Cardinality of a finite set

This defines the cardinality of a Finset and provides induction principles for finsets.

Main declarations

• Finset.card: s.card : ℕ returns the cardinality of s : Finset α.

Induction principles

• Finset.strongInduction: Strong induction
• Finset.strongInductionOn
• Finset.strongDownwardInduction
• Finset.strongDownwardInductionOn
• Finset.case_strong_induction_on

TODO

Should we add a noncomputable version?

def Finset.card {α : Type u_1} (s : Finset α) : ℕ

s.card is the number of elements of s, aka its cardinality.

Equations

• Finset.card s = ↑Multiset.card s.val

theorem Finset.card_def {α : Type u_1} (s : Finset α) : Finset.card s = ↑Multiset.card s.val

@[simp]

theorem Finset.card_mk {α : Type u_1} {m : Multiset α} {nodup : Multiset.Nodup m} : Finset.card { val := m, nodup := nodup } = ↑Multiset.card m

@[simp]

theorem Finset.card_empty {α : Type u_1} : Finset.card ∅ = 0

theorem Finset.card_le_of_subset {α : Type u_1} {s : Finset α} {t : Finset α} : s ⊆ t → Finset.card s ≤ Finset.card t

theorem Finset.card_mono {α : Type u_1} : Monotone Finset.card

@[simp]

theorem Finset.card_eq_zero {α : Type u_1} {s : Finset α} : Finset.card s = 0 ↔ s = ∅

theorem Finset.card_pos {α : Type u_1} {s : Finset α} : 0 < Finset.card s ↔ Finset.Nonempty s

theorem Finset.Nonempty.card_pos {α : Type u_1} {s : Finset α} : Finset.Nonempty s → 0 < Finset.card s

Alias of the reverse direction of Finset.card_pos.

theorem Finset.card_ne_zero_of_mem {α : Type u_1} {s : Finset α} {a : α} (h : a ∈ s) : Finset.card s ≠ 0

@[simp]

theorem Finset.card_singleton {α : Type u_1} (a : α) : Finset.card {a} = 1

theorem Finset.card_singleton_inter {α : Type u_1} {s : Finset α} {a : α} [DecidableEq α] : Finset.card ({a} ∩ s) ≤ 1

@[simp]

theorem Finset.card_cons {α : Type u_1} {s : Finset α} {a : α} (h : ¬a ∈ s) : Finset.card (Finset.cons a s h) = Finset.card s + 1

@[simp]

theorem Finset.card_insert_of_not_mem {α : Type u_1} {s : Finset α} {a : α} [DecidableEq α] (h : ¬a ∈ s) : Finset.card (insert a s) = Finset.card s + 1

theorem Finset.card_insert_of_mem {α : Type u_1} {s : Finset α} {a : α} [DecidableEq α] (h : a ∈ s) : Finset.card (insert a s) = Finset.card s

theorem Finset.card_insert_le {α : Type u_1} [DecidableEq α] (a : α) (s : Finset α) : Finset.card (insert a s) ≤ Finset.card s + 1

theorem Finset.card_insert_eq_ite {α : Type u_1} {s : Finset α} {a : α} [DecidableEq α] : Finset.card (insert a s) = if a ∈ s then Finset.card s else Finset.card s + 1

If a ∈ s is known, see also Finset.card_insert_of_mem and Finset.card_insert_of_not_mem.

@[simp]

theorem Finset.card_doubleton {α : Type u_1} {a : α} {b : α} [DecidableEq α] (h : a ≠ b) : Finset.card {a, b} = 2

@[simp]

theorem Finset.card_erase_of_mem {α : Type u_1} {s : Finset α} {a : α} [DecidableEq α] : a ∈ s → Finset.card (Finset.erase s a) = Finset.card s - 1

$#(s \setminus {a}) = #s - 1$ if $a \in s$.

