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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 | import OAI.Combinatorics.EditApproximation.Refinement.ComputedChargedRandomBudget
import OAI.Combinatorics.EditApproximation.Geometry.ChargedRandomProgress
import OAI.Combinatorics.EditApproximation.Geometry.ChargedQueryAsymptotics
import OAI.Combinatorics.EditApproximation.Geometry.SourceRankedWork
namespace OAI
namespace EditApproximation
section Expansion
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)
variable
(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)
local notation "mixedProgram" => 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
local notation "program" => SourceRanked.projectedBody ChargedPhysicalRequest.rank mixedProgram sourceAnswer
local notation "values" => 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
local notation "depth" => @ChargedPhysicalRequest.depth M J target.length passes copies T S
local notation "index" => @ChargedPhysicalRequest.index M J target.length passes copies T S
variable (hbody : ∀ (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).value
(Sum.elim sourceAnswer (fun child : {child : ChargedPhysicalRequest M J target.length passes copies T S //
child.rank < request.rank} => 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)) =
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 request)
include hbody
theorem chargedRandomRequestList_progress (root child : ChargedPhysicalRequest M J target.length passes copies T S)
(hchild : child ∈ queryRequestList depth index program root) :
(depth child < depth root ∧ index child ≤ index root) ∨
(depth child ≤ depth root ∧ index child < index root) := by
have hvalue := rankedProgramValue_eq ChargedPhysicalRequest.rank program values
(SourceRanked.projectedBody_value ChargedPhysicalRequest.rank mixedProgram sourceAnswer values hbody)
change child ∈ ((program root).keys
(fun key => rankedProgramValue ChargedPhysicalRequest.rank program key.val)).map Subtype.val at hchild
simp only [hvalue] at hchild
obtain ⟨key, hk, rfl⟩ := List.mem_map.mp hchild
apply FiniteQuery.eraseSources_keys_all sourceAnswer
(fun child => values child.val) (mixedProgram root)
(fun child => ChargedPhysicalRequest.DependencyProgress root child.val) ?_ key hk
intro next hn
have hp := chargedRandomGlobalQuery_keys_progress 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 values root next hn
cases next <;> exact hp
noncomputable def chargedRandomDependencyTree (root : ChargedPhysicalRequest M J target.length passes copies T S) :
DependencyExpansion :=
queryDependencyExpansion depth index program
(chargedRandomRequestList_progress source target N P H T S passes copies factor multiplier
tau eta kappa delta group integer initial readGroup readInteger sourceAnswer hbody) root
variable (hN : 0 < N) (hell : 200 ≤ inputSmallLog N)
(hM : M = (integerParameters N).M) (hP : P = (integerParameters N).P)
(hT : T = (integerParameters N).T) (hS : S = (integerParameters N).S)
(hcopies : copies ≤ inputHeight N ^ 2)
(hAmin : ∀ pass, 1 ≤ factor pass)
(hAmax : ∀ pass, factor pass ≤ ((integerParameters N).initialFactor : ℚ))
include hN hell hM hP hT hS hcopies hAmin hAmax in
theorem chargedRandomDependencyTree_valid (root : ChargedPhysicalRequest M J target.length passes copies T S) :
(chargedRandomDependencyTree source target N P H T S passes copies factor multiplier
tau eta kappa delta group integer initial readGroup readInteger sourceAnswer hbody root).Valid
(M * inputSmallLog N ^ 257)
(inputHeight N ^ (512 * (coarseExponent (inputSmallLog N) + 1))) := by
apply queryDependencyExpansion_valid
· intro key
rw [querySameRequests_length_eq_cost depth index program values
(SourceRanked.projectedBody_value ChargedPhysicalRequest.rank mixedProgram sourceAnswer values hbody)]
change (FiniteQuery.eraseSources sourceAnswer (mixedProgram key)).cost
(fun child => values child.val) (fun child => chargedSameWeight key child.val) ≤ _
rw [FiniteQuery.eraseSources_cost]
have h := computed_chargedRandomGlobalQuery_current_reads source target N P H T S passes copies
factor multiplier tau eta kappa delta group integer initial readGroup readInteger sourceAnswer hN hell hM hAmax hP hT hS key
apply h.trans
calc
2 * M * inputSmallLog N ^ 256 ≤ inputSmallLog N * M * inputSmallLog N ^ 256 := by
gcongr
omega
_ = M * inputSmallLog N ^ 257 := by ring
· intro key
rw [queryEarlierRequests_length_eq_cost depth index program values
(SourceRanked.projectedBody_value ChargedPhysicalRequest.rank mixedProgram sourceAnswer values hbody)]
change (FiniteQuery.eraseSources sourceAnswer (mixedProgram key)).cost
(fun child => values child.val) (fun child => chargedEarlierWeight key child.val) ≤ _
rw [FiniteQuery.eraseSources_cost]
exact computed_chargedRandomGlobalQuery_earlier_reads source target N P H T S passes copies
factor multiplier tau eta kappa delta group integer initial readGroup readInteger sourceAnswer hN hell hM hAmax hP hT hS hcopies hAmin key
include hN hell hM hP hT hS hcopies hAmin hAmax in
theorem chargedRandomDependencyTree_widthWork_le
(hJ : J = intervalTreeDepth M source.length)
(hpasses : passes = computedSeedPassCount N + 1)
(root : ChargedPhysicalRequest M J target.length passes copies T S)
(hroot : depth root = J) (overhead : ℝ) (hoverhead : 0 ≤ overhead)
(hpath : ∀ j ≤ intervalTreeDepth M source.length,
(dependencyPathBudget j (computedDependencyIndexCount N)
(M * inputSmallLog N ^ 257)
(inputHeight N ^ (512 * (coarseExponent (inputSmallLog N) + 1))) : ℝ) ≤
(M : ℝ) ^ j * overhead) :
((chargedRandomDependencyTree source target N P H T S passes copies factor multiplier
tau eta kappa delta group integer initial readGroup readInteger sourceAnswer hbody root).widthWork M source.length : ℝ) ≤
(((intervalTreeDepth M source.length + 1) * source.length +
4 * M * max 1 source.length : ℕ) : ℝ) * overhead := by
apply DependencyExpansion.widthWork_le M source.length _ _ (computedDependencyIndexCount N)
(by rw [hM]; exact (computed_branching_bounds N hN hell).1) _
(chargedRandomDependencyTree_valid source target N P H T S passes copies factor multiplier
tau eta kappa delta group integer initial readGroup readInteger sourceAnswer hbody hN hell hM hP hT hS hcopies hAmin hAmax root)
· exact (queryDependencyExpansion_depth depth index program
(chargedRandomRequestList_progress source target N P H T S passes copies factor multiplier
tau eta kappa delta group integer initial readGroup readInteger sourceAnswer hbody) root).trans (hroot.trans hJ)
· have hi := chargedRequest_index_le root
have hb : index root ≤ computedDependencyIndexCount N := by
apply hi.trans_eq
unfold computedDependencyIndexCount
rw [hpasses, hT, hS]
exact (queryDependencyExpansion_index depth index program
(chargedRandomRequestList_progress source target N P H T S passes copies factor multiplier
tau eta kappa delta group integer initial readGroup readInteger sourceAnswer hbody) root).le.trans hb
· exact hoverhead
· exact hpath
end Expansion
end EditApproximation
end OAI
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