YES

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

Consider the left-linear TRS R:

  H(F(x,y)) -> F(H(R(x)),y)
  F(x,K(y,z)) -> G(P(y),Q(z,x))
  H(Q(x,y)) -> Q(x,H(R(y)))
  Q(x,H(R(y))) -> H(Q(x,y))
  H(G(x,y)) -> G(x,H(y))

Let C be the following subset of R:

  F(x,K(y,z)) -> G(P(y),Q(z,x))
  Q(x,H(R(y))) -> H(Q(x,y))
  H(G(x,y)) -> G(x,H(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:

  F(x,K(y,z)) -> G(P(y),Q(z,x))
  Q(x,H(R(y))) -> H(Q(x,y))
  H(G(x,y)) -> G(x,H(y))

Let C be the following subset of R:

  Q(x,H(R(y))) -> H(Q(x,y))
  H(G(x,y)) -> G(x,H(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:

  Q(x,H(R(y))) -> H(Q(x,y))
  H(G(x,y)) -> G(x,H(y))

Let C be the following subset of R:

  Q(x,H(R(y))) -> H(Q(x,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:

  Q(x,H(R(y))) -> H(Q(x,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.