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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 | import OAI.Combinatorics.EditApproximation.Queries.QueryFamilyMemo
import OAI.Combinatorics.EditApproximation.Binary.BitMemoWorkOn
namespace OAI
/-! The complete mixed memo is normalized on the physical
table grid. Its actual ranked binary values and adaptive reference trace
agree on every coarse seed and every finite scalar draw. -/
namespace EditApproximation
private theorem familyBase_pos (ell : ℕ) (base : List Bool)
(hbase : bitWordValue base = ell ^ 10 + 1) : 0 < bitWordValue base := by
rw [hbase]
omega
section Family
variable {M J R passes copies T S : ℕ} [NeZero M] [NeZero R]
(source target : List ℕ) (N P F exponent H Q eM eF : ℕ)
(hP : 0 < P) (hM : M = 2 ^ eM) (hF : F = 2 ^ eF)
(A a : ℕ → BinaryFraction) (tau eta kappa delta : BinaryFraction) (multiplier : ℕ → ℕ)
(occurrence : BinaryMemo PositionCounts) (symbolBits positionBits : ℕ)
(coarse : PhysicalEntry M J target.length → BinaryFraction × ℕ)
(draw : PhysicalFamilyScalarDraw (M := M) (J := J) (R := R)
target N P F exponent H Q passes copies)
(initial : PhysicalEntry M J target.length → ℕ)
(ha : ∀ pass, pass ≤ passes → (a pass).value = refinementFactor (BinaryFraction.value (A pass)) F)
(hs : ∀ z ∈ source, z < 2 ^ symbolBits) (ht : ∀ z ∈ target, z < 2 ^ symbolBits)
(hp : target.length ≤ 2 ^ positionBits)
(htable : occurrence = occurrenceBuild symbolBits positionBits target.zipIdx)
(hc : ∀ entry, (coarse entry).1.value = initial entry)
(ell : ℕ) (hell : 0 < ell) (hdelta : delta.value = 1 / (ell : ℚ) ^ 10)
(base : List Bool) (hbase : bitWordValue base = ell ^ 10 + 1)
local notation "raw" => queryFamilyMemoRaw source target N P F exponent H Q eM eF hP hM hF
A a tau eta kappa delta multiplier occurrence symbolBits positionBits coarse draw
local notation "reference" => SourceRanked.body ChargedPhysicalRequest.rank
(physicalFamilyProgram source target N P F exponent H Q T S passes copies
(fun pass => BinaryFraction.value (A pass)) multiplier tau.value eta.value kappa.value delta.value initial)
(physicalFamilyAnswer target N P F exponent H Q passes copies draw)
local notation "semantic" => SourceRanked.value
(physicalFamilyAnswer target N P F exponent H Q passes copies draw)
(physicalFamilyRequestValue source target N P F exponent H Q T S passes copies
(fun pass => BinaryFraction.value (A pass)) multiplier tau.value eta.value kappa.value delta.value initial hP draw)
local notation "normalized" => queryFamilyMemoNormalized source target N P F exponent H Q eM eF hP hM hF
A a tau eta kappa delta multiplier occurrence symbolBits positionBits coarse draw
base (familyBase_pos ell base hbase)
include ha hs ht hp htable hc hell hdelta in
theorem queryFamilyMemoNormalized_correct
(key : QueryFamilyMemoKey (M := M) (J := J) (R := R) (passes := passes) (copies := copies) (T := T) (S := S) target N P F exponent H Q) :
(rankedBitProgramValue (SourceRanked.rank ChargedPhysicalRequest.rank) normalized key).value =
semantic key ∧
BitQuery.RefinesOn
(fun child => rankedBitProgramValue (SourceRanked.rank ChargedPhysicalRequest.rank)
normalized child.val)
(fun word value => word.value = value) (normalized key) (reference key) := by
have hraw (key : QueryFamilyMemoKey (M := M) (J := J) (R := R) (passes := passes) (copies := copies) (T := T) (S := S) target N P F exponent H Q) :
BitQuery.Refines (fun word value => word.value = value) (raw key) (reference key) :=
