YES

# parallel critical pair closing system (Shintani and Hirokawa 2022)

Consider the left-linear TRS R:

  a(x) -> b(x)
  a(b(x)) -> b(a(c(a(x))))
  b(x) -> c(x)
  c(c(x)) -> x

Let C be the following subset of R:

  a(x) -> b(x)
  b(x) -> c(x)
  c(c(x)) -> x

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

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

Consider the left-linear TRS R:

  a(x) -> b(x)
  b(x) -> c(x)
  c(c(x)) -> x

Let C be the following subset of R:

  b(x) -> c(x)
  c(c(x)) -> x

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.

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

Consider the left-linear TRS R:

  b(x) -> c(x)
  c(c(x)) -> x

Let C be the following subset of R:

  (empty)

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.

# emptiness

The empty TRS is confluent.