Rank nonsquare eq deg of deg le
Vandermonde.rank_nonsquare_eq_deg_of_deg_le
Plain-language statement
The rank of a non-square Vandermonde matrix with more rows than columns is the number of columns.
Exact Lean statement
lemma rank_nonsquare_eq_deg_of_deg_le (inj : Function.Injective α) (h : n ≤ m) :
(Vandermonde.nonsquare (ι' := n) α).rank = nFormal artifact
Lean source
lemma rank_nonsquare_eq_deg_of_deg_le (inj : Function.Injective α) (h : n ≤ m) : (Vandermonde.nonsquare (ι' := n) α).rank = n := by suffices ((Vandermonde.nonsquare (ι' := n) α).subUpFull (Fin.castLE h)).rank = n by exact Matrix.rank_eq_if_subUpFull_eq h this rw[ ←subUpFull_of_vandermonde_is_vandermonde, Matrix.rank_eq_if_det_ne_zero ] rw [@Matrix.det_vandermonde_ne_zero_iff F _ n _ (α ∘ Fin.castLE h)] apply Function.Injective.comp <;> aesop (add simp Fin.castLE_injective)- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/Matrix/Vandermonde.lean:58-67
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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.
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.