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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 | import OAI.Probability.MatroidProphet.Pivots.Guarantee
import OAI.Probability.MatroidProphet.Pivots.Existence
namespace OAI
namespace MatroidProphet
namespace Pivots
open Set Finset
variable {α : Type*} [Fintype α] {r q n : ℕ}
lemma recordsPivots_exists
{α : Type u_1} [Fintype α] {r : ℕ} {q : ℕ} {n : ℕ} (M : Matroid α) (label : Occurrence r q → α)
(test : Fin n → α) (time : Occurrence r q → ℕ)
(oldMark : Fin r → Bool) (movableMark : Fin q → Bool)
(hnonloop : ∀ f, test f ∉ M.closure ∅)
(hspan : ∀ f, test f ∈ M.closure (Set.range label)) :
∃ t, RecordsPivots M label test time oldMark movableMark t := by
classical
choose pivot hpivot using fun f => exists_pivot M label time (hnonloop f) (hspan f)
refine ⟨Finset.univ.filter (fun f => occurrenceMark oldMark movableMark (pivot f) = true), ?_⟩
intro f
exact ⟨pivot f, hpivot f, by simp⟩
lemma recorded_pattern_exists_of_old_spanning (M : Matroid α) (b : ℕ → α) (a : Fin q → α)
(test : Fin n → α) (oldMark : Fin r → Bool)
(holdspan : M.closure (oldPrefix b r) = M.E)
(htest : ∀ f, test f ∈ M.E ∧ test f ∉ M.closure ∅)
(time : Occurrence r q → ℕ) (horder : OldOrdered time) (htime : Function.Injective time)
(movableMark : Fin q → Bool) :
∃ t ∈ markedPivotPatterns M b a test oldMark,
RecordsPivots M (occurrenceLabel b a) test time oldMark movableMark t := by
classical
have hspan : ∀ f, test f ∈ M.closure (Set.range (occurrenceLabel (r := r) b a)) := by
intro f
have htest' : test f ∈ M.closure (oldPrefix b r) := holdspan.symm ▸ (htest f).1
apply M.closure_subset_closure ?_ htest'
rintro _ ⟨k, hk, rfl⟩
exact ⟨Sum.inl ⟨k, hk⟩, rfl⟩
obtain ⟨t, ht⟩ := recordsPivots_exists M (occurrenceLabel b a) test time oldMark movableMark
(fun f => (htest f).2) hspan
exact ⟨t, Finset.mem_filter.mpr ⟨Finset.mem_univ _, time, horder, htime, movableMark, ht⟩, ht⟩
end Pivots
end MatroidProphet
end OAI
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