Square roots explicit
Domain.CosetFftDomainClass.square_roots_explicit
Plain-language statement
The square roots of x inside the ith subdomain are exactly y and -y, for any square root y of x.
Exact Lean statement
lemma square_roots_explicit [DecidableEq F] {i : ℕ} (hi : i < n) {y : F}
(hx : x ∈ subdomain ω (i + 1)) (hy : y ^ 2 = x) :
{y ∈ (subdomain ω i).toFinset | y ^ 2 = x} = {y, -y}Formal artifact
Lean source
lemma square_roots_explicit [DecidableEq F] {i : ℕ} (hi : i < n) {y : F} (hx : x ∈ subdomain ω (i + 1)) (hy : y ^ 2 = x) : {y ∈ (subdomain ω i).toFinset | y ^ 2 = x} = {y, -y} := by have : NeZero (n - i) := ⟨by omega⟩ apply Finset.Subset.antisymm · intro z hz simp_all only [Finset.mem_filter, Finset.mem_insert, Finset.mem_singleton] exact eq_or_eq_neg_of_sq_eq_sq _ _ <| by rw [hz.2, hy] · have hy_mem : y ∈ subdomain ω i := sq_root_mem_subdomain hi hx hy simp_all [Finset.subset_iff]- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/Domain/CosetFftDomain/Subdomain.lean:447-456
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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.