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5d5048e | 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 | import OAI.Combinatorics.LowSpaceSubsetSum.Storage.ResiduePreparationCells
import OAI.Combinatorics.LowSpaceSubsetSum.Representation.RepresentationExtraPreparation
namespace OAI
/-! # Full prepared-state memory through the bounded extra-prime draw -/
noncomputable section
namespace LowSpaceSubsetSum.ExtraPreparationCells
variable {n : ℕ}
def run (I : Instance n) (hn : 10000 ≤ n)
(hlog : Real.logb 2 ((mixerSize n : ℝ)+1)+1 ≤ (n : ℝ)/1000)
(s : ((canonicalRepresentationData I).zeroSetup.extraHistoryCoupling (by omega)).Seed) :
(Option (RepresentationExtraPreparation.Data n) × ℕ) × ℕ :=
let prepared := ResiduePreparationCells.run I hn hlog s.1
match h : prepared.1.1 with
| none => ((none, prepared.1.2+1), prepared.2+9*n^3+512)
| some data =>
let hs : (RepresentationResiduePreparation.run I hn hlog s.1).1 = some data :=
(congrArg Prod.fst (ResiduePreparationCells.value I hn hlog s.1)).symm.trans h
let out := RepresentationScalarCells.extra (canonicalRepresentationData I).zeroSetup
(((canonicalRepresentationData I).zeroSetup.trialHistoryCoupling.ideal s.1).1.1.1.1)
(by omega) data.primes.stored.stored.layout.parameters.widths[3]
(RepresentationExtraPreparation.width_ideal I hn hlog s.1 data hs) s.2
((out.1.1.map (fun q => ⟨data,q⟩), prepared.1.2+out.1.2+1),
prepared.2+out.2+9*n^3+512)
theorem value (I : Instance n) (hn : 10000 ≤ n)
(hlog : Real.logb 2 ((mixerSize n : ℝ)+1)+1 ≤ (n : ℝ)/1000)
(s : ((canonicalRepresentationData I).zeroSetup.extraHistoryCoupling (by omega)).Seed) :
(run I hn hlog s).1 = RepresentationExtraPreparation.run I hn hlog s := by
have hp₁ := congrArg Prod.fst (ResiduePreparationCells.value I hn hlog s.1)
have hp₂ := congrArg Prod.snd (ResiduePreparationCells.value I hn hlog s.1)
simp only [run, hp₂, RepresentationExtraPreparation.run]
split
next h =>
split
· rfl
next data hs =>
have he := (hp₁.symm.trans h).symm.trans hs
cases he
next data h =>
simp only [RepresentationScalarCells.extra_value]
split
next hs =>
have he := (hp₁.symm.trans h).symm.trans hs
cases he
next data' hs =>
have he : data = data' := Option.some.inj ((hp₁.symm.trans h).symm.trans hs)
subst data'
rfl
theorem cells_add (I : Instance n) (hn : 10000 ≤ n)
(hlog : Real.logb 2 ((mixerSize n : ℝ)+1)+1 ≤ (n : ℝ)/1000)
(s : ((canonicalRepresentationData I).zeroSetup.extraHistoryCoupling (by omega)).Seed) :
(run I hn hlog s).2 ≤ (ResiduePreparationCells.run I hn hlog s.1).2+12*n^3+608 := by
unfold run
dsimp only
split
· omega
next data h =>
have hs : (RepresentationResiduePreparation.run I hn hlog s.1).1 = some data :=
(congrArg Prod.fst (ResiduePreparationCells.value I hn hlog s.1)).symm.trans h
have hx := RepresentationScalarCells.extra_cells (canonicalRepresentationData I).zeroSetup
(((canonicalRepresentationData I).zeroSetup.trialHistoryCoupling.ideal s.1).1.1.1.1)
(by omega) data.primes.stored.stored.layout.parameters.widths[3]
(RepresentationExtraPreparation.width_ideal I hn hlog s.1 data hs) s.2
change _+_+9*n^3+512 ≤ _
omega
theorem cells (I : Instance n) (hn : 10000 ≤ n)
(hlog : Real.logb 2 ((mixerSize n : ℝ)+1)+1 ≤ (n : ℝ)/1000)
(s : ((canonicalRepresentationData I).zeroSetup.extraHistoryCoupling (by omega)).Seed) :
(run I hn hlog s).2 ≤ inputScale I^13*(compressionDictionaryCapacity n+1) := by
have h1 := cells_add I hn hlog s
have h2 := ResiduePreparationCells.cells_add I hn hlog s.1
have h3 := PrimePreparationCells.cells_add I hn hlog s.1.1
have hbase := RepresentationStoredPreparationCells.cells I hn s.1.1.1
have htotal : (run I hn hlog s).2 ≤ (RepresentationStoredPreparationCells.run I s.1.1.1).2+(48*n^3+2240) := by
calc
_ ≤ (ResiduePreparationCells.run I hn hlog s.1).2+12*n^3+608 := h1
_ ≤ ((PrimePreparationCells.run I hn hlog s.1.1).2+18*n^3+800)+12*n^3+608 :=
Nat.add_le_add_right (Nat.add_le_add_right h2 (12*n^3)) 608
_ ≤ (((RepresentationStoredPreparationCells.run I s.1.1.1).2+18*n^3+832)+18*n^3+800)+12*n^3+608 :=
Nat.add_le_add_right (Nat.add_le_add_right
(Nat.add_le_add_right (Nat.add_le_add_right h3 (18*n^3)) 800) (12*n^3)) 608
_ = _ := by ring
exact htotal.trans (ScalarStageCells.aggregate n (inputScale I) (compressionDictionaryCapacity n)
(RepresentationStoredPreparationCells.run I s.1.1.1).2 (48*n^3+2240)
(by unfold inputScale; omega) (by unfold inputScale; omega) hbase le_rfl)
end LowSpaceSubsetSum.ExtraPreparationCells
end
end OAI
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