YES

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

Consider the left-linear TRS R:

  or(x,T()) -> T()
  or(x,F()) -> x
  or(T(),x) -> T()
  or(F(),x) -> x
  or(or(x,y),z) -> or(x,or(y,z))
  or(x,y) -> or(y,x)

Let C be the following subset of R:

  or(x,T()) -> T()
  or(x,F()) -> x
  or(T(),x) -> T()
  or(F(),x) -> x
  or(or(x,y),z) -> or(x,or(y,z))
  or(x,y) -> or(y,x)

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.

# Parallel rule labeling (Zankl et al. 2015).

Consider the left-linear TRS R:

  or(x,T()) -> T()
  or(x,F()) -> x
  or(T(),x) -> T()
  or(F(),x) -> x
  or(or(x,y),z) -> or(x,or(y,z))
  or(x,y) -> or(y,x)

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

  phi(or(x,T()) -> T()) = 5
  phi(or(x,F()) -> x) = 7
  phi(or(T(),x) -> T()) = 6
  phi(or(F(),x) -> x) = 6
  phi(or(or(x,y),z) -> or(x,or(y,z))) = 10
  phi(or(x,y) -> or(y,x)) = 10

  psi(or(x,T()) -> T()) = 7
  psi(or(x,F()) -> x) = 7
  psi(or(T(),x) -> T()) = 11
  psi(or(F(),x) -> x) = 9
  psi(or(or(x,y),z) -> or(x,or(y,z))) = 8
  psi(or(x,y) -> or(y,x)) = 8