File size: 1,599 Bytes
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 | import OAI.Algebra.DepthFive.UpperBlockData
import OAI.Algebra.DepthFive.MatrixPathSemantics
namespace OAI
noncomputable section
namespace Problem335.BlockData
variable {n : ℕ}
/-- The lower-gate monomial belonging to an enumerated block path. -/
def lowerMonomial (K : Type*) [CommSemiring K]
(blocks : List (List (Fin n))) (p : Lower blocks) :
MvPolynomial (Fin n × Fin n × Fin n) K :=
((lowerInputs blocks p).map MvPolynomial.X).prod
/-- The weighted middle input list computes exactly the corresponding block matrix entry. -/
theorem middleInputs_value (K : Type*) [CommSemiring K]
(blocks : List (List (Fin n))) (m : Middle blocks) :
((middleInputs K blocks m).map (fun e =>
MvPolynomial.C e.1 * lowerMonomial K blocks e.2)).sum =
(((blocks.get m.1).map (immLayer K n)).prod) m.2.1 m.2.2 := by
simp only [middleInputs, List.map_ofFn, Function.comp_def, lowerMonomial,
lowerInputs, MvPolynomial.C_1, one_mul, List.sum_ofFn]
exact matrixEntryPaths_fin_sum_prod (immLayer K n) (blocks.get m.1) m.2.1 m.2.2
/-- Substitute lower-gate semantics into the middle gate. -/
theorem middleInputs_value_of_lower
(K : Type*) [CommSemiring K] (blocks : List (List (Fin n)))
(v : Lower blocks → MvPolynomial (Fin n × Fin n × Fin n) K)
(hv : ∀ p, v p = lowerMonomial K blocks p) (m : Middle blocks) :
((middleInputs K blocks m).map (fun e => MvPolynomial.C e.1 * v e.2)).sum =
(((blocks.get m.1).map (immLayer K n)).prod) m.2.1 m.2.2 := by
simp_rw [hv]
exact middleInputs_value K blocks m
end Problem335.BlockData
end
end OAI
|