| import Mathlib.Algebra.BigOperators.Ring.Finset |
| import Mathlib.Algebra.Order.BigOperators.Group.Finset |
| import Mathlib.Algebra.Order.Field.Rat |
|
|
| namespace OAI |
|
|
| universe uΩ uI |
|
|
| namespace PeriodicTilingThree |
|
|
| open scoped BigOperators |
|
|
| theorem exists_avoiding_of_sum_card_lt {Ω : Type uΩ} {I : Type uI} [Fintype Ω] |
| (events : Finset I) (bad : I → Finset Ω) |
| (hsmall : (∑ i ∈ events, (bad i).card) < Fintype.card Ω) : |
| ∃ ω, ∀ i ∈ events, ω ∉ bad i := by |
| classical |
| have hUnion : (events.biUnion bad).card < (Finset.univ : Finset Ω).card := by |
| rw [Finset.card_univ] |
| exact lt_of_le_of_lt Finset.card_biUnion_le hsmall |
| obtain ⟨ω, _, hω⟩ := Finset.exists_mem_notMem_of_card_lt_card hUnion |
| refine ⟨ω, ?_⟩ |
| intro i hi hbad |
| exact hω (Finset.mem_biUnion.mpr ⟨i, hi, hbad⟩) |
|
|
| theorem exists_avoiding_of_uniform_ratio_lt {Ω : Type uΩ} {I : Type uI} [Fintype Ω] |
| (events : Finset I) (bad : I → Finset Ω) (r : ℚ) |
| (hΩ : 0 < Fintype.card Ω) |
| (hbound : ∀ i ∈ events, ((bad i).card : ℚ) / (Fintype.card Ω : ℚ) ≤ r) |
| (hsmall : (events.card : ℚ) * r < 1) : |
| ∃ ω, ∀ i ∈ events, ω ∉ bad i := by |
| have hpos : (0 : ℚ) < (Fintype.card Ω : ℚ) := Nat.cast_pos.mpr hΩ |
| have hsum : (∑ i ∈ events, ((bad i).card : ℚ)) < (Fintype.card Ω : ℚ) := by |
| calc |
| (∑ i ∈ events, ((bad i).card : ℚ)) |
| ≤ ∑ i ∈ events, r * (Fintype.card Ω : ℚ) := |
| Finset.sum_le_sum fun i hi => (div_le_iff₀ hpos).mp (hbound i hi) |
| _ = (events.card : ℚ) * r * (Fintype.card Ω : ℚ) := by |
| simp only [Finset.sum_const, nsmul_eq_mul, mul_assoc] |
| _ < 1 * (Fintype.card Ω : ℚ) := mul_lt_mul_of_pos_right hsmall hpos |
| _ = (Fintype.card Ω : ℚ) := one_mul _ |
| apply exists_avoiding_of_sum_card_lt events bad |
| apply (Nat.cast_lt (α := ℚ)).mp |
| simpa only [Nat.cast_sum] using hsum |
|
|
| end PeriodicTilingThree |
|
|
| end OAI |
|
|
