Documentation

Mathlib.Combinatorics.DoubleCounting

Double countings #

This file gathers a few double counting arguments.

Bipartite graphs #

In a bipartite graph (considered as a relation r : α → β → Prop), we can bound the number of edges between s : Finset α and t : Finset β by the minimum/maximum of edges over all a ∈ s times the the size of s. Similarly for t. Combining those two yields inequalities between the sizes of s and t.

Bipartite graph #

def Finset.bipartiteBelow {α : Type u_1} {β : Type u_2} (r : αβProp) (s : Finset α) (b : β) [inst : (a : α) → Decidable (r a b)] :

Elements of s which are "below" b according to relation r.

Equations
def Finset.bipartiteAbove {α : Type u_1} {β : Type u_2} (r : αβProp) (t : Finset β) (a : α) [inst : DecidablePred (r a)] :

Elements of t which are "above" a according to relation r.

Equations
theorem Finset.bipartiteBelow_swap {α : Type u_2} {β : Type u_1} (r : αβProp) (t : Finset β) (a : α) [inst : DecidablePred (r a)] :
theorem Finset.bipartiteAbove_swap {α : Type u_1} {β : Type u_2} (r : αβProp) (s : Finset α) (b : β) [inst : (a : α) → Decidable (r a b)] :
@[simp]
theorem Finset.coe_bipartiteBelow {α : Type u_1} {β : Type u_2} (r : αβProp) (s : Finset α) (b : β) [inst : (a : α) → Decidable (r a b)] :
↑(Finset.bipartiteBelow r s b) = { a | a s r a b }
@[simp]
theorem Finset.coe_bipartiteAbove {α : Type u_2} {β : Type u_1} (r : αβProp) (t : Finset β) (a : α) [inst : DecidablePred (r a)] :
↑(Finset.bipartiteAbove r t a) = { b | b t r a b }
@[simp]
theorem Finset.mem_bipartiteBelow {α : Type u_1} {β : Type u_2} (r : αβProp) {s : Finset α} {b : β} [inst : (a : α) → Decidable (r a b)] {a : α} :
a Finset.bipartiteBelow r s b a s r a b
@[simp]
theorem Finset.mem_bipartiteAbove {α : Type u_2} {β : Type u_1} (r : αβProp) {t : Finset β} {a : α} [inst : DecidablePred (r a)] {b : β} :
b Finset.bipartiteAbove r t a b t r a b
theorem Finset.sum_card_bipartiteAbove_eq_sum_card_bipartiteBelow {α : Type u_1} {β : Type u_2} (r : αβProp) {s : Finset α} {t : Finset β} [inst : (a : α) → (b : β) → Decidable (r a b)] :
theorem Finset.card_mul_le_card_mul {α : Type u_1} {β : Type u_2} (r : αβProp) {s : Finset α} {t : Finset β} {m : } {n : } [inst : (a : α) → (b : β) → Decidable (r a b)] (hm : ∀ (a : α), a sm Finset.card (Finset.bipartiteAbove r t a)) (hn : ∀ (b : β), b tFinset.card (Finset.bipartiteBelow r s b) n) :

Double counting argument. Considering r as a bipartite graph, the LHS is a lower bound on the number of edges while the RHS is an upper bound.

theorem Finset.card_mul_le_card_mul' {α : Type u_2} {β : Type u_1} (r : αβProp) {s : Finset α} {t : Finset β} {m : } {n : } [inst : (a : α) → (b : β) → Decidable (r a b)] (hn : ∀ (b : β), b tn Finset.card (Finset.bipartiteBelow r s b)) (hm : ∀ (a : α), a sFinset.card (Finset.bipartiteAbove r t a) m) :
theorem Finset.card_mul_eq_card_mul {α : Type u_1} {β : Type u_2} (r : αβProp) {s : Finset α} {t : Finset β} {m : } {n : } [inst : (a : α) → (b : β) → Decidable (r a b)] (hm : ∀ (a : α), a sFinset.card (Finset.bipartiteAbove r t a) = m) (hn : ∀ (b : β), b tFinset.card (Finset.bipartiteBelow r s b) = n) :
theorem Finset.card_le_card_of_forall_subsingleton {α : Type u_1} {β : Type u_2} (r : αβProp) {s : Finset α} {t : Finset β} (hs : ∀ (a : α), a sb, b t r a b) (ht : ∀ (b : β), b tSet.Subsingleton { a | a s r a b }) :
theorem Finset.card_le_card_of_forall_subsingleton' {α : Type u_2} {β : Type u_1} (r : αβProp) {s : Finset α} {t : Finset β} (ht : ∀ (b : β), b ta, a s r a b) (hs : ∀ (a : α), a sSet.Subsingleton { b | b t r a b }) :
theorem Fintype.card_le_card_of_leftTotal_unique {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst : Fintype β] {r : αβProp} (h₁ : Relator.LeftTotal r) (h₂ : Relator.LeftUnique r) :
theorem Fintype.card_le_card_of_rightTotal_unique {α : Type u_1} {β : Type u_2} [inst : Fintype α] [inst : Fintype β] {r : αβProp} (h₁ : Relator.RightTotal r) (h₂ : Relator.RightUnique r) :