File size: 5,226 Bytes
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 | import OAI.Combinatorics.EditApproximation.Geometry.ChargedRandomCost
namespace OAI
namespace EditApproximation
section Costs
variable {σ α : Type*} [DecidableEq α] {M J R : ℕ} [NeZero M] [NeZero R]
(source target : List α) (N P F exponent H Q T S passes copies : ℕ)
(factor : ℕ → ℚ) (multiplier : ℕ → ℕ) (tau eta kappa delta : ℚ)
(readGroup : ℕ → ℕ → PhysicalInternalNode M J → Fin R → GroupScalarKey M F Q → FiniteQuery σ)
(readInteger : ℕ → ℕ → PhysicalInternalNode M J → Fin R → (ℕ × TargetInterval target.length) → ℕ → FiniteQuery σ)
(sourceAnswer : σ → ℚ) (answer : PhysicalTableRequest M J target.length → ℚ)
(initial : PhysicalEntry M J target.length → ℕ)
(parent : {r : PhysicalTableRequest M J target.length // r.Bounded passes copies T S})
(hindex : T < parent.val.index) (hbelow : parent.val.node.1.val < J)
(hshort : ¬ (physicalSourceInterval source.length parent.val.node).hi -
(physicalSourceInterval source.length parent.val.node).lo ≤ 1)
include hindex hbelow hshort
theorem chargedRandomGlobalQuery_current_cost :
(chargedRandomGlobalQuery source target N P F exponent H Q T S passes copies factor multiplier
tau eta kappa delta readGroup readInteger initial (Sum.inr parent)).cost
(Sum.elim sourceAnswer (fun child => Sum.elim (fun entry => (initial entry : ℚ))
(fun r => answer r.val) child.val))
(Sum.elim (fun _ => 0) (fun child => chargedSameWeight (Sum.inr parent) child.val)) =
(physicalRandomDataRequestQuery source target N P F exponent H Q tau
(refinementFactor (factor parent.val.pass) F) eta kappa delta parent.val.pass parent.val.copy T S
(parent.val.index - T) (Nat.sub_pos_of_lt hindex) parent.val.node hbelow
(readGroup parent.val.pass parent.val.copy ⟨parent.val.node, hbelow⟩ (Fin.ofNat R (parent.val.index - T - 1)))
(readInteger parent.val.pass parent.val.copy ⟨parent.val.node, hbelow⟩ (Fin.ofNat R (parent.val.index - T - 1)))
parent.val.state).cost
(Sum.elim sourceAnswer (fun child => answer child.val))
(Sum.elim (fun _ => 0) (fun child => physicalCurrentReadWeight (T + (parent.val.index - T)) child.val)) := by
simp only [chargedSameWeight_table]
rw [chargedRandomGlobalQuery_refinement_cost source target N P F exponent H Q T S passes copies
factor multiplier tau eta kappa delta readGroup readInteger initial sourceAnswer answer (fun _ => 0) _ _ parent hindex hbelow hshort]
have hrank : (PhysicalTableRequest.refinement parent.val.pass parent.val.copy parent.val.node
T (parent.val.index - T) parent.val.state).rank T S = parent.val.rank T S := by
unfold PhysicalTableRequest.refinement PhysicalTableRequest.rank
dsimp only
congr 1
omega
rw [← hrank]
exact physicalRandomDataRequest_same_rank_cost source target N P F exponent H Q tau
(refinementFactor (factor parent.val.pass) F) eta kappa delta parent.val.pass parent.val.copy T S
(parent.val.index - T) (Nat.sub_pos_of_lt hindex) parent.val.node hbelow _ _ sourceAnswer parent.val.state answer
theorem chargedRandomGlobalQuery_earlier_cost :
(chargedRandomGlobalQuery source target N P F exponent H Q T S passes copies factor multiplier
tau eta kappa delta readGroup readInteger initial (Sum.inr parent)).cost
(Sum.elim sourceAnswer (fun child => Sum.elim (fun entry => (initial entry : ℚ))
(fun r => answer r.val) child.val))
(Sum.elim (fun _ => 0) (fun child => chargedEarlierWeight (Sum.inr parent) child.val)) =
(physicalRandomDataRequestQuery source target N P F exponent H Q tau
(refinementFactor (factor parent.val.pass) F) eta kappa delta parent.val.pass parent.val.copy T S
(parent.val.index - T) (Nat.sub_pos_of_lt hindex) parent.val.node hbelow
(readGroup parent.val.pass parent.val.copy ⟨parent.val.node, hbelow⟩ (Fin.ofNat R (parent.val.index - T - 1)))
(readInteger parent.val.pass parent.val.copy ⟨parent.val.node, hbelow⟩ (Fin.ofNat R (parent.val.index - T - 1)))
parent.val.state).cost
(Sum.elim sourceAnswer (fun child => answer child.val))
(Sum.elim (fun _ => 0) (fun child => physicalEarlierReadWeight (T + (parent.val.index - T)) child.val)) := by
simp only [chargedEarlierWeight_table]
rw [chargedRandomGlobalQuery_refinement_cost source target N P F exponent H Q T S passes copies
factor multiplier tau eta kappa delta readGroup readInteger initial sourceAnswer answer (fun _ => 0) _ _ parent hindex hbelow hshort]
have hrank : (PhysicalTableRequest.refinement parent.val.pass parent.val.copy parent.val.node
T (parent.val.index - T) parent.val.state).rank T S = parent.val.rank T S := by
unfold PhysicalTableRequest.refinement PhysicalTableRequest.rank
dsimp only
congr 1
omega
rw [← hrank]
exact physicalRandomDataRequest_earlier_rank_cost source target N P F exponent H Q tau
(refinementFactor (factor parent.val.pass) F) eta kappa delta parent.val.pass parent.val.copy T S
(parent.val.index - T) (Nat.sub_pos_of_lt hindex) parent.val.node hbelow _ _ sourceAnswer parent.val.state answer
end Costs
end EditApproximation
end OAI
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