Exists nonzero solution gen
GuruswamiSudan.exists_nonzero_solution_gen
Plain-language statement
Generalized existence: non-zero kernel element for arbitrary degree bound D, given numVars k D > numConstraints n m.
Exact Lean statement
lemma exists_nonzero_solution_gen (k n m : ℕ) (ωs : Fin n ↪ F) (f : Fin n → F) (D : ℕ)
(hD : numVars k D > numConstraints n m) :
∃ c : (weigthBoundIndices k D) → F,
c ≠ 0 ∧ constraintMap k n m ωs f D c = 0Formal artifact
Lean source
lemma exists_nonzero_solution_gen (k n m : ℕ) (ωs : Fin n ↪ F) (f : Fin n → F) (D : ℕ) (hD : numVars k D > numConstraints n m) : ∃ c : (weigthBoundIndices k D) → F, c ≠ 0 ∧ constraintMap k n m ωs f D c = 0 := by have h_kernel_nontrivial : Module.finrank F ((weigthBoundIndices k D) → F) > Module.finrank F ((Fin n → constraintIndices m → F)) := by convert hD using 1 · simp [numVars] · simp [numConstraints] norm_num [Module.finrank] have h_inj : ¬ Function.Injective (constraintMap k n m ωs f D) := by intro h_inj exact h_kernel_nontrivial.not_ge (LinearMap.finrank_range_of_inj h_inj ▸ Submodule.finrank_le _) contrapose! h_inj exact LinearMap.ker_eq_bot.mp (eq_bot_iff.mpr fun x hx ↦ by_contra fun hx' ↦ h_inj x hx' <| by simpa using hx)- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/CodingTheory/GuruswamiSudan/Basic.lean:391-407
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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 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.