@[simp]

theorem Finset.cast_card_erase_of_mem {α : Type u_1} {a : α} [DecidableEq α] {R : Type u_3} [AddGroupWithOne R] {s : Finset α} (hs : a ∈ s) : ↑(Finset.card (Finset.erase s a)) = ↑(Finset.card s) - 1

$#(s \setminus {a}) = #s - 1$ if $a \in s$. This result is casted to any additive group with 1, so that we don't have to work with ℕ-subtraction.

@[simp]

theorem Finset.card_erase_add_one {α : Type u_1} {s : Finset α} {a : α} [DecidableEq α] : a ∈ s → Finset.card (Finset.erase s a) + 1 = Finset.card s

theorem Finset.card_erase_lt_of_mem {α : Type u_1} {s : Finset α} {a : α} [DecidableEq α] : a ∈ s → Finset.card (Finset.erase s a) < Finset.card s

theorem Finset.card_erase_le {α : Type u_1} {s : Finset α} {a : α} [DecidableEq α] : Finset.card (Finset.erase s a) ≤ Finset.card s

theorem Finset.pred_card_le_card_erase {α : Type u_1} {s : Finset α} {a : α} [DecidableEq α] : Finset.card s - 1 ≤ Finset.card (Finset.erase s a)

theorem Finset.card_erase_eq_ite {α : Type u_1} {s : Finset α} {a : α} [DecidableEq α] : Finset.card (Finset.erase s a) = if a ∈ s then Finset.card s - 1 else Finset.card s

If a ∈ s is known, see also Finset.card_erase_of_mem and Finset.erase_eq_of_not_mem.

@[simp]

theorem Finset.card_range (n : ℕ) : Finset.card (Finset.range n) = n

@[simp]

theorem Finset.card_attach {α : Type u_1} {s : Finset α} : Finset.card (Finset.attach s) = Finset.card s

theorem Multiset.card_toFinset {α : Type u_1} [DecidableEq α] (m : Multiset α) : Finset.card (Multiset.toFinset m) = ↑Multiset.card (Multiset.dedup m)

theorem Multiset.toFinset_card_le {α : Type u_1} [DecidableEq α] (m : Multiset α) : Finset.card (Multiset.toFinset m) ≤ ↑Multiset.card m

theorem Multiset.toFinset_card_of_nodup {α : Type u_1} [DecidableEq α] {m : Multiset α} (h : Multiset.Nodup m) : Finset.card (Multiset.toFinset m) = ↑Multiset.card m

theorem List.card_toFinset {α : Type u_1} [DecidableEq α] (l : List α) : Finset.card (List.toFinset l) = List.length (List.dedup l)

theorem List.toFinset_card_le {α : Type u_1} [DecidableEq α] (l : List α) : Finset.card (List.toFinset l) ≤ List.length l

theorem List.toFinset_card_of_nodup {α : Type u_1} [DecidableEq α] {l : List α} (h : List.Nodup l) : Finset.card (List.toFinset l) = List.length l

@[simp]

theorem Finset.length_toList {α : Type u_1} (s : Finset α) : List.length (Finset.toList s) = Finset.card s

theorem Finset.card_image_le {α : Type u_1} {β : Type u_2} {s : Finset α} {f : α → β} [DecidableEq β] : Finset.card (Finset.image f s) ≤ Finset.card s

theorem Finset.card_image_of_injOn {α : Type u_1} {β : Type u_2} {s : Finset α} {f : α → β} [DecidableEq β] (H : Set.InjOn f ↑s) : Finset.card (Finset.image f s) = Finset.card s

theorem Finset.injOn_of_card_image_eq {α : Type u_1} {β : Type u_2} {s : Finset α} {f : α → β} [DecidableEq β] (H : Finset.card (Finset.image f s) = Finset.card s) : Set.InjOn f ↑s

theorem Finset.card_image_iff {α : Type u_1} {β : Type u_2} {s : Finset α} {f : α → β} [DecidableEq β] : Finset.card (Finset.image f s) = Finset.card s ↔ Set.InjOn f ↑s

theorem Finset.card_image_of_injective {α : Type u_1} {β : Type u_2} {f : α → β} [DecidableEq β] (s : Finset α) (H : Function.Injective f) : Finset.card (Finset.image f s) = Finset.card s

theorem Finset.fiber_card_ne_zero_iff_mem_image {α : Type u_1} {β : Type u_2} (s : Finset α) (f : α → β) [DecidableEq β] (y : β) : Finset.card (Finset.filter (fun x => f x = y) s) ≠ 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.card (Finset.map f s) = Finset.card s

