Split Nth of sum comp
Polynomial.splitNth_of_sum_comp
Plain-language statement
splitNth is the left inverse of the n-way recombination: splitting the polynomial ∑ j, X^j * (u j)(X^n) recovers u i for each component i.
Exact Lean statement
@[simp]
lemma splitNth_of_sum_comp {n : ℕ} [inst : NeZero n] (u : Fin n → 𝔽[X]) (i : Fin n) :
splitNth (∑ j : Fin n, X ^ (j : ℕ) * (u j).comp (X ^ n)) n i = u iFormal artifact
Lean source
@[simp]lemma splitNth_of_sum_comp {n : ℕ} [inst : NeZero n] (u : Fin n → 𝔽[X]) (i : Fin n) : splitNth (∑ j : Fin n, X ^ (j : ℕ) * (u j).comp (X ^ n)) n i = u i := by have hn : 0 < n := Nat.pos_of_ne_zero inst.out ext e rw [splitNth_coeff, finsetSum_coeff, Finset.sum_eq_single i] · aesop (add unsafe (by rw [←expand_eq_comp_X_pow])) · intro j _ hj rw [coeff_X_pow_mul'] by_cases hle : (j : ℕ) ≤ e * n + i · rw [if_pos hle, ←expand_eq_comp_X_pow, coeff_expand hn, if_neg] intro hdvd have hmod := (Nat.modEq_iff_dvd' hle).mpr hdvd aesop (add safe cases Fin) (add simp [Nat.ModEq, Nat.mod_eq_of_lt]) · simp_all · aesop- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/Polynomial/SplitFold.lean:305-322
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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.