Core Interaction Oracle Reduction perfect Completeness
Binius.BinaryBasefold.CoreInteraction.coreInteractionOracleReduction_perfectCompleteness
Plain-language statement
Perfect completeness for the core interaction oracle reduction
Exact Lean statement
theorem coreInteractionOracleReduction_perfectCompleteness :
OracleReduction.perfectCompleteness
(pSpec := pSpecCoreInteraction 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate))
(relIn := roundRelation (mp := BBF_SumcheckMultiplierParam) 𝔽q β (ϑ:=ϑ)
(h_ℓ_add_R_rate := h_ℓ_add_R_rate) 0)
(relOut := finalSumcheckRelOut 𝔽q β (ϑ:=ϑ) (h_ℓ_add_R_rate := h_ℓ_add_R_rate))
(oracleReduction := coreInteractionOracleReduction 𝔽q β (ϑ:=ϑ) )
(init := init)
(impl := impl)Formal artifact
Lean source
theorem coreInteractionOracleReduction_perfectCompleteness : OracleReduction.perfectCompleteness (pSpec := pSpecCoreInteraction 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate)) (relIn := roundRelation (mp := BBF_SumcheckMultiplierParam) 𝔽q β (ϑ:=ϑ) (h_ℓ_add_R_rate := h_ℓ_add_R_rate) 0) (relOut := finalSumcheckRelOut 𝔽q β (ϑ:=ϑ) (h_ℓ_add_R_rate := h_ℓ_add_R_rate)) (oracleReduction := coreInteractionOracleReduction 𝔽q β (ϑ:=ϑ) ) (init := init) (impl := impl) := by unfold coreInteractionOracleReduction pSpecCoreInteraction apply OracleReduction.append_perfectCompleteness · -- Perfect completeness of sumcheckFoldOracleReduction exact sumcheckFoldOracleReduction_perfectCompleteness 𝔽q β (ϑ:=ϑ) (h_ℓ_add_R_rate := h_ℓ_add_R_rate) (mp := BBF_SumcheckMultiplierParam) (init := init) (impl := impl) · -- Perfect completeness of finalSumcheckOracleReduction exact finalSumcheckOracleReduction_perfectCompleteness 𝔽q β (ϑ:=ϑ) init impl- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/ProofSystem/Binius/BinaryBasefold/CoreInteractionPhase.lean:754-770
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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...
Source project: ArkLib
Person-level attribution pending.
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ArkLib.Lattices.Ajtai.gadgetDecompose_coeff
Plain-language statement
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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.