@[simp]

theorem Finset.card_subtype {α : Type u_1} (p : α → Prop) [DecidablePred p] (s : Finset α) : Finset.card (Finset.subtype p s) = Finset.card (Finset.filter p s)

theorem Finset.card_filter_le {α : Type u_1} (s : Finset α) (p : α → Prop) [DecidablePred p] : Finset.card (Finset.filter p s) ≤ Finset.card s

theorem Finset.eq_of_subset_of_card_le {α : Type u_1} {s : Finset α} {t : Finset α} (h : s ⊆ t) (h₂ : Finset.card t ≤ Finset.card s) : s = t

theorem Finset.eq_of_superset_of_card_ge {α : Type u_1} {s : Finset α} {t : Finset α} (hst : s ⊆ t) (hts : Finset.card t ≤ Finset.card s) : t = s

theorem Finset.subset_iff_eq_of_card_le {α : Type u_1} {s : Finset α} {t : Finset α} (h : Finset.card t ≤ Finset.card s) : s ⊆ t ↔ s = t

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} [DecidablePred p] (h : Finset.card (Finset.filter p s) = Finset.card s) (x : α) (hx : x ∈ s) : p x

theorem Finset.card_lt_card {α : Type u_1} {s : Finset α} {t : Finset α} (h : s ⊂ t) : Finset.card s < Finset.card t

theorem Finset.card_eq_of_bijective {α : Type u_1} {s : Finset α} {n : ℕ} (f : (i : ℕ) → i < n → α) (hf : ∀ (a : α), a ∈ s → ∃ i h, f i h = a) (hf' : ∀ (i : ℕ) (h : i < n), f i h ∈ s) (f_inj : ∀ (i j : ℕ) (hi : i < n) (hj : j < n), f i hi = f j hj → i = j) : Finset.card s = n

theorem Finset.card_congr {α : Type u_1} {β : Type u_2} {s : Finset α} {t : Finset β} (f : (a : α) → a ∈ s → β) (h₁ : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t) (h₂ : ∀ (a b : α) (ha : a ∈ s) (hb : b ∈ s), f a ha = f b hb → a = b) (h₃ : ∀ (b : β), b ∈ t → ∃ a ha, f a ha = b) : Finset.card s = Finset.card t

theorem Finset.card_le_card_of_inj_on {α : Type u_1} {β : Type u_2} {s : Finset α} {t : Finset β} (f : α → β) (hf : ∀ (a : α), a ∈ s → f a ∈ t) (f_inj : ∀ (a₁ : α), a₁ ∈ s → ∀ (a₂ : α), a₂ ∈ s → f a₁ = f a₂ → a₁ = a₂) : Finset.card s ≤ Finset.card t

theorem Finset.exists_ne_map_eq_of_card_lt_of_maps_to {α : Type u_1} {β : Type u_2} {s : Finset α} {t : Finset β} (hc : Finset.card t < Finset.card s) {f : α → β} (hf : ∀ (a : α), a ∈ s → f a ∈ t) : ∃ x, x ∈ s ∧ ∃ y, y ∈ s ∧ x ≠ y ∧ f x = f y

If there are more pigeons than pigeonholes, then there are two pigeons in the same pigeonhole.

theorem Finset.le_card_of_inj_on_range {α : Type u_1} {s : Finset α} {n : ℕ} (f : ℕ → α) (hf : ∀ (i : ℕ), i < n → f i ∈ s) (f_inj : ∀ (i : ℕ), i < n → ∀ (j : ℕ), j < n → f i = f j → i = j) : n ≤ Finset.card s

