openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Separation /ComplexFiniteMMRealization.lean
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ComplexAnalyticValue | |
| import OAI.LinearAlgebra.MatrixMultiplication.Rectangular.ComplexASI | |
| namespace OAI | |
| /-! Finite type counts, hierarchy separation and tensor execution bounds. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.Foundation | |
| structure FiniteMMRealization where | |
| occurrences : ℕ | |
| multiplicity : ℕ | |
| auxiliaryRank : ℕ | |
| rows : ℕ | |
| inner : ℕ | |
| columns : ℕ | |
| occurrences_pos : 0 < occurrences | |
| multiplicity_pos : 0 < multiplicity | |
| auxiliaryRank_pos : 0 < auxiliaryRank | |
| rows_pos : 0 < rows | |
| inner_pos : 0 < inner | |
| columns_pos : 0 < columns | |
| order : ℕ | |
| degree : ℕ | |
| approximation : Tensor.PolynomialApproximation | |
| (Tensor.directSum (fun _ : Fin multiplicity => | |
| Tensor.matrixMultiplication rows inner columns)) | |
| (3 ^ occurrences * auxiliaryRank) order degree | |
| namespace FiniteMMRealization | |
| def rates (W : FiniteMMRealization) : FiniteConstructionRates where | |
| occurrences := W.occurrences | |
| multiplicity := W.multiplicity | |
| auxiliaryRank := W.auxiliaryRank | |
| volume := W.rows * W.inner * W.columns | |
| occurrences_pos := W.occurrences_pos | |
| multiplicity_pos := W.multiplicity_pos | |
| auxiliaryRank_pos := W.auxiliaryRank_pos | |
| volume_pos := Nat.mul_pos (Nat.mul_pos W.rows_pos W.inner_pos) W.columns_pos | |
| theorem score_mono (W : FiniteMMRealization) {β γ : ℝ} (hβγ : β ≤ γ) : | |
| W.rates.score β ≤ W.rates.score γ := by | |
| have hN : (0 : ℝ) < W.rates.occurrences := by | |
| exact_mod_cast W.rates.occurrences_pos | |
| have hv : (1 : ℝ) ≤ W.rates.volume := by | |
| exact_mod_cast (Nat.succ_le_iff.mpr W.rates.volume_pos) | |
| have hlog := Real.log_nonneg hv | |
| dsimp [FiniteConstructionRates.score] | |
| apply (div_le_div_iff_of_pos_right hN).2 | |
| nlinarith | |
| theorem score_omega_le_log_three (W : FiniteMMRealization) : | |
| W.rates.score Arithmetic.omega ≤ Real.log 3 := by | |
| have h := Arithmetic.equalRectangular_inequality_of_polynomialApproximation | |
| (Nat.succ_le_iff.mpr W.rows_pos) (Nat.succ_le_iff.mpr W.inner_pos) | |
| (Nat.succ_le_iff.mpr W.columns_pos) W.approximation | |
| have hL : (0 : ℝ) < W.multiplicity := by exact_mod_cast W.multiplicity_pos | |
| have hR : (0 : ℝ) < W.auxiliaryRank := by exact_mod_cast W.auxiliaryRank_pos | |
| have hN : (0 : ℝ) < W.occurrences := by exact_mod_cast W.occurrences_pos | |
| have hv : (0 : ℝ) < W.rates.volume := by exact_mod_cast W.rates.volume_pos | |
| have hp : (0 : ℝ) < (W.rates.volume : ℝ) ^ (Arithmetic.omega / 3) := | |
| Real.rpow_pos_of_pos hv _ | |
| change (W.multiplicity : ℝ) * (W.rates.volume : ℝ) ^ (Arithmetic.omega / 3) ≤ | |
| ((3 ^ W.occurrences * W.auxiliaryRank : ℕ) : ℝ) at h | |
| have hlog := Real.log_le_log (mul_pos hL hp) h | |
| rw [Real.log_mul (ne_of_gt hL) (ne_of_gt hp), Real.log_rpow hv] at hlog | |
| have hcost : | |
| Real.log ((3 ^ W.occurrences * W.auxiliaryRank : ℕ) : ℝ) = | |
| (W.occurrences : ℝ) * Real.log 3 + Real.log W.auxiliaryRank := by | |
| push_cast | |
| rw [Real.log_mul (pow_ne_zero _ (by norm_num)) (ne_of_gt hR), Real.log_pow] | |
| rw [hcost] at hlog | |
| change (Real.log W.multiplicity - Real.log W.auxiliaryRank + | |
| Arithmetic.omega / 3 * Real.log W.rates.volume) / W.occurrences ≤ Real.log 3 | |
| apply (div_le_iff₀ hN).2 | |
| nlinarith | |
| theorem omega_lt_of_score_gt (W : FiniteMMRealization) {β : ℝ} | |
| (hscore : Real.log 3 < W.rates.score β) : Arithmetic.omega < β := by | |
| by_contra h | |
| have hmono := W.score_mono (le_of_not_gt h) | |
| exact (not_lt_of_ge (hmono.trans W.score_omega_le_log_three)) hscore | |
| end FiniteMMRealization | |
| def realizedFamily {C X : Type*} (β : ℝ) (law : C → X) | |
| (realization : C → FiniteMMRealization) : AttainableFamily C X := | |
| finiteConstructionFamily β law (fun c => (realization c).rates) | |
| theorem omega_lt_of_realized_value_gt {C X : Type*} [TopologicalSpace X] | |
| (β : ℝ) (law : C → X) (realization : C → FiniteMMRealization) (x : X) | |
| (hne : ((realizedFamily β law realization).fiber x).Nonempty) | |
| (hgap : Real.log 3 < (realizedFamily β law realization).value x) : | |
| Arithmetic.omega < β := by | |
| obtain ⟨c, hc⟩ := (realizedFamily β law realization).exists_finite_gt hne hgap | |
| exact (realization c).omega_lt_of_score_gt hc | |
| end MatrixMultiplication.Foundation | |
| end | |
| end OAI | |