YES

# Compositional parallel critical pair system (Shintani and Hirokawa 2022).

Consider the left-linear TRS R:

  from(x) -> :(x,from(s(x)))
  sel(0(),:(y,z)) -> y
  sel(s(x),:(y,z)) -> sel(x,z)

Let C be the following subset of R:

  from(x) -> :(x,from(s(x)))
  sel(0(),:(y,z)) -> y
  sel(s(x),:(y,z)) -> sel(x,z)

The parallel critical pair system PCPS(R,C) is:

  (empty)

All pairs in PCP(R) are joinable and PCPS(R,C)/R is terminating.
Therefore, the confluence of R follows from that of C.

# Parallel rule labeling (Zankl et al. 2015).

Consider the left-linear TRS R:

  from(x) -> :(x,from(s(x)))
  sel(0(),:(y,z)) -> y
  sel(s(x),:(y,z)) -> sel(x,z)

All parallel critical peaks (except C's) are decreasing wrt rule labeling:

  phi(from(x) -> :(x,from(s(x)))) = 1
  phi(sel(0(),:(y,z)) -> y) = 1
  phi(sel(s(x),:(y,z)) -> sel(x,z)) = 1

  psi(from(x) -> :(x,from(s(x)))) = 1
  psi(sel(0(),:(y,z)) -> y) = 1
  psi(sel(s(x),:(y,z)) -> sel(x,z)) = 1