Guruswami-Sudan divisibility

[ster-oc]

In this PR with @Aristotle-Harmonic, we prove the divisibility property for a bivariate polynomial fulfilling Guruswami-Sudan set of conditions. There is an arithmetical lemma left to prove, which will be there soon.
A bit of golfing and refactor will follow shortly!

Depends on : #299
Closes : #224

[github-actions]

🤖 Gemini PR Summary

This Pull Request implements the divisibility property for the Guruswami-Sudan list decoding algorithm, a critical component for Reed-Solomon code verification. The work establishes the formal existence of interpolation polynomials and refines how bivariate root multiplicity is handled within the library.

Features

• Guruswami-Sudan Implementation: Introduced the formal implementation of the Guruswami-Sudan list decoding algorithm, including proofs for the existence of bivariate interpolation polynomials.
• Divisibility Property: Proved that codewords within the specified decoding radius are roots of the interpolation polynomial (the "divisibility property").
• Bivariate Polynomial Enhancements: Redefined root multiplicity for bivariate polynomials using the "order of vanishing" (minimum total degree of nonzero coefficients after shifting) and added shifting helper functions.
• List Utilities: Added a new lemma to Data.List.Lemmas regarding the lower bounds of minimum values in lists produced via filterMap.

Refactoring

• Module Reorganization: Decoupled core Guruswami-Sudan definitions into a dedicated Basic module to improve maintainability and build logic.
• Predicate Renaming: Updated the Condition predicate to Conditions within the Proximity Gap lemmas (BCIKS20.lean) for better naming consistency.
• Top-level Imports: Updated ArkLib.lean to include the new Guruswami-Sudan modules.

Maintenance

• Dependency Update: Updated the lake-manifest.json to track a newer revision of the VCVio dependency.

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Notes for Reviewers:

• This PR closes Proof obligation for guruswami_sudan_for_proximity_gap_property in ArkLib/Data/CodingTheory/GuruswamiSudan/GuruswamiSudan.lean #224 and depends on Guruswami-Sudan existence #299.
• The authors have noted that one arithmetical lemma remains to be proven, and further code "golfing" (optimization/shortening) is planned in subsequent commits.

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Analysis of Changes

Metric	Count
📝 Files Changed	7
✅ Lines Added	1018
❌ Lines Removed	80

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sorry Tracking

✅ **Removed:** 2 `sorry`(s)

• lemma guruswami_sudan_for_proximity_gap_existence {k m : ℕ} {ωs : Fin n ↪ F} {f : Fin n → F} : in ArkLib/Data/CodingTheory/GuruswamiSudan/GuruswamiSudan.lean
• lemma guruswami_sudan_for_proximity_gap_property {k m : ℕ} {ωs : Fin n ↪ F} in ArkLib/Data/CodingTheory/GuruswamiSudan/GuruswamiSudan.lean

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🎨 **Style Guide Adherence**

The following code changes violate the ArkLib style guide:

ArkLib/Data/CodingTheory/GuruswamiSudan/Basic.lean

• Line 34: noncomputable def proximity_gap_degree_bound : ℕ :=
  • Violation: "Functions and Terms: lowerCamelCase (e.g., binarySearch, isPrime)."
• Line 41: noncomputable def proximity_gap_johnson : ℝ :=
  • Violation: "Functions and Terms: lowerCamelCase (e.g., binarySearch, isPrime)."
• Line 60: def weigthBoundIndices (k D : ℕ) : Finset (ℕ × ℕ) :=
  • Violation: "Spelling: Use American English for declaration names (e.g., analyze not analyse, color not colour)." (Note: Spelling error "weigth").
• Line 66: lemma card_weigthBoundIndices_eq_sum (D : ℕ) (hk : 1 < k) :
  • Violation: "Theorems and Proofs: snake_case (e.g., add_comm, list_reverse_id)." (Contains camelCase segments and spelling error).
• Line 304: lemma coeff_linearCombination_monomial (c : ℕ × ℕ →₀ F) (i j : ℕ) :
  • Violation: "Theorems and Proofs: snake_case (e.g., add_comm, list_reverse_id)."
• Line 313: lemma linearIndependent_monomials :
  • Violation: "Theorems and Proofs: snake_case (e.g., add_comm, list_reverse_id)."
• Line 333: lemma natWeightedDegree_monomial (i j u v : ℕ) :
  • Violation: "Theorems and Proofs: snake_case (e.g., add_comm, list_reverse_id)."
• Line 345: lemma natWeightedDegree_monomial_eq (i j u v : ℕ) :
  • Violation: "Theorems and Proofs: snake_case (e.g., add_comm, list_reverse_id)."
• Line 349: lemma natWeightedDegree_add_le (p q : F[X][Y]) (u v : ℕ) :
  • Violation: "Theorems and Proofs: snake_case (e.g., add_comm, list_reverse_id)."
• Line 368: lemma natWeightedDegree_sum_le {ι : Type*} (s : Finset ι) (f : ι → F[X][Y]) (u v : ℕ) :
  • Violation: "Theorems and Proofs: snake_case (e.g., add_comm, list_reverse_id)."
• Line 383: lemma natWeightedDegree_smul_le {F : Type} [Semiring F] (a : F) (p : F[X][Y]) (u v : ℕ) :
  • Violation: "Theorems and Proofs: snake_case (e.g., add_comm, list_reverse_id)."
• Line 394: lemma natWeightedDegree_coeffsToPoly_le (k D : ℕ) (c : (weigthBoundIndices k D) → F) :
  • Violation: "Theorems and Proofs: snake_case (e.g., add_comm, list_reverse_id)."
• Line 466: lemma rootMultiplicity_le_of_coeff_ne_zero [DecidableEq F] {Q : F[X][Y]} {x y : F} {s t : ℕ}
  • Violation: "Theorems and Proofs: snake_case (e.g., add_comm, list_reverse_id)."
• Line 505: lemma rootMultiplicity_ge_of_shift_zero [DecidableEq F] {f : F[X][Y]} {x y : F}
  • Violation: "Theorems and Proofs: snake_case (e.g., add_comm, list_reverse_id)."
• Line 568: lemma degree_eval_le_weightedDegree (Q : F[X][Y]) (P : F[X]) (k : ℕ) (hP : P.natDegree ≤ k - 1) :
  • Violation: "Theorems and Proofs: snake_case (e.g., add_comm, list_reverse_id)."
• Line 585: lemma dvd_eval_of_rootMultiplicity_zero (Q : F[X][Y]) (P : F[X]) (m : ℕ)
  • Violation: "Theorems and Proofs: snake_case (e.g., add_comm, list_reverse_id)."
• Line 610: def HasOrderAt (Q : F[X][Y]) (x y : F) (m : ℕ) : Prop :=
  • Violation: "Functions and Terms: lowerCamelCase (e.g., binarySearch, isPrime)."
• Line 613: lemma orderAt_eval_ge (Q : F[X][Y]) (P : F[X]) (x : F) (m : ℕ)
  • Violation: "Theorems and Proofs: snake_case (e.g., add_comm, list_reverse_id)."
• Line 672: lemma codewordToPoly_degree_le (hk : k + 1 ≤ n) (p : code ωs k) :
  • Violation: "Theorems and Proofs: snake_case (e.g., add_comm, list_reverse_id)."

