YES

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

Consider the left-linear TRS R:

  0(1(2(3(4(x))))) -> 0(2(1(3(4(x)))))
  0(5(1(2(4(3(x)))))) -> 0(5(2(1(4(3(x))))))
  0(5(2(4(1(3(x)))))) -> 0(1(5(2(4(3(x))))))
  0(5(3(1(2(4(x)))))) -> 0(1(5(3(2(4(x))))))
  0(5(4(1(3(2(x)))))) -> 0(5(4(3(1(2(x))))))

Let C be the following subset of R:

  0(1(2(3(4(x))))) -> 0(2(1(3(4(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:

  0(1(2(3(4(x))))) -> 0(2(1(3(4(x)))))

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.

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

Consider the left-linear TRS R:

  (empty)

Let C be the following subset of R:

  (empty)

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




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.