Full row rank via rank sub Left Full
Matrix.full_row_rank_via_rank_subLeftFull
Plain-language statement
An m×n matrix has full rank if the submatrix consisting of columns 1 through m has rank m.
Exact Lean statement
lemma full_row_rank_via_rank_subLeftFull (h : m ≤ n) :
(subLeftFull U (Fin.castLE h)).rank = m → U.rank = mFormal artifact
Lean source
lemma full_row_rank_via_rank_subLeftFull (h : m ≤ n) : (subLeftFull U (Fin.castLE h)).rank = m → U.rank = m := by intro h_sub_mat_rank rw[ Matrix.rank_eq_finrank_span_cols, ← Matrix.cRank_toNat_eq_finrank ] have h_cRank : U.cRank = ↑m := by apply le_antisymm · calc U.cRank ≤ ↑(Fintype.card (Fin m)) := Matrix.cRank_le_card_height U _ = ↑m := by rw[Fintype.card_fin] · calc ↑m = ↑((subLeftFull U (Fin.castLE h)).rank) := by rw[h_sub_mat_rank] _ = (subLeftFull U (Fin.castLE h)).cRank := by exact cRank_rank_conversion _ ≤ U.cRank := by exact Matrix.cRank_submatrix_le U id (Fin.castLE h) simp [h_cRank]- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/CodingTheory/Prelims.lean:106-120
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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.