openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /FieldConstruction /OrderedLimits.lean
| 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 | |