Eval property of folding polynomial
Polynomial.FoldingPolynomial.eval_property_of_folding_polynomial
Plain-language statement
A means to evaluate the original polynomial in terms of the folding polynomial.
Exact Lean statement
lemma eval_property_of_folding_polynomial {q f : F[X]} {x : F} :
((foldingPolynomial q f).map (Polynomial.evalRingHom (q.eval x))).eval x = f.eval xFormal artifact
Lean source
lemma eval_property_of_folding_polynomial {q f : F[X]} {x : F} : ((foldingPolynomial q f).map (Polynomial.evalRingHom (q.eval x))).eval x = f.eval x := by have h_subst : ((Polynomial.FoldingPolynomial.foldingPolynomial q f).map (Polynomial.compRingHom q)).eval X = f := substitution_property_of_folding_polynomial generalize_proofs at * (replace h_subst := congr_arg (Polynomial.eval x) h_subst simp_all only [eval_map] convert h_subst using 1 simp +decide [Polynomial.eval₂_eq_sum_range] ring_nf simp +decide [Polynomial.eval_finsetSum])- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/Polynomial/FoldingPolynomial.lean:378-389
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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.