Research proof index

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...

n-ary base case. The identity verifier is tree-special-sound for any shape S, with input relation equal to output relation: the empty protocol !p[] has no challenge rounds, so this is the no-challenge bridge with the extractor that noncomputably picks a witness of stmtIn whenever one exists.

Converse of pure_accepting_of_mem: if a verifier deterministically outputs out on (stmt, tr) and its run is accepted into lang with probability one, then out ∈ lang.

n-ary CWSS composition. The coordinate-wise special-soundness wrapper of seqCompose_treeSpecialSound, obtained by unfolding coordinateWiseSpecialSound to tree-soundness of the induced shape and rewriting with CWSSStructure.toShape_seqCompose.