YES

# parallel critical pair closing system (Shintani and Hirokawa 2022, Section 8 in LMCS 2023)

Consider the left-linear TRS R:

  a() -> b()
  a() -> c()
  a() -> e()
  b() -> d()
  c() -> a()
  d() -> a()
  d() -> e()
  g(x) -> h(a())
  h(x) -> e()

Let C be the following subset of R:

  a() -> c()
  c() -> a()
  d() -> a()
  b() -> d()
  a() -> b()
  a() -> e()
  d() -> e()

The TRS R is left-linear and all parallel critical pairs are joinable by C.
Therefore, the confluence of R is equivalent to that of C.

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

Consider the left-linear TRS R:

  a() -> c()
  c() -> a()
  d() -> a()
  b() -> d()
  a() -> b()
  a() -> e()
  d() -> e()

Let C be the following subset of R:

  a() -> c()
  c() -> a()
  d() -> a()
  b() -> d()
  a() -> b()
  a() -> e()
  d() -> e()

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:

  a() -> c()
  c() -> a()
  d() -> a()
  b() -> d()
  a() -> b()
  a() -> e()
  d() -> e()

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

  phi(a() -> c()) = 5
  phi(c() -> a()) = 1
  phi(d() -> a()) = 3
  phi(b() -> d()) = 1
  phi(a() -> b()) = 4
  phi(a() -> e()) = 2
  phi(d() -> e()) = 1

  psi(a() -> c()) = 5
  psi(c() -> a()) = 1
  psi(d() -> a()) = 3
  psi(b() -> d()) = 1
  psi(a() -> b()) = 4
  psi(a() -> e()) = 2
  psi(d() -> e()) = 1