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 : (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 : Finset.card t ≤ Finset.card s) (b : β) : b ∈ t → ∃ a ha, b = f a ha

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 : (a : α) → a ∈ s → β) (hf : ∀ (a : α) (ha : a ∈ s), f a ha ∈ t) (hsurj : ∀ (b : β), b ∈ t → ∃ a ha, b = f a ha) (hst : Finset.card s ≤ Finset.card t) ⦃a₁ : α⦄ ⦃a₂ : α⦄ (ha₁ : a₁ ∈ s) (ha₂ : a₂ ∈ s) (ha₁a₂ : f a₁ ha₁ = f a₂ ha₂) : a₁ = a₂

@[simp]

theorem Finset.card_disjUnion {α : Type u_1} (s : Finset α) (t : Finset α) (h : Disjoint s t) : Finset.card (Finset.disjUnion s t h) = Finset.card s + Finset.card t

Lattice structure

theorem Finset.card_union_add_card_inter {α : Type u_1} [DecidableEq α] (s : Finset α) (t : Finset α) : Finset.card (s ∪ t) + Finset.card (s ∩ t) = Finset.card s + Finset.card t

theorem Finset.card_inter_add_card_union {α : Type u_1} [DecidableEq α] (s : Finset α) (t : Finset α) : Finset.card (s ∩ t) + Finset.card (s ∪ t) = Finset.card s + Finset.card t

theorem Finset.card_union_le {α : Type u_1} [DecidableEq α] (s : Finset α) (t : Finset α) : Finset.card (s ∪ t) ≤ Finset.card s + Finset.card t

theorem Finset.card_union_eq {α : Type u_1} {s : Finset α} {t : Finset α} [DecidableEq α] (h : Disjoint s t) : Finset.card (s ∪ t) = Finset.card s + Finset.card t

@[simp]

theorem Finset.card_disjoint_union {α : Type u_1} {s : Finset α} {t : Finset α} [DecidableEq α] (h : Disjoint s t) : Finset.card (s ∪ t) = Finset.card s + Finset.card t

theorem Finset.card_sdiff {α : Type u_1} {s : Finset α} {t : Finset α} [DecidableEq α] (h : s ⊆ t) : Finset.card (t \ s) = Finset.card t - Finset.card s

theorem Finset.card_sdiff_add_card_eq_card {α : Type u_1} [DecidableEq α] {s : Finset α} {t : Finset α} (h : s ⊆ t) : Finset.card (t \ s) + Finset.card s = Finset.card t

theorem Finset.le_card_sdiff {α : Type u_1} [DecidableEq α] (s : Finset α) (t : Finset α) : Finset.card t - Finset.card s ≤ Finset.card (t \ s)

theorem Finset.card_le_card_sdiff_add_card {α : Type u_1} {s : Finset α} {t : Finset α} [DecidableEq α] : Finset.card s ≤ Finset.card (s \ t) + Finset.card t

theorem Finset.card_sdiff_add_card {α : Type u_1} {s : Finset α} {t : Finset α} [DecidableEq α] : Finset.card (s \ t) + Finset.card t = Finset.card (s ∪ t)

theorem Finset.card_sdiff_comm {α : Type u_1} {s : Finset α} {t : Finset α} [DecidableEq α] (h : Finset.card s = Finset.card t) : Finset.card (s \ t) = Finset.card (t \ s)

theorem Finset.filter_card_add_filter_neg_card_eq_card {α : Type u_1} {s : Finset α} (p : α → Prop) [DecidablePred p] [(x : α) → Decidable ¬p x] : Finset.card (Finset.filter p s) + Finset.card (Finset.filter (fun a => ¬p a) s) = Finset.card s

theorem Finset.exists_intermediate_set {α : Type u_1} {A : Finset α} {B : Finset α} (i : ℕ) (h₁ : i + Finset.card B ≤ Finset.card A) (h₂ : B ⊆ A) : ∃ C, B ⊆ C ∧ C ⊆ A ∧ Finset.card C = i + Finset.card B

Given a set A and a set B inside it, we can shrink A to any appropriate size, and keep B inside it.

