Coeff linear Combination monomial
GuruswamiSudan.coeff_linearCombination_monomial
Plain-language statement
The coefficient of X^i Y^j in a linear combination of monomials is the coefficient of the combination.
Exact Lean statement
lemma coeff_linearCombination_monomial (c : ℕ × ℕ →₀ F) (i j : ℕ) :
((linearCombination F (fun p ↦ monomial (F := F) p.1 p.2) c).coeff j).coeff i = c (i, j)Formal artifact
Lean source
lemma coeff_linearCombination_monomial (c : ℕ × ℕ →₀ F) (i j : ℕ) : ((linearCombination F (fun p ↦ monomial (F := F) p.1 p.2) c).coeff j).coeff i = c (i, j) := by simp only [linearCombination_apply, Finsupp.sum, finsetSum_coeff, coeff_smul, smul_eq_mul] rw [Finset.sum_eq_single (i, j)] <;> simp +contextual only [Finsupp.mem_support_iff, ne_eq, mul_eq_zero, false_or, Prod.forall, Prod.mk.injEq, not_and] · erw [coeff_monomial, if_pos rfl]; aesop · intro a b rw [monomial] by_cases ha : a = i <;> by_cases hb : b = j <;> simp_all [coeff_monomial] · exact fun h ↦ by left; exact Function.notMem_support.mp h- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/CodingTheory/GuruswamiSudan/Basic.lean:426-435
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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
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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.
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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.