| import OAI.Algebra.DepthFive.ProductNormalization |
| import OAI.Algebra.DepthFive.Homogeneity |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace Problem335 |
|
|
| open MvPolynomial |
|
|
| universe u v w |
|
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| / |
| is a sublist of the original gate occurrences, so sharing and multiplicity |
| are preserved. -/ |
| theorem homogeneous_indexed_product_positive_factors |
| {K : Type u} [CommSemiring K] {σ : Type v} {ι : Type w} |
| (l : List ι) (p : ι → MvPolynomial σ K) (d : ι → ℕ) |
| (hl : ∀ i ∈ l, (p i).IsHomogeneous (d i)) : |
| ∃ a : K, ∃ factors : List ι, |
| factors.Sublist l ∧ |
| (∀ i ∈ factors, 0 < d i) ∧ |
| (factors.map d).sum = (l.map d).sum ∧ |
| (l.map p).prod = C a * (factors.map p).prod ∧ |
| factors.length ≤ (l.map d).sum := by |
| obtain ⟨a, fs, hf, hd, hv, hs, hn⟩ := |
| homogeneous_product_positive_factors (l.map (fun i => (p i, d i))) (by |
| intro q hq |
| obtain ⟨i, hi, rfl⟩ := List.mem_map.mp hq |
| exact hl i hi) |
| obtain ⟨factors, hsub, rfl⟩ := List.sublist_map_iff.mp hs |
| refine ⟨a, factors, hsub, ?_, ?_, ?_, ?_⟩ |
| · intro i hi |
| exact (hf (p i, d i) (List.mem_map.mpr ⟨i, hi, rfl⟩)).1 |
| · simpa only [List.map_map, Function.comp_def] using hd |
| · simpa only [List.map_map, Function.comp_def] using hv |
| · simpa only [List.length_map, List.map_map, Function.comp_def] using hn |
|
|
| / |
| middle-gate occurrences whose degrees add to the original formal degree. -/ |
| theorem upperValue_positive_factors |
| {K : Type u} [CommSemiring K] {n : ℕ} |
| (c : Depth5Circuit K n) (i : Fin c.upperCount) : |
| ∃ a : K, ∃ factors : List (Fin c.middleCount), |
| factors.Sublist (c.upperInputs i) ∧ |
| (∀ j ∈ factors, 0 < c.middleDegree j) ∧ |
| (factors.map c.middleDegree).sum = |
| ((c.upperInputs i).map c.middleDegree).sum ∧ |
| upperValue c i = C a * (factors.map (middleValue c)).prod ∧ |
| factors.length ≤ ((c.upperInputs i).map c.middleDegree).sum := by |
| exact homogeneous_indexed_product_positive_factors |
| (c.upperInputs i) (middleValue c) c.middleDegree |
| (fun j _ => middleValue_isHomogeneous c j) |
|
|
| / |
| sum to the circuit output degree, hence there are at most that many factors. -/ |
| theorem output_term_positive_factors |
| {K : Type u} [CommSemiring K] {n : ℕ} |
| (c : Depth5Circuit K n) (entry : K × Fin c.upperCount) |
| (hentry : entry ∈ c.outputInputs) : |
| ∃ a : K, ∃ factors : List (Fin c.middleCount), |
| factors.Sublist (c.upperInputs entry.2) ∧ |
| (∀ j ∈ factors, 0 < c.middleDegree j) ∧ |
| (factors.map c.middleDegree).sum = c.outputDegree ∧ |
| upperValue c entry.2 = C a * (factors.map (middleValue c)).prod ∧ |
| factors.length ≤ c.outputDegree := by |
| simpa only [c.outputHomogeneous entry hentry] using |
| upperValue_positive_factors c entry.2 |
|
|
| end Problem335 |
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| end |
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| end OAI |
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|
