Trace H sum
ArkLib.Lattices.CyclotomicModulus.traceH_sum
Plain-language statement
Tr_H distributes over finite sums (it is additive).
Exact Lean statement
theorem traceH_sum (α k : ℕ) {ι : Type*} (s : Finset ι)
(f : ι → Rq (powTwoCyclotomic (R := R) α)) :
traceH α k (∑ j ∈ s, f j) = ∑ j ∈ s, traceH α k (f j)Formal artifact
Lean source
theorem traceH_sum (α k : ℕ) {ι : Type*} (s : Finset ι) (f : ι → Rq (powTwoCyclotomic (R := R) α)) : traceH α k (∑ j ∈ s, f j) = ∑ j ∈ s, traceH α k (f j) := by classical induction s using Finset.induction with | empty => simp only [Finset.sum_empty]; exact traceOver_zero _ _ | insert a s ha ih => rw [Finset.sum_insert ha, show traceH α k (f a + ∑ j ∈ s, f j) = traceH α k (f a) + traceH α k (∑ j ∈ s, f j) from traceOver_add _ _ _ _, ih, Finset.sum_insert ha]- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/Lattices/CyclotomicRing/Subfield/TraceInnerProduct.lean:213-223
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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.
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ArkLib.Lattices.Ajtai.gadgetDecompose_coeff
Plain-language statement
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Source project: ArkLib
Person-level attribution pending.
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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.