| import OAI.Combinatorics.EditApproximation.Geometry.ChargedPhysicalRequest |
| import OAI.Combinatorics.EditApproximation.Refinement.PhysicalMedianQuery |
| import OAI.Combinatorics.EditApproximation.Definitions |
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| namespace OAI |
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| /-! A pass-zero seed read requests a cached coarse entry. Later seed reads |
| request the previous pass's final table value in each distinct copy. |
| Thus neither kind of seed value enters the charged program as a free oracle. -/ |
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| namespace EditApproximation |
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| theorem chargedSeedQuery_value {M J ny passes copies T S : ℕ} (N : ℕ) (multiplier : ℕ → ℕ) |
| (parent : {r : PhysicalTableRequest M J ny // r.Bounded passes copies T S}) |
| (q : TargetInterval ny) (answer : ChargedPhysicalRequest M J ny passes copies T S → ℚ) : |
| (chargedSeedQuery N multiplier parent (parent.val.node, q)).value (fun child => answer child.val) = |
| if parent.val.pass = 0 then answer (Sum.inl (parent.val.node, q)) else |
| (min N (upperMedian (List.ofFn fun copy : Fin copies => |
| ⌈(multiplier (parent.val.pass - 1) : ℚ) * |
| answer (Sum.inr (chargedPreviousPass parent q copy))⌉₊)) : ℕ) := by |
| unfold chargedSeedQuery |
| rw [dite_eq_left rfl] |
| split_ifs with hp |
| · rfl |
| · rw [FiniteQuery.collect_value] |
| simp only [chargedPreviousRead, FiniteQuery.value, List.map_map, |
| List.ofFn_eq_map, Function.comp_def] |
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| theorem chargedSeedQuery_cost {M J ny passes copies T S : ℕ} (N : ℕ) (multiplier : ℕ → ℕ) |
| (parent : {r : PhysicalTableRequest M J ny // r.Bounded passes copies T S}) |
| (state : PhysicalEntry M J ny) |
| (oracle : {child : ChargedPhysicalRequest M J ny passes copies T S // |
| child.rank < ChargedPhysicalRequest.rank (Sum.inr parent)} → ℚ) : |
| (chargedSeedQuery N multiplier parent state).cost oracle (fun _ => 1) = |
| if state.1 = parent.val.node then if parent.val.pass = 0 then 1 else copies else 0 := by |
| unfold chargedSeedQuery |
| split_ifs with hnode hpass |
| · rfl |
| · rw [FiniteQuery.cost_collect] |
| simp only [chargedPreviousRead, FiniteQuery.cost, Nat.add_zero, List.map_const', |
| List.length_finRange, List.sum_replicate, smul_eq_mul, Nat.mul_one] |
| · rfl |
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| end EditApproximation |
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| end OAI |
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