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e63eb7f | 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 | import OAI.Combinatorics.EditApproximation.Refinement.ComputedEarlierInputBudget
namespace OAI
/-! A query gathers only its preceding history, rather than allocating
the full time range used to bound that history's cardinality. -/
namespace EditApproximation
theorem wideEarlierInputCover_mono_history {nx ny M : ℕ} (parent : TargetInterval nx)
(P small large : ℕ) (representatives : List (ℕ × TargetInterval ny)) (h : small ≤ large) :
wideEarlierInputCover (M := M) parent P small representatives ⊆
wideEarlierInputCover parent P large representatives := by
intro input hinput
obtain ⟨br, hbr, hinput⟩ := Finset.mem_biUnion.mp hinput
obtain ⟨i, hi, hinput⟩ := Finset.mem_biUnion.mp hinput
obtain ⟨t, ht, hinput⟩ := Finset.mem_biUnion.mp hinput
exact Finset.mem_biUnion.mpr ⟨br, hbr, Finset.mem_biUnion.mpr ⟨i, hi,
Finset.mem_biUnion.mpr ⟨t, (Finset.mem_range.mpr ((Finset.mem_range.mp ht).trans_le h)), hinput⟩⟩⟩
theorem computed_queryEarlierHistory_card {nx ny : ℕ}
(parent : TargetInterval nx) (N P history : ℕ) (hN : 0 < N)
(hell : 200 ≤ inputSmallLog N) (hPeq : P = (integerParameters N).P)
(hhistory : history ≤ (integerParameters N).T + (integerParameters N).S + 1)
(A : ℚ) (hA : A ≤ ((integerParameters N).initialFactor : ℚ))
(initial : TargetInterval ny → ℚ) (q : TargetInterval ny) (hw : 0 ≤ initial q) :
(queryEarlierInputCover (M := (integerParameters N).M) parent N P (inputSmallLog N)
(40 * smallLogExponent N) history (refinementFactor A (inputSmallLog N)) initial q).card ≤
inputHeight N ^ (256 * (coarseExponent (inputSmallLog N) + 1)) := by
apply (Finset.card_le_card (wideEarlierInputCover_mono_history parent P history
((integerParameters N).T + (integerParameters N).S + 1) _ hhistory)).trans
exact computed_queryEarlierInputCover_card parent N P hN hell hPeq A hA initial q hw
end EditApproximation
end OAI
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