YES

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

Consider the left-linear TRS R:

  F(H(x),y) -> F(H(x),I(I(y)))
  F(x,G(y)) -> F(I(x),G(y))
  I(x) -> x

Let C be the following subset of R:

  I(x) -> x
  F(H(x),y) -> F(H(x),I(I(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:

  I(x) -> x
  F(H(x),y) -> F(H(x),I(I(y)))

Let C be the following subset of R:

  I(x) -> 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:

  I(x) -> 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.

# emptiness

The empty TRS is confluent.