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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 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 | import OAI.Combinatorics.EditApproximation.Refinement.ComputedGlobalSourceCost
import OAI.Combinatorics.EditApproximation.Geometry.ChargedRandomCost
import OAI.Combinatorics.EditApproximation.Geometry.ChargedTerminalCost
namespace OAI
namespace EditApproximation
section Cost
variable {σ α : Type*} [DecidableEq α] {M J R : ℕ} [NeZero M] [NeZero R]
(source target : List α) (N P H T S passes copies : ℕ)
(factor : ℕ → ℚ) (multiplier : ℕ → ℕ) (tau eta kappa delta : ℚ)
(group : ℕ → ℕ → PhysicalInternalNode M J → Fin R →
CellGroupDraws M (inputSmallLog N) (inputSmallLog N ^ 80))
(integer : ℕ → ℕ → PhysicalInternalNode M J → Fin R → ℕ → TargetInterval target.length → ℕ)
(initial : PhysicalEntry M J target.length → ℕ)
(readGroup : ℕ → ℕ → PhysicalInternalNode M J → Fin R →
GroupScalarKey M (inputSmallLog N) (inputSmallLog N ^ 80) → FiniteQuery σ)
(readInteger : ℕ → ℕ → PhysicalInternalNode M J → Fin R →
(ℕ × TargetInterval target.length) → ℕ → FiniteQuery σ)
(sourceAnswer : σ → ℚ) (weight : σ → ℕ) (bound : ℕ)
(hN : 0 < N) (hell : 200 ≤ inputSmallLog N) (hM : M = (integerParameters N).M)
(hA : ∀ pass, factor pass ≤ ((integerParameters N).initialFactor : ℚ))
(hgroup : ∀ pass copy node time key, (readGroup pass copy node time key).cost sourceAnswer weight ≤ bound)
(honline : ∀ pass copy node time key range,
(readInteger pass copy node time key range).cost sourceAnswer weight ≤ bound)
local notation "tableValue" => localRequestValue source target N P (inputSmallLog N)
(40 * smallLogExponent N) H (inputSmallLog N ^ 80) T S copies factor multiplier
tau eta kappa delta group integer initial
include hN hell hM hA hgroup honline in
theorem computed_chargedRandomGlobalQuery_source_cost
(request : ChargedPhysicalRequest M J target.length passes copies T S) :
(chargedRandomGlobalQuery source target N P (inputSmallLog N) (40 * smallLogExponent N)
H (inputSmallLog N ^ 80) T S passes copies factor multiplier tau eta kappa delta
readGroup readInteger initial request).cost
(Sum.elim sourceAnswer (fun child => chargedRequestValue source target N P (inputSmallLog N)
(40 * smallLogExponent N) H (inputSmallLog N ^ 80) T S passes copies factor multiplier
tau eta kappa delta group integer initial child.val))
(Sum.elim weight (fun _ => 0)) ≤ inputHeight N ^ 91 * bound := by
cases request with
| inl entry => exact Nat.zero_le _
| inr parent =>
by_cases hi : parent.val.index ≤ T
· unfold chargedRandomGlobalQuery
dsimp only
rw [ite_eq_left hi, FiniteQuery.mapKeys_cost]
simp only [Function.comp_def, Sum.elim_inr]
rw [FiniteQuery.cost_zero]
exact Nat.zero_le _
· by_cases hb : parent.val.node.1.val < J
· by_cases hs : (physicalSourceInterval source.length parent.val.node).hi -
(physicalSourceInterval source.length parent.val.node).lo ≤ 1
· unfold chargedRandomGlobalQuery
dsimp only
rw [ite_eq_right hi, ite_eq_left hb, ite_eq_left hs, FiniteQuery.mapKeys_cost]
simp only [Function.comp_def, Sum.elim_inr]
rw [FiniteQuery.cost_zero]
exact Nat.zero_le _
· have hanswer : chargedRequestValue source target N P (inputSmallLog N)
(40 * smallLogExponent N) H (inputSmallLog N ^ 80) T S passes copies factor multiplier
tau eta kappa delta group integer initial =
Sum.elim (fun entry => (initial entry : ℚ)) (fun r => tableValue r.val) := by
funext key
cases key <;> rfl
have hz : (fun _ : ChargedPhysicalRequest M J target.length passes copies T S => 0) =
chargedTableWeight 0 (fun _ => 0) := by
funext key
cases key <;> rfl
rw [hanswer]
have hcost := chargedRandomGlobalQuery_refinement_cost source target N P (inputSmallLog N)
(40 * smallLogExponent N) H (inputSmallLog N ^ 80) T S passes copies factor multiplier
tau eta kappa delta readGroup readInteger initial sourceAnswer tableValue weight 0 (fun _ => 0)
parent (by omega) hb hs
simp only [← hz] at hcost
rw [hcost]
rw [← physicalRandomGlobalQuery_refinement_cost source target N P (inputSmallLog N)
(40 * smallLogExponent N) H (inputSmallLog N ^ 80) T S copies factor multiplier
tau eta kappa delta readGroup readInteger initial sourceAnswer tableValue weight (fun _ => 0)
parent.val (by omega) hb hs]
apply computed_physicalRandomGlobalQuery_source_cost source target N P H T S copies
hN hell hM factor multiplier tau eta kappa delta hA readGroup readInteger initial
sourceAnswer tableValue weight bound hgroup honline
intro pass copy node q
rw [localRequestValue_initial]
exact stageInitial_nonneg_all_seeds _ _ _ _ _ _ _ _
· unfold chargedRandomGlobalQuery
dsimp only
rw [ite_eq_right hi, ite_eq_right hb, FiniteQuery.mapKeys_cost]
simp only [Function.comp_def, Sum.elim_inr]
rw [FiniteQuery.cost_zero]
exact Nat.zero_le _
end Cost
end EditApproximation
end OAI
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