Choose s conflict insert eval ne zero
KZG.CommitmentScheme.choose_s_conflict_insert_eval_ne_zero
Plain-language statement
The conflict-branch adjoined vanishing product is nonzero at τ.
Exact Lean statement
lemma choose_s_conflict_insert_eval_ne_zero (hn : 1 ≤ n) (α τ : ZMod p)
(srs : Vector G₁ (n + 1) × Vector G₂ 2)
(hsrs : srs = Groups.PowerSrs.generate (g₁ := g₁) (g₂ := g₂) n τ) (hατ : α ≠ τ) :
(∏ s ∈ insert α (chooseSConflict α srs hn),
(X - C s : CPolynomial (ZMod p))).eval τ ≠ 0Formal artifact
Lean source
lemma choose_s_conflict_insert_eval_ne_zero (hn : 1 ≤ n) (α τ : ZMod p) (srs : Vector G₁ (n + 1) × Vector G₂ 2) (hsrs : srs = Groups.PowerSrs.generate (g₁ := g₁) (g₂ := g₂) n τ) (hατ : α ≠ τ) : (∏ s ∈ insert α (chooseSConflict α srs hn), (X - C s : CPolynomial (ZMod p))).eval τ ≠ 0 := by have hτS : τ ∉ chooseSConflict α srs hn := choose_s_conflict_tau hn α τ srs hsrs have hτS_insert : τ ∉ insert α (chooseSConflict α srs hn) := by simp only [Finset.mem_insert, not_or] exact ⟨Ne.symm hατ, hτS⟩ exact prod_x_sub_c_eval_ne_zero hτS_insert- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/Commitments/Functional/KZG/FunctionBinding/EvaluationBindingConflict.lean:296-306
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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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Plain-language statement
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Source project: ArkLib
Person-level attribution pending.