Sum cube succ
SumcheckDomain.sum_cube_succ
Plain-language statement
Telescoping identity (the core sum-check completeness step): summing over the (k+1)-coordinate cube equals summing coordinate 0 over its domain, then the rest over the tail cube. This is the piFinset "cons decomposition" ๐ป^{k+1} โ ๐ปโ ร ๐ป^k.
Exact Lean statement
theorem sum_cube_succ {M : Type*} [AddCommMonoid M] (D : SumcheckDomain R (k + 1))
(f : (Fin (k + 1) โ R) โ M) :
โ x โ D.cube, f x = โ b โ D.points 0, โ y โ D.tail.cube, f (Fin.cons b y)Formal artifact
Lean source
theorem sum_cube_succ {M : Type*} [AddCommMonoid M] (D : SumcheckDomain R (k + 1)) (f : (Fin (k + 1) โ R) โ M) : โ x โ D.cube, f x = โ b โ D.points 0, โ y โ D.tail.cube, f (Fin.cons b y) := by rw [โ Finset.sum_product'] have htail : D.tail.cube = Fintype.piFinset (Fin.tail D.points) := by congr 1 have hcube : D.cube = (D.points 0 รหข D.tail.cube).map (Fin.consEquiv (fun _ : Fin (k + 1) => R)).toEmbedding := by rw [htail] simpa [cube] using Finset.filter_piFinset_eq_map_consEquiv (S := D.points) (fun _ => True) rw [hcube, Finset.sum_map] rfl- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/ProofSystem/Sumcheck/Domain.lean:150-162
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Project documentation
This lemma proves the final algebraic step in the DG25 Theorem 3.1 proof. It shows that if R > e + 1, then e * (R / (R - 1)) < e + 1. The intuition is that the fraction R / (R - 1) is always greater than 1, but as R gets larger, it gets closer to 1. The hypothesis R > e + 1 provides a strong enough bound to ensure the product e * (fraction) do...
Source project: ArkLib
Person-level attribution pending.
Gadget Decompose coeff
ArkLib.Lattices.Ajtai.gadgetDecompose_coeff
Plain-language statement
The k-th coefficient (k < deg ฯ) of a gadget-decomposition block is exactly the corresponding digit of the corresponding input coefficient.
Source project: ArkLib
Person-level attribution pending.
Gadget Decompose lawful
ArkLib.Lattices.Ajtai.gadgetDecompose_lawful
Plain-language statement
The base-b gadget decomposition is a lawful gadget decomposition.
Source project: ArkLib
Person-level attribution pending.