Right Prefix concat
ProtocolSpec.ChallengeTree.rightPrefix_concat
Plain-language statement
rightPrefix commutes with extending the right prefix by one round. The rightPrefix/Fin.snoc dites are split by split_ifs; contradictory combinations close by omega (with idx's bound), matching ones by cast_eq_cast_of_heq (stripping casts to a base HEq, then rfl/index omega).
Exact Lean statement
theorem rightPrefix_concat (tr₁ : FullTranscript pSpec₁) {i : Fin n}
(pre₂ : Transcript i.castSucc pSpec₂) (x : pSpec₂.«Type» i) :
rightPrefix tr₁ (pre₂.concat x) =
(rightPrefix tr₁ pre₂).concat (cast (by simp only [Fin.vappend_eq_append,
Fin.append_right]) x : (pSpec₁ ++ₚ pSpec₂).«Type» (Fin.natAdd m i))Formal artifact
Lean source
theorem rightPrefix_concat (tr₁ : FullTranscript pSpec₁) {i : Fin n} (pre₂ : Transcript i.castSucc pSpec₂) (x : pSpec₂.«Type» i) : rightPrefix tr₁ (pre₂.concat x) = (rightPrefix tr₁ pre₂).concat (cast (by simp only [Fin.vappend_eq_append, Fin.append_right]) x : (pSpec₁ ++ₚ pSpec₂).«Type» (Fin.natAdd m i)) := by funext idx have hidx : idx.val < m + i.val + 1 := by have := idx.isLt; simp only [rightRound, Fin.val_natAdd, Fin.val_succ] at this; omega have hi : i.val < n := i.isLt simp only [rightPrefix, Transcript.concat, Fin.snoc, Fin.val_castLT, Fin.val_castSucc, Fin.val_succ, Fin.val_natAdd] split_ifs <;> first | (exfalso; omega) | rfl | (apply cast_eq_cast_of_heq try simp only [cast_heq_iff_heq] first | rfl | exact HEq.rfl | (exact (heq_cast_iff_heq _ _ _).mpr HEq.rfl))- Project
- ArkLib
- License
- Apache-2.0
- Commit
- fad5cbf80877
- Source
- ArkLib/OracleReduction/Security/TranscriptTree/Composition.lean:819-839
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
Related declarations
Affine gaps lifted to interleaved codes
affine_gaps_lifted_to_interleaved_codes
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
ArkLib.Lattices.Ajtai.gadgetDecompose_coeff
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.
Gadget Decompose lawful
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.