File size: 3,404 Bytes
cdec422 | 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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.ConstructionFamily
namespace OAI
/-! Tensor extraction over arbitrary fields and its asymptotic rate. -/
noncomputable section
namespace MatrixMultiplication.AllFieldOrderedLimits
open AllFieldHistory AllFieldNativeCapacity AllFieldScheduledYield
open AllFieldConstructionRates Filter
open scoped Topology
variable (F : Type*) [Field F]
theorem fixed_lot_constraint (K : ℕ) (hK : 2 ≤ K)
(hgap : nativeB < nativeHigh)
(hfeas : ∀ i, nativeCstar - (nativeLA i + nativeLB i) < nativeHigh)
(ha : ∀ i, 0 < nativeLA i) (hb : ∀ i, 0 < nativeLB i)
(hc : ∀ i, 0 < nativeLC nativeLambda i) :
AllFieldInitialEntropy.nativeH0 + nativeCstar - nativeBeta / (K : ℝ) +
(Arithmetic.omega F / 3) * AllFieldTerminalRates.S ≤ 8 * Real.log 7 := by
obtain ⟨allocation, halloc⟩ := exists_balanced_allocation_sequence hgap hfeas
have hcap := eventually_capacity_pos hK allocation halloc ha hb hc
have hy := (physicalYield_tendsto_balanced K hK hgap allocation halloc).div_const (K : ℝ)
have hH0 : Tendsto (fun _ : ℕ => AllFieldInitialEntropy.nativeH0) atTop
(𝓝 AllFieldInitialEntropy.nativeH0) := tendsto_const_nhds
have hlimit := (hH0.add hy).add
(tendsto_const_nhds : Tendsto (fun _ : ℕ =>
(Arithmetic.omega F / 3) * AllFieldTerminalRates.S) atTop (𝓝 _))
have hineq : AllFieldInitialEntropy.nativeH0 +
((K : ℝ) * nativeCstar - nativeBeta) / (K : ℝ) +
(Arithmetic.omega F / 3) * AllFieldTerminalRates.S ≤ 8 * Real.log 7 := by
apply le_of_tendsto_of_tendsto hlimit tendsto_const_nhds
filter_upwards [hcap] with j hj
exact AllFieldConstructionFamily.fixed_allocation_constraint F (by omega)
(allocation j) hj
have hk : (K : ℝ) ≠ 0 := by exact_mod_cast (show K ≠ 0 by omega)
have heq : ((K : ℝ) * nativeCstar - nativeBeta) / (K : ℝ) =
nativeCstar - nativeBeta / (K : ℝ) := by
rw [sub_div, mul_div_cancel_left₀ _ hk]
rw [heq] at hineq
simpa only [← add_sub_assoc] using hineq
theorem native_rank_constraint
(hgap : nativeB < nativeHigh)
(hfeas : ∀ i, nativeCstar - (nativeLA i + nativeLB i) < nativeHigh)
(ha : ∀ i, 0 < nativeLA i) (hb : ∀ i, 0 < nativeLB i)
(hc : ∀ i, 0 < nativeLC nativeLambda i) :
AllFieldInitialEntropy.nativeH0 + nativeCstar +
(Arithmetic.omega F / 3) * AllFieldTerminalRates.S ≤ 8 * Real.log 7 := by
have hfixed (K : ℕ) (hK : 2 ≤ K) :
(AllFieldInitialEntropy.nativeH0 + nativeCstar) - nativeBeta / (K : ℝ) +
Arithmetic.omega F * AllFieldTerminalRates.S / 3 ≤ 8 * Real.log 7 := by
have h := fixed_lot_constraint F K hK hgap hfeas ha hb hc
nlinarith
have h := FiniteSchedule.fixed_lot_boundary_limit hfixed
nlinarith
theorem omega_le_native_ratio
(hgap : nativeB < nativeHigh)
(hfeas : ∀ i, nativeCstar - (nativeLA i + nativeLB i) < nativeHigh)
(ha : ∀ i, 0 < nativeLA i) (hb : ∀ i, 0 < nativeLB i)
(hc : ∀ i, 0 < nativeLC nativeLambda i)
(hS : 0 < AllFieldTerminalRates.S) :
Arithmetic.omega F ≤
3 * (8 * Real.log 7 -
(AllFieldInitialEntropy.nativeH0 + nativeCstar)) / AllFieldTerminalRates.S := by
apply (le_div_iff₀ hS).mpr
have h := native_rank_constraint F hgap hfeas ha hb hc
nlinarith
end MatrixMultiplication.AllFieldOrderedLimits
end
end OAI
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