queryFamilyMemoRaw_refines source target N P F exponent H Q eM eF hP hM hF
A a tau eta kappa delta multiplier occurrence symbolBits positionBits coarse draw
ha hs ht hp htable initial hc key
have hbody (key : QueryFamilyMemoKey (M := M) (J := J) (R := R) (passes := passes) (copies := copies) (T := T) (S := S) target N P F exponent H Q) :
(reference key).value (fun child => semantic child.val) = semantic key := by
apply SourceRanked.body_value
intro request
exact physicalFamilyProgram_value source target N P F exponent H Q T S passes copies
(fun pass => BinaryFraction.value (A pass)) multiplier tau.value eta.value kappa.value delta.value initial hP draw request
have hd (key : QueryFamilyMemoKey (M := M) (J := J) (R := R) (passes := passes) (copies := copies) (T := T) (S := S) target N P F exponent H Q) :
(semantic key).den ∣ bitWordValue base ^ queryFamilyMemoRemaining (M := M) (J := J) (R := R) (passes := passes) (copies := copies) (T := T) (S := S) target N P F exponent H Q key := by
cases key with
| inl source =>
change ((draw source).val : ℚ).den ∣ _
rw [Rat.den_natCast]
exact one_dvd _
| inr request =>
change (physicalFamilyRequestValue source target N P F exponent H Q T S passes copies
(fun pass => BinaryFraction.value (A pass)) multiplier tau.value eta.value kappa.value delta.value initial hP draw
request).den ∣ bitWordValue base ^ chargedPhysicalRemaining request
unfold physicalFamilyRequestValue
rw [hdelta, hbase]
exact chargedRequestValue_den_dvd source target N P F exponent H Q T S (passes + 1) copies ell
(fun pass => BinaryFraction.value (A pass)) multiplier tau.value eta.value kappa.value _ _ initial hell request
exact rankedNormalizeBody_correct (SourceRanked.rank ChargedPhysicalRequest.rank) raw reference
semantic hraw hbody base (familyBase_pos ell base hbase)
(queryFamilyMemoRemaining (M := M) (J := J) (R := R) (passes := passes) (copies := copies) (T := T) (S := S) target N P F exponent H Q) hd key
include ha hs ht hp htable hc hell hdelta in
theorem queryFamilyMemoNormalized_width
(key : QueryFamilyMemoKey (M := M) (J := J) (R := R) (passes := passes) (copies := copies) (T := T) (S := S) target N P F exponent H Q)
(B V : ℕ) (hB : base.length ≤ B) (hV : |semantic key| ≤ (V : ℚ)) :
(rankedBitProgramValue (SourceRanked.rank ChargedPhysicalRequest.rank) normalized key).width ≤
Nat.size V + 2 * (queryFamilyMemoRemaining (M := M) (J := J) (R := R)
(passes := passes) (copies := copies) (T := T) (S := S) target N P F exponent H Q key * B + 1) + 1 := by
have hraw (key : QueryFamilyMemoKey (M := M) (J := J) (R := R) (passes := passes) (copies := copies) (T := T) (S := S) target N P F exponent H Q) :
BitQuery.Refines (fun word value => word.value = value) (raw key) (reference key) :=
queryFamilyMemoRaw_refines source target N P F exponent H Q eM eF hP hM hF
A a tau eta kappa delta multiplier occurrence symbolBits positionBits coarse draw
ha hs ht hp htable initial hc key
have hbody (key : QueryFamilyMemoKey (M := M) (J := J) (R := R) (passes := passes) (copies := copies) (T := T) (S := S) target N P F exponent H Q) :
(reference key).value (fun child => semantic child.val) = semantic key := by
apply SourceRanked.body_value
intro request
exact physicalFamilyProgram_value source target N P F exponent H Q T S passes copies
(fun pass => BinaryFraction.value (A pass)) multiplier tau.value eta.value kappa.value delta.value initial hP draw request
have hd (key : QueryFamilyMemoKey (M := M) (J := J) (R := R) (passes := passes) (copies := copies) (T := T) (S := S) target N P F exponent H Q) :