theorem Finset.exists_smaller_set {α : Type u_1} (A : Finset α) (i : ℕ) (h₁ : i ≤ Finset.card A) : ∃ B, B ⊆ A ∧ Finset.card B = i

We can shrink A to any smaller size.

theorem Finset.exists_subset_or_subset_of_two_mul_lt_card {α : Type u_1} [DecidableEq α] {X : Finset α} {Y : Finset α} {n : ℕ} (hXY : 2 * n < Finset.card (X ∪ Y)) : ∃ C, n < Finset.card C ∧ (C ⊆ X ∨ C ⊆ Y)

Explicit description of a finset from its card

theorem Finset.card_eq_one {α : Type u_1} {s : Finset α} : Finset.card s = 1 ↔ ∃ a, s = {a}

theorem Finset.exists_eq_insert_iff {α : Type u_1} [DecidableEq α] {s : Finset α} {t : Finset α} : (∃ a x, insert a s = t) ↔ s ⊆ t ∧ Finset.card s + 1 = Finset.card t

theorem Finset.card_le_one {α : Type u_1} {s : Finset α} : Finset.card s ≤ 1 ↔ ∀ (a : α), a ∈ s → ∀ (b : α), b ∈ s → a = b

theorem Finset.card_le_one_iff {α : Type u_1} {s : Finset α} : Finset.card s ≤ 1 ↔ ∀ {a b : α}, a ∈ s → b ∈ s → a = b

theorem Finset.card_le_one_iff_subsingleton_coe {α : Type u_1} {s : Finset α} : Finset.card s ≤ 1 ↔ Subsingleton { x // x ∈ s }

theorem Finset.card_le_one_iff_subset_singleton {α : Type u_1} {s : Finset α} [Nonempty α] : Finset.card s ≤ 1 ↔ ∃ x, s ⊆ {x}

theorem Finset.card_le_one_of_subsingleton {α : Type u_1} [Subsingleton α] (s : Finset α) : Finset.card s ≤ 1

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

theorem Finset.one_lt_card {α : Type u_1} {s : Finset α} : 1 < Finset.card s ↔ ∃ a, a ∈ s ∧ ∃ b, b ∈ s ∧ a ≠ b

theorem Finset.one_lt_card_iff {α : Type u_1} {s : Finset α} : 1 < Finset.card s ↔ ∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b

theorem Finset.one_lt_card_iff_nontrivial_coe {α : Type u_1} {s : Finset α} : 1 < Finset.card s ↔ Nontrivial { x // x ∈ s }

theorem Finset.two_lt_card_iff {α : Type u_1} {s : Finset α} : 2 < Finset.card s ↔ ∃ a b c, a ∈ s ∧ b ∈ s ∧ c ∈ s ∧ a ≠ b ∧ a ≠ c ∧ b ≠ c

theorem Finset.two_lt_card {α : Type u_1} {s : Finset α} : 2 < Finset.card s ↔ ∃ a, a ∈ s ∧ ∃ b, b ∈ s ∧ ∃ c, c ∈ s ∧ a ≠ b ∧ a ≠ c ∧ b ≠ c

theorem Finset.exists_ne_of_one_lt_card {α : Type u_1} {s : Finset α} (hs : 1 < Finset.card s) (a : α) : ∃ b, b ∈ s ∧ b ≠ a

theorem Finset.exists_of_one_lt_card_pi {ι : Type u_3} {α : ι → Type u_4} [(i : ι) → DecidableEq (α i)] {s : Finset ((i : ι) → α i)} (h : 1 < Finset.card s) : ∃ i, 1 < Finset.card (Finset.image (fun x => x i) s) ∧ ∀ (ai : α i), Finset.filter (fun x => x i = ai) s ⊂ s

If a Finset in a Pi type is nontrivial (has at least two elements), then its projection to some factor is nontrivial, and the fibers of the projection are proper subsets.

