YES

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

Consider the left-linear TRS R:

  F(x,y) -> c(y)
  G(x) -> x
  f(x) -> g(x)
  g(x) -> c(x)

Let C be the following subset of R:

  F(x,y) -> c(y)
  G(x) -> x
  g(x) -> c(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:

  F(x,y) -> c(y)
  G(x) -> x
  g(x) -> c(x)

Let C be the following subset of R:

  F(x,y) -> c(y)
  G(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:

  F(x,y) -> c(y)
  G(x) -> x

Let C be the following subset of R:

  F(x,y) -> c(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:

  F(x,y) -> c(y)

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.