Research proof index

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.

If all verifiers in a sequence satisfy knowledge soundness with respective knowledge errors, then their sequential composition also satisfies knowledge soundness. The knowledge error of the seqComposed verifier is the sum of the individual errors.

If all verifiers in a sequence satisfy soundness with respective soundness errors, then their sequential composition also satisfies soundness. The soundness error of the seqComposed verifier is the sum of the individual errors.

n-ary generic tree-soundness composition. If each factor verifier is pure (IsPure) and tree-special-sound for the seam relations rel i.castSucc ↦ rel i.succ, then the sequential composition Verifier.seqCompose is tree-special-sound for the sequentially-composed shape from rel 0 to rel (Fin.last m). The induction's base case is `Verifier.id...

Core inequality: W†W ≤ I. This is the key step, following from the isometry argument: V_rho ⊗ I_C and I_A ⊗ V_sigma are isometries, their cross product has norm ≤ 1, and the result can be related to W_mat through the MES computation (Eq. 6 in Lin-Kim-Hsieh).

For two $n$-times continuously differentiable functions $f$ and $g$, the $n$th iterated derivative distributes over subtraction: $(f-g)^{(n)}=f^{(n)}-g^{(n)}.$