To Poly div By Monic
CompPoly.CPolynomial.toPoly_divByMonic
Project documentation
Construct a canonical polynomial from a coefficient function Fin n → R. The coefficients are stored in an array (index i gives the coefficient of X^i) and then trimmed to remove trailing zeros. -/ def ofFn [Zero R] [BEq R] [LawfulBEq R] {n : ℕ} (f : Fin n → R) : CPolynomial R := ⟨(Raw.mk (Array.ofFn f)).trim, Raw.Trim.isCanonical_trim _⟩ section Div...
Exact Lean statement
theorem toPoly_divByMonic (fp fq : CPolynomial R) (hq : fq.toPoly.Monic) :
(fp.divByMonic fq).toPoly = fp.toPoly /ₘ fq.toPolyFormal artifact
Lean source
theorem toPoly_divByMonic (fp fq : CPolynomial R) (hq : fq.toPoly.Monic) : (fp.divByMonic fq).toPoly = fp.toPoly /ₘ fq.toPoly := by set fuel := fp.val.size have heq := divModByMonicAux_go_eq fuel fp.val fq.val have hdeg := divModByMonicAux_go_degree_bound fuel fp.val fq.val (trim_eq fp) (trim_eq fq) hq (by have hq_size_pos : 0 < fq.val.size := Raw.size_pos_of_toPoly_ne_zero hq.ne_zero omega) set quot := (Raw.divModByMonicAux.go fuel fp.val fq.val).1 set rem := (Raw.divModByMonicAux.go fuel fp.val fq.val).2 have hd : (fp.divByMonic fq).toPoly = quot.toPoly := by change (Raw.divByMonic fp.val fq.val).toPoly = quot.toPoly change (Raw.divModByMonicAux fp.val fq.val).1.toPoly = quot.toPoly simp only [Raw.divModByMonicAux, fuel, quot] have huniq := @Polynomial.div_modByMonic_unique R _ fp.toPoly fq.toPoly quot.toPoly rem.toPoly hq ⟨by rw [_root_.add_comm]; exact heq, hdeg⟩ rw [hd] exact huniq.1.symm- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/ToCompPoly/Univariate/Basic.lean:238-256
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Related declarations
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.