YES

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

Consider the left-linear TRS R:

  nats() -> :(0(),inc(nats()))
  inc(:(x,y)) -> :(s(x),inc(y))
  inc(tl(nats())) -> tl(inc(nats()))
  hd(:(x,y)) -> x
  tl(:(x,y)) -> y
  d(:(x,y)) -> :(x,:(x,d(y)))

Let C be the following subset of R:

  inc(tl(nats())) -> tl(inc(nats()))
  nats() -> :(0(),inc(nats()))
  inc(:(x,y)) -> :(s(x),inc(y))
  tl(:(x,y)) -> y

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)

Consider the left-linear TRS R:

  inc(tl(nats())) -> tl(inc(nats()))
  nats() -> :(0(),inc(nats()))
  inc(:(x,y)) -> :(s(x),inc(y))
  tl(:(x,y)) -> y

Let C be the following subset of R:

  nats() -> :(0(),inc(nats()))
  inc(:(x,y)) -> :(s(x),inc(y))
  tl(:(x,y)) -> y

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:

  nats() -> :(0(),inc(nats()))
  inc(:(x,y)) -> :(s(x),inc(y))
  tl(:(x,y)) -> y

Let C be the following subset of R:

  inc(:(x,y)) -> :(s(x),inc(y))
  tl(:(x,y)) -> y

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:

  inc(:(x,y)) -> :(s(x),inc(y))
  tl(:(x,y)) -> y

Let C be the following subset of R:

  inc(:(x,y)) -> :(s(x),inc(y))

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:

  inc(:(x,y)) -> :(s(x),inc(y))

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.