Final Sumcheck Oracle Verifier rbr Knowledge Soundness
Binius.BinaryBasefold.CoreInteraction.finalSumcheckOracleVerifier_rbrKnowledgeSoundness
Plain-language statement
Round-by-round knowledge soundness for the final sumcheck step
Exact Lean statement
theorem finalSumcheckOracleVerifier_rbrKnowledgeSoundness [Fintype L] {σ : Type}
(init : ProbComp σ) (impl : QueryImpl []ₒ (StateT σ ProbComp)) :
(finalSumcheckVerifier 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate)).rbrKnowledgeSoundness init impl
(relIn := roundRelation 𝔽q β (ϑ := ϑ)
(mp := BBF_SumcheckMultiplierParam) (Fin.last ℓ))
(relOut := finalSumcheckRelOut 𝔽q β (ϑ := ϑ) (h_ℓ_add_R_rate := h_ℓ_add_R_rate))
(rbrKnowledgeError := finalSumcheckKnowledgeError)Formal artifact
Lean source
theorem finalSumcheckOracleVerifier_rbrKnowledgeSoundness [Fintype L] {σ : Type} (init : ProbComp σ) (impl : QueryImpl []ₒ (StateT σ ProbComp)) : (finalSumcheckVerifier 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate)).rbrKnowledgeSoundness init impl (relIn := roundRelation 𝔽q β (ϑ := ϑ) (mp := BBF_SumcheckMultiplierParam) (Fin.last ℓ)) (relOut := finalSumcheckRelOut 𝔽q β (ϑ := ϑ) (h_ℓ_add_R_rate := h_ℓ_add_R_rate)) (rbrKnowledgeError := finalSumcheckKnowledgeError) := by use FinalSumcheckWit (L := L) 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) (ℓ := ℓ) use finalSumcheckRbrExtractor 𝔽q β (h_ℓ_add_R_rate := h_ℓ_add_R_rate) use finalSumcheckKnowledgeStateFunction 𝔽q β (ϑ := ϑ) (h_ℓ_add_R_rate := h_ℓ_add_R_rate) init impl intro stmtIn witIn prover j exact absurd j.2 (by simp [pSpecFinalSumcheckStep])- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/ProofSystem/Binius/BinaryBasefold/Steps.lean:1071-1083
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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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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.