(semantic key).den ∣ bitWordValue base ^ queryFamilyMemoRemaining (M := M) (J := J) (R := R) (passes := passes) (copies := copies) (T := T) (S := S) target N P F exponent H Q key := by
cases key with
| inl source =>
change ((draw source).val : ℚ).den ∣ _
rw [Rat.den_natCast]
exact one_dvd _
| inr request =>
change (physicalFamilyRequestValue source target N P F exponent H Q T S passes copies
(fun pass => BinaryFraction.value (A pass)) multiplier tau.value eta.value kappa.value delta.value initial hP draw
request).den ∣ bitWordValue base ^ chargedPhysicalRemaining request
unfold physicalFamilyRequestValue
rw [hdelta, hbase]
exact chargedRequestValue_den_dvd source target N P F exponent H Q T S (passes + 1) copies ell
(fun pass => BinaryFraction.value (A pass)) multiplier tau.value eta.value kappa.value _ _ initial hell request
exact rankedNormalizeBody_width (SourceRanked.rank ChargedPhysicalRequest.rank) raw reference
semantic hraw hbody base (familyBase_pos ell base hbase)
(queryFamilyMemoRemaining (M := M) (J := J) (R := R) (passes := passes) (copies := copies) (T := T) (S := S) target N P F exponent H Q)
hd key B V hB hV
include ha hs ht hp htable hc hell hdelta in
theorem queryFamilyMemoNormalized_decode (bits : ℕ)
(encode : QueryFamilyMemoKey (M := M) (J := J) (R := R) (passes := passes) (copies := copies)
(T := T) (S := S) target N P F exponent H Q → ℕ)
(hinjective : Function.Injective encode) (hbounded : ∀ key, encode key < 2 ^ bits)
(key : QueryFamilyMemoKey (M := M) (J := J) (R := R) (passes := passes) (copies := copies)
(T := T) (S := S) target N P F exponent H Q) :
(rankedBitMemoEvaluate (SourceRanked.rank ChargedPhysicalRequest.rank) normalized bits encode key
.empty).decode =
rankedMemoEvaluate (SourceRanked.rank ChargedPhysicalRequest.rank) reference bits encode key .empty := by
apply rankedBitMemoEvaluate_decode_on _ _ _ _ bits encode hinjective hbounded key .empty
(bitMemoSound_empty _ _ _)
intro next
exact (queryFamilyMemoNormalized_correct source target N P F exponent H Q eM eF hP hM hF
A a tau eta kappa delta multiplier occurrence symbolBits positionBits coarse draw initial
ha hs ht hp htable hc ell hell hdelta base hbase next).2
include ha hs ht hp htable hc hell hdelta in
theorem queryFamilyMemoNormalized_work_transfer (bits : ℕ)
(encode : QueryFamilyMemoKey (M := M) (J := J) (R := R) (passes := passes) (copies := copies)
(T := T) (S := S) target N P F exponent H Q → ℕ)
(hinjective : Function.Injective encode) (hbounded : ∀ key, encode key < 2 ^ bits)
(key : QueryFamilyMemoKey (M := M) (J := J) (R := R) (passes := passes) (copies := copies)
(T := T) (S := S) target N P F exponent H Q) :
(rankedBitMemoEvaluate (SourceRanked.rank ChargedPhysicalRequest.rank) normalized bits encode key
.empty).localWork =
rankedMemoKernelWork (SourceRanked.rank ChargedPhysicalRequest.rank) reference bits encode
(rankedBitLocalWork (SourceRanked.rank ChargedPhysicalRequest.rank) normalized) key .empty := by
apply rankedBitMemoEvaluate_work_transfer_on _ _ _ _ bits encode hinjective hbounded key .empty
(bitMemoSound_empty _ _ _)
intro next
exact (queryFamilyMemoNormalized_correct source target N P F exponent H Q eM eF hP hM hF
A a tau eta kappa delta multiplier occurrence symbolBits positionBits coarse draw initial
ha hs ht hp htable hc ell hell hdelta base hbase next).2
end Family
end EditApproximation
end OAI
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