| import Mathlib.Order.Interval.Finset.Fin |
| import OAI.Combinatorics.BalancedRyser.Probability.ExposureSchedule |
|
|
| namespace OAI |
|
|
| namespace Balanced.ExposureInterleaving |
|
|
| theorem previous_card_eq_time (hγ : 0 < γ) |
| (t : Fin (n + m)) : |
| (previousD γ (timeEquiv hγ t)).card + |
| (previousC γ (timeEquiv hγ t)).card = t.val := by |
| classical |
| rw [← Finset.card_disjSum] |
| have heq : (previousD γ (timeEquiv hγ t)).disjSum |
| (previousC γ (timeEquiv hγ t)) = |
| (Finset.Iio t).map (timeEquiv hγ).toEmbedding := by |
| ext s |
| rcases s with i | j |
| · simp only [Finset.inl_mem_disjSum, mem_previousD_iff_time hγ, Finset.mem_map, |
| Finset.mem_Iio, Equiv.toEmbedding_apply] |
| constructor |
| · intro hi |
| exact ⟨deletionTime hγ i, hi, by simp [deletionTime]⟩ |
| · rintro ⟨u, hu, he⟩ |
| have h : u = deletionTime hγ i := by |
| apply (timeEquiv hγ).injective |
| simpa [deletionTime] using he |
| simpa [h] using hu |
| · simp only [Finset.inr_mem_disjSum, mem_previousC_iff_time hγ, Finset.mem_map, |
| Finset.mem_Iio, Equiv.toEmbedding_apply] |
| constructor |
| · intro hj |
| exact ⟨coverTime hγ j, hj, by simp [coverTime]⟩ |
| · rintro ⟨u, hu, he⟩ |
| have h : u = coverTime hγ j := by |
| apply (timeEquiv hγ).injective |
| simpa [coverTime] using he |
| simpa [h] using hu |
| rw [heq, Finset.card_map, Fin.card_Iio] |
|
|
| theorem remaining_card_eq_time (hγ : 0 < γ) |
| (t : Fin (n + m)) : |
| (remainingD γ (timeEquiv hγ t)).card + |
| (remainingC γ (timeEquiv hγ t)).card = n + m - t.val := by |
| have hD := previousD_add_remainingD γ (timeEquiv hγ t) |
| have hC := previousC_add_remainingC γ (timeEquiv hγ t) |
| have hT := previous_card_eq_time hγ t |
| omega |
| end Balanced.ExposureInterleaving |
|
|
| end OAI |
|
|
