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2c7d975 | 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 | import OAI.Algebra.DepthFive.ProductNormalization
import OAI.Algebra.DepthFive.Homogeneity
namespace OAI
noncomputable section
namespace Problem335
open MvPolynomial
universe u v w
/-- An indexed version of positive-factor normalization. The resulting list
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
/-- Every upper product is a scalar multiple of positive-formal-degree
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)
/-- For an upper gate occurring at the output, the positive factor degrees
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
end
end OAI
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