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f18e7de | 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 | 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
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