| import OAI.Probability.MatroidProphet.Pivots.Patterns |
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| namespace OAI |
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| namespace MatroidProphet |
| namespace Pivots |
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| open Set Finset |
| open scoped BigOperators |
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| variable {α : Type*} [Fintype α] {r q n : ℕ} |
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| noncomputable def markedPivotPatterns (M : Matroid α) (b : ℕ → α) (a : Fin q → α) |
| (test : Fin n → α) (oldMark : Fin r → Bool) : Finset (Finset (Fin n)) := by |
| classical |
| exact Finset.univ.filter (fun t => ∃ time : Occurrence r q → ℕ, |
| OldOrdered time ∧ Function.Injective time ∧ ∃ movableMark : Fin q → Bool, |
| RecordsPivots M (occurrenceLabel b a) test time oldMark movableMark t) |
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| lemma card_markedPivotPatterns_le (M : Matroid α) (b : ℕ → α) (a : Fin q → α) |
| (test : Fin n → α) (oldMark : Fin r → Bool) (s : ℕ) (hqs : q ≤ s) |
| (hground : ∀ o : Occurrence r q, occurrenceLabel b a o ∈ M.E) : |
| (markedPivotPatterns M b a test oldMark).card ≤ |
| 4 ^ s * ∑ k ∈ Finset.Iic s, n.choose k := by |
| classical |
| let family := retainedFamily (r := r) M b a |
| let affine := fun J : Finset (Occurrence r q) => affinePatterns |
| (fun f : Fin n => patternLinear M (occurrenceLabel b a) J (test f)) |
| (fun f : Fin n => patternConstant M (occurrenceLabel b a) J (test f) s oldMark) |
| have hsub : markedPivotPatterns M b a test oldMark ⊆ family.biUnion affine := by |
| intro t ht |
| obtain ⟨time, horder, htime, movableMark, hrecords⟩ := (Finset.mem_filter.mp ht).2 |
| apply Finset.mem_biUnion.mpr |
| refine ⟨retained M b a time, ?_, ?_⟩ |
| · exact Finset.mem_filter.mpr ⟨Finset.mem_univ _, time, horder, rfl⟩ |
| · apply mem_affinePatterns.mpr |
| exact recordsPivots_affine M b a test s hqs oldMark movableMark time horder htime hground t hrecords |
| have hcard (J : Finset (Occurrence r q)) : |
| (affine J).card ≤ ∑ k ∈ Finset.Iic q, n.choose k := by |
| simpa [affine] using card_affinePatterns_le |
| (fun f : Fin n => patternLinear M (occurrenceLabel b a) J (test f)) |
| (fun f : Fin n => patternConstant M (occurrenceLabel b a) J (test f) s oldMark) |
| have hfamily : family.card ≤ 4 ^ s := |
| (card_retainedFamily_arbitrary_old M b a |
| (fun k hk => hground (Sum.inl ⟨k, hk⟩))).trans |
| (Nat.pow_le_pow_right (by decide) hqs) |
| have hsum : (∑ k ∈ Finset.Iic q, n.choose k) ≤ ∑ k ∈ Finset.Iic s, n.choose k := |
| Finset.sum_le_sum_of_subset_of_nonneg (Finset.Iic_subset_Iic.mpr hqs) (fun _ _ _ => Nat.zero_le _) |
| calc |
| (markedPivotPatterns M b a test oldMark).card ≤ (family.biUnion affine).card := Finset.card_le_card hsub |
| _ ≤ ∑ J ∈ family, (affine J).card := Finset.card_biUnion_le |
| _ ≤ ∑ _J ∈ family, ∑ k ∈ Finset.Iic q, n.choose k := Finset.sum_le_sum (fun J _ => hcard J) |
| _ = family.card * (∑ k ∈ Finset.Iic q, n.choose k) := by simp |
| _ ≤ 4 ^ s * ∑ k ∈ Finset.Iic s, n.choose k := Nat.mul_le_mul hfamily hsum |
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| end Pivots |
| end MatroidProphet |
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| end OAI |
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