Tensor Generator eq Tensor Generator Explicit
CoreDefinitions.TensorGenerator_eq_TensorGenerator_Explicit
Plain-language statement
The tensor product generator TensorGenerator and the explicit componentwise generator TensorGenerator_Explicit agree under the canonical isomorphism between F^ℓ ⊗ F^ℓ′ and (ℓ × ℓ') → F.
Exact Lean statement
theorem TensorGenerator_eq_TensorGenerator_Explicit {ℓ' : Type} [Fintype ℓ'] [DecidableEq ℓ]
[DecidableEq ℓ'] {S S' : Type} (G : Generator S ℓ F) (G' : Generator S' ℓ' F) (p : S × S') :
tensorProductPiFunEquiv F ℓ ℓ' (TensorGenerator G G' p) = TensorGenerator_Explicit G G' pFormal artifact
Lean source
theorem TensorGenerator_eq_TensorGenerator_Explicit {ℓ' : Type} [Fintype ℓ'] [DecidableEq ℓ] [DecidableEq ℓ'] {S S' : Type} (G : Generator S ℓ F) (G' : Generator S' ℓ' F) (p : S × S') : tensorProductPiFunEquiv F ℓ ℓ' (TensorGenerator G G' p) = TensorGenerator_Explicit G G' p := by unfold tensorProductPiFunEquiv TensorGenerator TensorGenerator_Explicit convert (Pi.basisFun F ℓ).tensorProduct (Pi.basisFun F ℓ') |> fun b => b.equivFun_apply ( G p.1 ⊗ₜ[F] G' p.2 ) using 1 ext ⟨i, j⟩ simp only [Module.Basis.tensorProduct_repr_tmul_apply, Pi.basisFun_repr, smul_eq_mul] ring- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Data/CodingTheory/ProximityGap/ProximityGenerators.lean:129-137
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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...
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Person-level attribution pending.
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Source project: ArkLib
Person-level attribution pending.
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Plain-language statement
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Source project: ArkLib
Person-level attribution pending.