YES

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

Consider the left-linear TRS R:

  P(x) -> Q(Q(p(x)))
  p(p(x)) -> q(q(x))
  p(Q(Q(x))) -> Q(Q(p(x)))
  Q(p(q(x))) -> q(p(Q(x)))
  q(q(p(x))) -> p(q(q(x)))
  q(Q(x)) -> x
  Q(q(x)) -> x
  p(P(x)) -> x
  P(p(x)) -> x

Let C be the following subset of R:

  Q(q(x)) -> x
  p(p(x)) -> q(q(x))
  p(Q(Q(x))) -> Q(Q(p(x)))
  P(x) -> Q(Q(p(x)))
  P(p(x)) -> x
  Q(p(q(x))) -> q(p(Q(x)))
  q(q(p(x))) -> p(q(q(x)))
  q(Q(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:

  Q(q(x)) -> x
  p(p(x)) -> q(q(x))
  p(Q(Q(x))) -> Q(Q(p(x)))
  P(x) -> Q(Q(p(x)))
  P(p(x)) -> x
  Q(p(q(x))) -> q(p(Q(x)))
  q(q(p(x))) -> p(q(q(x)))
  q(Q(x)) -> x

Let C be the following subset of R:

  Q(q(x)) -> x
  p(p(x)) -> q(q(x))
  P(x) -> Q(Q(p(x)))
  p(Q(Q(x))) -> Q(Q(p(x)))
  Q(p(q(x))) -> q(p(Q(x)))
  q(q(p(x))) -> p(q(q(x)))
  q(Q(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:

  Q(q(x)) -> x
  p(p(x)) -> q(q(x))
  P(x) -> Q(Q(p(x)))
  p(Q(Q(x))) -> Q(Q(p(x)))
  Q(p(q(x))) -> q(p(Q(x)))
  q(q(p(x))) -> p(q(q(x)))
  q(Q(x)) -> x

Let C be the following subset of R:

  Q(q(x)) -> x
  p(p(x)) -> q(q(x))
  p(Q(Q(x))) -> Q(Q(p(x)))
  Q(p(q(x))) -> q(p(Q(x)))
  q(q(p(x))) -> p(q(q(x)))
  q(Q(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.

# Compositional parallel rule labeling (Shintani and Hirokawa 2022).

Consider the left-linear TRS R:

  Q(q(x)) -> x
  p(p(x)) -> q(q(x))
  p(Q(Q(x))) -> Q(Q(p(x)))
  Q(p(q(x))) -> q(p(Q(x)))
  q(q(p(x))) -> p(q(q(x)))
  q(Q(x)) -> x

Let C be the following subset of R:

  Q(q(x)) -> x
  p(p(x)) -> q(q(x))
  p(Q(Q(x))) -> Q(Q(p(x)))
  Q(p(q(x))) -> q(p(Q(x)))
  q(q(p(x))) -> p(q(q(x)))
  q(Q(x)) -> x

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

  phi(Q(q(x)) -> x) = 0
  phi(p(p(x)) -> q(q(x))) = 0
  phi(p(Q(Q(x))) -> Q(Q(p(x)))) = 0
  phi(Q(p(q(x))) -> q(p(Q(x)))) = 0
  phi(q(q(p(x))) -> p(q(q(x)))) = 0
  phi(q(Q(x)) -> x) = 0

  psi(Q(q(x)) -> x) = 0
  psi(p(p(x)) -> q(q(x))) = 0
  psi(p(Q(Q(x))) -> Q(Q(p(x)))) = 0
  psi(Q(p(q(x))) -> q(p(Q(x)))) = 0
  psi(q(q(p(x))) -> p(q(q(x)))) = 0
  psi(q(Q(x)) -> x) = 0


Therefore, the confluence of R follows from that of C.

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

Consider the left-linear TRS R:

  Q(q(x)) -> x
  p(p(x)) -> q(q(x))
  p(Q(Q(x))) -> Q(Q(p(x)))
  Q(p(q(x))) -> q(p(Q(x)))
  q(q(p(x))) -> p(q(q(x)))
  q(Q(x)) -> x

Let C be the following subset of R:

  (empty)

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

  Q(q(q(p(x1_1)))) -> Q(p(q(q(x1_1))))
  Q(q(q(p(x1_1)))) -> q(p(x1_1))
  p(p(p(x1_1))) -> p(q(q(x1_1)))
  p(p(p(x1_1))) -> q(q(p(x1_1)))
  p(p(Q(Q(x1_1)))) -> p(Q(Q(p(x1_1))))
  p(p(Q(Q(x1_1)))) -> q(q(Q(Q(x1_1))))
  p(Q(Q(q(x2_1)))) -> p(Q(x2_1))
  p(Q(Q(q(x2_1)))) -> Q(Q(p(q(x2_1))))
  p(Q(Q(p(q(x2_1))))) -> p(Q(q(p(Q(x2_1)))))
  p(Q(Q(p(q(x2_1))))) -> Q(Q(p(p(q(x2_1)))))
  Q(p(q(q(p(x2_1))))) -> Q(p(p(q(q(x2_1)))))
  Q(p(q(q(p(x2_1))))) -> q(p(Q(q(p(x2_1)))))
  Q(p(q(Q(x2_1)))) -> Q(p(x2_1))
  Q(p(q(Q(x2_1)))) -> q(p(Q(Q(x2_1))))
  q(q(p(p(x2_1)))) -> q(q(q(q(x2_1))))
  q(q(p(p(x2_1)))) -> p(q(q(p(x2_1))))
  q(q(p(Q(Q(x2_1))))) -> q(q(Q(Q(p(x2_1)))))
  q(q(p(Q(Q(x2_1))))) -> p(q(q(Q(Q(x2_1)))))
  q(Q(p(q(x1_1)))) -> q(q(p(Q(x1_1))))
  q(Q(p(q(x1_1)))) -> p(q(x1_1))

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

# emptiness

The empty TRS is confluent.