Split Data snd At is Structured

theorem SplitData.sndAt_isStructured :    {r : Fin (m + 1)} → (S : SplitData S₁.arity S₂.arity r) →    S.src.IsStructured (S₁.append S₂) → (path : LeafPath S.fst) →    (S.sndAt path).IsStructured S₂  | _, .boundary rp, hS, _ => by      have hround : leftRound (n := n) (Fin.last m) =          rightRound (m := m) (0 : Fin (n + 1)) := by        apply Fin.ext        simp [leftRound, rightRound]      simp only [SplitData.src] at hS      apply rp.tree_isStructured      convert hS using 1      · exact hround.symm      · rfl  | _, .msg i h m₁ child, hS, path => by      have hround : (Fin.castAdd n i).succ = leftRound i.succ := by        apply Fin.ext        rfl      simp only [SplitData.src, ChallengeTree.IsStructured] at hS      have hS' : child.src.IsStructured (S₁.append S₂) := by        convert hS using 1        · exact hround.symm        · rfl      exact SplitData.sndAt_isStructured child hS' (peelMsg path)  | _, .chal i h chals children, hS, path => by      have hApp : (pSpec₁ ++ₚ pSpec₂).dir (Fin.castAdd n i) = .V_to_P := by        simpa [ProtocolSpec.append, Fin.vappend_eq_append, Fin.append_left] using h      have hIdx : (⟨Fin.castAdd n i, hApp⟩ : (pSpec₁ ++ₚ pSpec₂).ChallengeIdx)          = ChallengeIdx.inl ⟨i, h⟩ := by ext; rfl      have hAr : appendArity S₁.arity S₂.arity ⟨Fin.castAdd n i, hApp⟩ = S₁.arity ⟨i, h⟩ := by        rw [hIdx]; simpa [appendArity] using          congrArg (Sum.elim S₁.arity S₂.arity)            (ChallengeIdx.sumEquiv_symm_inl (pSpec₂ := pSpec₂) ⟨i, h⟩)      have hS' := hS      simp only [SplitData.src, ChallengeTree.IsStructured] at hS'      exact SplitData.sndAt_isStructured (children (chalPeel path).1)        (hS'.2 (Fin.cast hAr.symm (chalPeel path).1)) (chalPeel path).2