Possible Deltas subset rel Hamming Dist Range
DivergenceOfSets.possibleDeltas_subset_relHammingDistRange
Plain-language statement
The set of possible relative Hamming distances between two sets is well-defined.
Exact Lean statement
@[simp]
lemma possibleDeltas_subset_relHammingDistRange :
possibleDeltas U V ⊆ relHammingDistRange ιFormal artifact
Lean source
@[simp]lemma possibleDeltas_subset_relHammingDistRange : possibleDeltas U V ⊆ relHammingDistRange ι := fun x hx_mem_deltas ↦ by simp only [possibleDeltas, Set.mem_setOf_eq] at hx_mem_deltas rcases hx_mem_deltas with ⟨u, hu_mem, h_dist_eq⟩ rw [←h_dist_eq] unfold relDistFromCode' have h_mem : (Finset.univ.image (fun (c : V) => relHammingDist u c)).min' (Finset.univ_nonempty.image _) ∈ Finset.univ.image (fun (c : V) => relHammingDist u c) := Finset.min'_mem _ (Finset.univ_nonempty.image _) obtain ⟨c, _, h_eq⟩ := Finset.mem_image.mp h_mem rw [←h_eq] exact relHammingDist_mem_relHammingDistRange- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/CodingTheory/DivergenceOfSets.lean:43-56
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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.
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Source project: ArkLib
Person-level attribution pending.
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Plain-language statement
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Source project: ArkLib
Person-level attribution pending.