Eval Split eq eval
ArkLib.Lattices.Hachi.evalSplit_eq_eval
Plain-language statement
The split. Evaluating a multilinear polynomial equals the vector–matrix–vector product of its reshaped coefficient matrix with the monomial bases of the low and high evaluation points (Hachi [NOZ26, §4]).
Exact Lean statement
theorem evalSplit_eq_eval (p : CMlPolynomial R (nl + nh)) (xl : Vector R nl) (xh : Vector R nh) :
evalSplit p xl xh = eval p (xl ++ xh)Formal artifact
Lean source
theorem evalSplit_eq_eval (p : CMlPolynomial R (nl + nh)) (xl : Vector R nl) (xh : Vector R nh) : evalSplit p xl xh = eval p (xl ++ xh) := by simp only [evalSplit, splitForm] rw [eval_eq_sum, ← Equiv.sum_comp (splitEquiv nl nh) (fun i => p.get i * (monomialBasis (xl ++ xh)).get i), Fintype.sum_prod_type, dot_eq_sum, Finset.sum_comm] apply Finset.sum_congr rfl intro y _ rw [matVecMul_apply, dot_eq_sum, Finset.mul_sum] apply Finset.sum_congr rfl intro x _ rw [monomialBasis_split] change (monomialBasis xl).get y * (toMatrix p y x * (monomialBasis xh).get x) = toMatrix p y x * ((monomialBasis xl).get y * (monomialBasis xh).get x) ring- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Commitments/Functional/Hachi/EvalSplit.lean:172-186
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Person-level attribution pending.
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Plain-language statement
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Source project: ArkLib
Person-level attribution pending.