ⊆ s → p t) → p (has_insert.insert a s)) :
p s
def
finset.strong_downward_induction
{α : Type u_1}
{p : finset α → Sort u_2}
{n : ℕ}
(H : Π (t₁ : finset α), (Π {t₂ : finset α}, t₂.card ≤ n → t₁ ⊂ t₂ → p t₂) → t₁.card ≤ n → p t₁)
(s : finset α) :
Suppose that, given that p t can be defined on all supersets of s of cardinality less than
n, one knows how to define p s. Then one can inductively define p s for all finsets s of
cardinality less than n, starting from finsets of card n and iterating. This
can be used either to define data, or to prove properties.
Equations
- finset.strong_downward_induction H s = H s (λ (t : finset α) (ht : t.card ≤ n) (h : s ⊂ t), finset.strong_downward_induction H t ht)
theorem
finset.strong_downward_induction_eq
{α : Type u_1}
{n : ℕ}
{p : finset α → Sort u_2}
(H : Π (t₁ : finset α), (Π {t₂ : finset α}, t₂.card ≤ n → t₁ ⊂ t₂ → p t₂) → t₁.card ≤ n → p t₁)
(s : finset α) :
finset.strong_downward_induction H s = H s (λ (t : finset α) (ht : t.card ≤ n) (hst : s ⊂ t), finset.strong_downward_induction H t ht)
theorem
finset.strong_downward_induction_on_eq
{α : Type u_1}
{n : ℕ}
{p : finset α → Sort u_2}
(s : finset α)
(H : Π (t₁ : finset α), (Π {t₂ : finset α}, t₂.card ≤ n → t₁ ⊂ t₂ → p t₂) → t₁.card ≤ n → p t₁) :
s.strong_downward_induction_on H = H s (λ (t : finset α) (ht : t.card ≤ n) (h : s ⊂ t), t.strong_downward_induction_on H ht)