YES

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

Consider the left-linear TRS R:

  +(x,+(y,z)) -> +(+(x,y),z)
  +(+(x,y),z) -> +(x,+(y,z))
  +(x,y) -> +(y,x)
  +(s(x),y) -> +(x,s(y))
  +(x,s(y)) -> +(s(x),y)
  *(x,s(y)) -> +(x,*(x,y))
  *(s(x),y) -> +(*(x,y),y)
  *(x,y) -> *(y,x)
  sq(x) -> *(x,x)
  sq(s(x)) -> +(*(x,x),s(+(x,x)))

Let C be the following subset of R:

  *(x,y) -> *(y,x)
  +(x,y) -> +(y,x)
  +(s(x),y) -> +(x,s(y))
  +(x,s(y)) -> +(s(x),y)
  +(+(x,y),z) -> +(x,+(y,z))
  +(x,+(y,z)) -> +(+(x,y),z)
  *(x,s(y)) -> +(x,*(x,y))
  *(s(x),y) -> +(*(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.

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

Consider the left-linear TRS R:

  *(x,y) -> *(y,x)
  +(x,y) -> +(y,x)
  +(s(x),y) -> +(x,s(y))
  +(x,s(y)) -> +(s(x),y)
  +(+(x,y),z) -> +(x,+(y,z))
  +(x,+(y,z)) -> +(+(x,y),z)
  *(x,s(y)) -> +(x,*(x,y))
  *(s(x),y) -> +(*(x,y),y)

Let C be the following subset of R:

  (empty)

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

  phi(*(x,y) -> *(y,x)) = 3
  phi(+(x,y) -> +(y,x)) = 1
  phi(+(s(x),y) -> +(x,s(y))) = 1
  phi(+(x,s(y)) -> +(s(x),y)) = 9
  phi(+(+(x,y),z) -> +(x,+(y,z))) = 3
  phi(+(x,+(y,z)) -> +(+(x,y),z)) = 4
  phi(*(x,s(y)) -> +(x,*(x,y))) = 10
  phi(*(s(x),y) -> +(*(x,y),y)) = 13

  psi(*(x,y) -> *(y,x)) = 11
  psi(+(x,y) -> +(y,x)) = 2
  psi(+(s(x),y) -> +(x,s(y))) = 7
  psi(+(x,s(y)) -> +(s(x),y)) = 8
  psi(+(+(x,y),z) -> +(x,+(y,z))) = 5
  psi(+(x,+(y,z)) -> +(+(x,y),z)) = 6
  psi(*(x,s(y)) -> +(x,*(x,y))) = 12
  psi(*(s(x),y) -> +(*(x,y),y)) = 12


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:

  (empty)

Let C be the following subset of R:

  (empty)

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.

# emptiness

The empty TRS is confluent.