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Project-declaredLean 4.31.0 · mathlib@fabf563a7c95

Degree Of aeval le

MvPolynomial.degreeOf_aeval_le

Plain-language statement

For a multilinear t (each variable has degreeOf ≤ 1), substituting t into a univariate Q : L[X] via Polynomial.aeval yields a multivariate polynomial whose degree in each variable is bounded by Q.natDegree. Used by the structured sumcheck to bound the degree of Q(witness) in the round polynomial H = P · Q(t).

Exact Lean statement

theorem degreeOf_aeval_le {L : Type*} [CommSemiring L] {σ : Type*} (i : σ)
    (Q : Polynomial L) (t : MvPolynomial σ L) (ht : degreeOf i t ≤ 1) :
    degreeOf i (Polynomial.aeval t Q) ≤ Q.natDegree

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
theorem degreeOf_aeval_le {L : Type*} [CommSemiring L] {σ : Type*} (i : σ)    (Q : Polynomial L) (t : MvPolynomial σ L) (ht : degreeOf i t  1) :    degreeOf i (Polynomial.aeval t Q)  Q.natDegree := by  rw [Polynomial.aeval_def, Polynomial.eval₂_eq_sum, Polynomial.sum]  refine le_trans (degreeOf_sum_le i Q.support _) ?_  refine Finset.sup_le fun e he => ?_  calc degreeOf i (algebraMap L (MvPolynomial σ L) (Q.coeff e) * t ^ e)       degreeOf i (algebraMap L (MvPolynomial σ L) (Q.coeff e)) + degreeOf i (t ^ e) :=        degreeOf_mul_le i _ _    _ = degreeOf i (t ^ e) := by rw [MvPolynomial.algebraMap_eq, degreeOf_C, zero_add]    _  e * degreeOf i t := degreeOf_pow_le i t e    _  e * 1 := by gcongr    _ = e := mul_one e    _  Q.natDegree := Polynomial.le_natDegree_of_mem_supp e he
Project
ArkLib
License
Apache-2.0
Commit
fad5cbf80877
Source
ArkLib/Data/MvPolynomial/RestrictDegree.lean:105-118

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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...

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Project-declaredLean 4.31.0

Gadget Decompose coeff

ArkLib.Lattices.Ajtai.gadgetDecompose_coeff

Plain-language statement

The k-th coefficient (k < deg φ) of a gadget-decomposition block is exactly the corresponding digit of the corresponding input coefficient.

cryptographyproof systemscoding theory

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Person-level attribution pending.

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Project-declaredLean 4.31.0

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ArkLib.Lattices.Ajtai.gadgetDecompose_lawful

Plain-language statement

The base-b gadget decomposition is a lawful gadget decomposition.

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Source project: ArkLib

Person-level attribution pending.

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