| import OAI.Algebra.DepthFive.CircuitDegreeBounds |
| import OAI.Algebra.DepthFive.Homogeneity |
| import OAI.Algebra.DepthFive.ProductNormalization |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace Problem335 |
| universe u |
| variable {K : Type u} [CommSemiring K] {n : ℕ} |
|
|
| /-- A lower product computes a scalar times exactly its formal degree many |
| homogeneous linear forms, with repeated inputs retained as repeated factors. -/ |
| theorem lowerValue_normal_form (c : Depth5Circuit K n) (i : Fin c.lowerCount) : |
| ∃ a : K, ∃ factors : List (MvPolynomial (Fin n × Fin n × Fin n) K), |
| factors.length = ((c.lowerInputs i).map c.bottomDegree).sum ∧ |
| (∀ p ∈ factors, p.IsHomogeneous 1) ∧ |
| lowerValue c i = MvPolynomial.C a * factors.prod := by |
| let l := (c.lowerInputs i).map fun j => (bottomValue c j, c.bottomDegree j) |
| have hl : ∀ q ∈ l, q.1.IsHomogeneous q.2 := by |
| intro q hq |
| obtain ⟨j, _, rfl⟩ := List.mem_map.mp hq |
| exact bottomValue_isHomogeneous c j |
| have hle : ∀ q ∈ l, q.2 ≤ 1 := by |
| intro q hq |
| obtain ⟨j, _, rfl⟩ := List.mem_map.mp hq |
| exact bottomDegree_le_one c j |
| obtain ⟨a, factors, hlen, hhom, hval⟩ := homogeneous_product_linear_factors l hl hle |
| exact ⟨a, factors, by simpa [l, Function.comp_def] using hlen, hhom, |
| by simpa [l, lowerValue, Function.comp_def] using hval⟩ |
|
|
| /-- A weighted term of a middle sum has a linear-factor form of the middle |
| sum's assigned degree, even for a zero coefficient or zero product. -/ |
| theorem middle_input_normal_form (c : Depth5Circuit K n) (i : Fin c.middleCount) |
| (entry : K × Fin c.lowerCount) (hentry : entry ∈ c.middleInputs i) : |
| ∃ a : K, ∃ factors : List (MvPolynomial (Fin n × Fin n × Fin n) K), |
| factors.length = c.middleDegree i ∧ |
| (∀ p ∈ factors, p.IsHomogeneous 1) ∧ |
| MvPolynomial.C entry.1 * lowerValue c entry.2 = |
| MvPolynomial.C a * factors.prod := by |
| obtain ⟨a, factors, hlen, hhom, hval⟩ := lowerValue_normal_form c entry.2 |
| refine ⟨entry.1 * a, factors, hlen.trans (c.middleHomogeneous i entry hentry), hhom, ?_⟩ |
| simp [hval, mul_assoc] |
|
|
| end Problem335 |
|
|
| end |
|
|
| end OAI |
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|
