Research proof index

The handler writes only the cells declared by demoWrites.

The append node predicate at a left-embedded index reduces to the left shape's predicate.

The append node predicate at a right-embedded index reduces to the right shape's predicate.

Successor unfolding of the sequentially-composed shape. ChallengeTreeShape.seqCompose of a family over m + 1 factors is the binary append of the head shape with the sequential composition of the tail. This is the shape-level analogue of ProtocolSpec.seqCompose_succ_eq_append, and is what lets the n-ary tree-soundness induction reduce its ste...

Perfect completeness of the pure pass-through CheckClaim oracle reduction. Because the verifier no longer checks P at runtime (it is a pure pass-through, with P living in oracleRelOut), completeness needs the explicit hypothesis hP that every relIn input already satisfies P. Under hP, the prover forwards ⟨stmt, oStmt⟩ unchanged and t...

Coordinate-wise special soundness of CheckClaim. The verifier is a pure pass-through with no challenge rounds, so CWSS collapses (via the oracle no-challenge bridge coordinateWiseSpecialSound_of_isEmpty_challengeIdx) to a transcript-level obligation. The extractor is trivial (e := fun _ _ => (), there is no witness); since the pass-through outpu...