ArkLib/Data/Polynomial/Bivariate.lean

• Line 103: lemma weightedDegree_ne_none {F : Type} [Semiring F] (f : F[X][Y]) (u v : ℕ) :
  • Violation: "Theorems and Proofs: snake_case (e.g., add_comm, list_reverse_id)."
• Line 380: map_smul' r x := by simp; rw[smul_monomial]; aesop
  • Violation: "We aim to adhere to the Lean community's contribution guidelines." (Specifically: missing space after rw and avoid using ; to separate tactics).
• Line 390: rw[smul_monomial, smul_monomial]
  • Violation: "We aim to adhere to the Lean community's contribution guidelines." (Specifically: missing space after rw).

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📄 **Per-File Summaries**

• ArkLib.lean: Added the GuruswamiSudan.Basic module to the library's top-level imports.
• ArkLib/Data/CodingTheory/GuruswamiSudan/Basic.lean: This file provides the formal implementation of the Guruswami-Sudan list decoding algorithm for Reed-Solomon codes, establishing the existence of a bivariate interpolation polynomial and proving that codewords within the decoding radius are roots of this polynomial.
• ArkLib/Data/CodingTheory/GuruswamiSudan/GuruswamiSudan.lean: Refactor the Guruswami-Sudan implementation by moving core definitions to a dedicated module and updating the existence and divisibility theorems for the proximity gap.
• ArkLib/Data/CodingTheory/ProximityGap/BCIKS20.lean: Rename the Condition predicate to Conditions in the Guruswami-Sudan proximity gap lemmas.
• ArkLib/Data/List/Lemmas.lean: Adds a lemma proving that if all elements produced by filterMap satisfy a lower bound, then the minimum of the resulting list also satisfies that bound.
• ArkLib/Data/Polynomial/Bivariate.lean: This update redefines bivariate root multiplicity as the order of vanishing (the minimum total degree of nonzero coefficients after shifting) and introduces a helper function for shifting polynomials.
• lake-manifest.json: Updates the revision hash for the VCVio dependency.

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Last updated: 2026-02-14 20:27 UTC.

[ster-oc]

I see a problem with the definition of proximity_gap_johnson.
In particular, I believe that in Lemma 5.3 of [BCIKS20] the second part of the statement really asks to bound the Hamming distance with the real number δ₀(ρ, m) and not with its floor as per the definition in GuruswamiSudan.lean.

The relative radius δ₀(ρ, m) is always <1, which means that its floor is always 0.
This means that in [BCIKS20] Δ is the relative Hamming distance and not the absolute one

[alexanderlhicks]

Hello! Thanks for the PR and sorry for the delay in getting to this. I'll be merging the 4.26 PR soon, which might break a few things, and get back to all the other PRs including this one.

[ster-oc]

I realized that the definition of rootMultiplicity₀ was not correct, with essentially a List.max? to be replaced by a List.min?.
(An example is the polynomial p(x) := X + Y² which has a zero of order 1 at (0,0), but List.max would give order 2 because of the monomial Y²)

This fix together with the correct hypothesis on the relative Hamming distance should now be fine for everything to go through.

There is still a sorry left because some code broke after changing rootMultiplicity.

[ster-oc]

Guruswami-Sudan is now sorry free!

I see you opened PR #306, be mindful that I put some of the new definitions in Bivariate.lean✌️