Rel Dist From Code le iff dist From Code le
Code.relDistFromCode_le_iff_distFromCode_le
Plain-language statement
A word u is relatively close to a code C within an relative error bound δ if and only if it is relatively close within the equivalent absolute error bound ⌊δ * n⌋.
Exact Lean statement
theorem relDistFromCode_le_iff_distFromCode_le {C : Set (ι → F)} (u : ι → F) (δ : ℝ≥0) :
δᵣ(u, C) ≤ δ ↔ Δ₀(u, C) ≤ Nat.floor (δ * Fintype.card ι)Formal artifact
Lean source
theorem relDistFromCode_le_iff_distFromCode_le {C : Set (ι → F)} (u : ι → F) (δ : ℝ≥0) : δᵣ(u, C) ≤ δ ↔ Δ₀(u, C) ≤ Nat.floor (δ * Fintype.card ι) := by have h_n_pos : 0 < Fintype.card ι := Fintype.card_pos have h_n_pos_nnreal : 0 < (Fintype.card ι : ℝ≥0) := by exact_mod_cast h_n_pos conv_rhs => rw [closeToCode_iff_closeToCodeword_of_minDist] conv_lhs => rw [relCloseToCode_iff_relCloseToCodeword_of_minDist (u := u) (C := C)] apply exists_congr intro v simp only [and_congr_right_iff] intro hv_mem rw [pairRelDist_le_iff_pairDist_le]- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/CodingTheory/Basic/RelativeDistance.lean:349-359
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.