YES

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

Consider the left-linear TRS R:

  E(+(x,y)) -> *(E(x),E(y))
  E(0()) -> 1()
  +(x,0()) -> x
  +(0(),x) -> x
  *(x,1()) -> x
  *(1(),x) -> x
  +(+(x,y),z) -> +(x,+(y,z))
  +(x,+(y,z)) -> +(+(x,y),z)
  +(x,y) -> +(y,x)
  *(*(x,y),z) -> *(x,*(y,z))
  *(x,*(y,z)) -> *(*(x,y),z)
  *(x,y) -> *(y,x)

Let C be the following subset of R:

  (empty)

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

  phi(E(+(x,y)) -> *(E(x),E(y))) = 13
  phi(E(0()) -> 1()) = 3
  phi(+(x,0()) -> x) = 17
  phi(+(0(),x) -> x) = 16
  phi(*(x,1()) -> x) = 6
  phi(*(1(),x) -> x) = 5
  phi(+(+(x,y),z) -> +(x,+(y,z))) = 13
  phi(+(x,+(y,z)) -> +(+(x,y),z)) = 14
  phi(+(x,y) -> +(y,x)) = 13
  phi(*(*(x,y),z) -> *(x,*(y,z))) = 11
  phi(*(x,*(y,z)) -> *(*(x,y),z)) = 11
  phi(*(x,y) -> *(y,x)) = 12

  psi(E(+(x,y)) -> *(E(x),E(y))) = 7
  psi(E(0()) -> 1()) = 1
  psi(+(x,0()) -> x) = 12
  psi(+(0(),x) -> x) = 11
  psi(*(x,1()) -> x) = 9
  psi(*(1(),x) -> x) = 10
  psi(+(+(x,y),z) -> +(x,+(y,z))) = 15
  psi(+(x,+(y,z)) -> +(+(x,y),z)) = 18
  psi(+(x,y) -> +(y,x)) = 15
  psi(*(*(x,y),z) -> *(x,*(y,z))) = 9
  psi(*(x,*(y,z)) -> *(*(x,y),z)) = 8
  psi(*(x,y) -> *(y,x)) = 8


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.