theorem Finset.card_eq_succ {α : Type u_1} {s : Finset α} {n : ℕ} [DecidableEq α] : Finset.card s = n + 1 ↔ ∃ a t, ¬a ∈ t ∧ insert a t = s ∧ Finset.card t = n

theorem Finset.card_eq_two {α : Type u_1} {s : Finset α} [DecidableEq α] : Finset.card s = 2 ↔ ∃ x y, x ≠ y ∧ s = {x, y}

theorem Finset.card_eq_three {α : Type u_1} {s : Finset α} [DecidableEq α] : Finset.card s = 3 ↔ ∃ x y z, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ s = {x, y, z}

Inductions

def Finset.strongInduction {α : Type u_1} {p : Finset α → Sort u_3} (H : (s : Finset α) → ((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.strongInduction H x = H x fun t h => let_fun this := (_ : Finset.card t < Finset.card x); Finset.strongInduction H t

theorem Finset.strongInduction_eq {α : Type u_1} {p : Finset α → Sort u_3} (H : (s : Finset α) → ((t : Finset α) → t ⊂ s → p t) → p s) (s : Finset α) : Finset.strongInduction H s = H s fun t x => Finset.strongInduction H t

def Finset.strongInductionOn {α : Type u_1} {p : Finset α → Sort u_3} (s : Finset α) : ((s : Finset α) → ((t : Finset α) → t ⊂ s → p t) → p s) → p s

Analogue of strongInduction with order of arguments swapped.

Equations

• Finset.strongInductionOn s H = Finset.strongInduction H s

theorem Finset.strongInductionOn_eq {α : Type u_1} {p : Finset α → Sort u_3} (s : Finset α) (H : (s : Finset α) → ((t : Finset α) → t ⊂ s → p t) → p s) : Finset.strongInductionOn s H = H s fun t x => Finset.strongInductionOn t H

theorem Finset.case_strong_induction_on {α : Type u_1} [DecidableEq 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 (insert a s)) : p s

def Finset.strongDownwardInduction {α : Type u_1} {p : Finset α → Sort u_3} {n : ℕ} (H : (t₁ : Finset α) → ({t₂ : Finset α} → Finset.card t₂ ≤ n → t₁ ⊂ t₂ → p t₂) → Finset.card t₁ ≤ n → p t₁) (s : Finset α) : Finset.card s ≤ n → p s

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.strongDownwardInduction H x = H x fun {t} ht h => let_fun this := (_ : n - Finset.card t < n - Finset.card x); Finset.strongDownwardInduction H t ht

theorem Finset.strongDownwardInduction_eq {α : Type u_1} {n : ℕ} {p : Finset α → Sort u_3} (H : (t₁ : Finset α) → ({t₂ : Finset α} → Finset.card t₂ ≤ n → t₁ ⊂ t₂ → p t₂) → Finset.card t₁ ≤ n → p t₁) (s : Finset α) : Finset.strongDownwardInduction H s = H s fun {t} ht x => Finset.strongDownwardInduction H t ht

def Finset.strongDownwardInductionOn {α : Type u_1} {n : ℕ} {p : Finset α → Sort u_3} (s : Finset α) (H : (t₁ : Finset α) → ({t₂ : Finset α} → Finset.card t₂ ≤ n → t₁ ⊂ t₂ → p t₂) → Finset.card t₁ ≤ n → p t₁) : Finset.card s ≤ n → p s

Analogue of strongDownwardInduction with order of arguments swapped.

Equations

• Finset.strongDownwardInductionOn s H = Finset.strongDownwardInduction H s

theorem Finset.strongDownwardInductionOn_eq {α : Type u_1} {n : ℕ} {p : Finset α → Sort u_3} (s : Finset α) (H : (t₁ : Finset α) → ({t₂ : Finset α} → Finset.card t₂ ≤ n → t₁ ⊂ t₂ → p t₂) → Finset.card t₁ ≤ n → p t₁) : (fun a => Finset.strongDownwardInductionOn s H a) = H s fun {t} ht x => Finset.strongDownwardInductionOn t H ht

theorem Finset.lt_wf {α : Type u_3} : WellFounded LT.lt