Prob eval zero le div
prob_eval_zero_le_div
Plain-language statement
Probability of a nonzero polynomial evaluating to zero over a uniform product distribution is at most d / m, where d bounds the total degree and m bounds below the cardinality of each factor. This bridges schwartz_zippel_counting with the probability formulation.
Exact Lean statement
lemma prob_eval_zero_le_div
{F : Type} [Field F]
{s : ℕ}
{S : Fin s → Set F} [∀ i, Fintype ↥(S i)] [∀ i, Nonempty ↥(S i)]
(f : MvPolynomial (Fin s) F) (hf : f ≠ 0)
(d m : ℕ) (hd : f.totalDegree ≤ d) (hm_pos : 0 < m)
(hm : ∀ i, m ≤ (S i).toFinset.card) :
Pr_{let x ←$ᵖ (∀ i, ↥(S i))}[MvPolynomial.eval (fun i => (↑(x i) : F)) f = 0] ≤ (d : ℝ≥0∞) / mFormal artifact
Lean source
lemma prob_eval_zero_le_div {F : Type} [Field F] {s : ℕ} {S : Fin s → Set F} [∀ i, Fintype ↥(S i)] [∀ i, Nonempty ↥(S i)] (f : MvPolynomial (Fin s) F) (hf : f ≠ 0) (d m : ℕ) (hd : f.totalDegree ≤ d) (hm_pos : 0 < m) (hm : ∀ i, m ≤ (S i).toFinset.card) : Pr_{let x ←$ᵖ (∀ i, ↥(S i))}[MvPolynomial.eval (fun i => (↑(x i) : F)) f = 0] ≤ (d : ℝ≥0∞) / m := by classical convert ENNReal.div_le_div_of_mul_le hm_pos _ _ using 1 · convert uniform_prob_eq_card_div _ · infer_instance · exact Fintype.card_pos_iff.mpr ⟨fun _ => Classical.arbitrary _⟩ · convert schwartz_zippel_counting f hf ( fun i => ( S i ).toFinset ) d m hd hm_pos hm using 1 · convert congr_arg₂ (· * ·) (card_filter_eval_subtype_eq_piFinset S f) rfl · rw [Fintype.card_pi] aesop- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/MvPolynomial/SchwartzZippelCounting.lean:126-143
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
Related declarations
Affine gaps lifted to interleaved codes
affine_gaps_lifted_to_interleaved_codes
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 φ) 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.