YES # parallel critical pair closing system (Shintani and Hirokawa 2022, Section 8 in LMCS 2023) Consider the left-linear TRS R: f(g(x)) -> h(x,x) g(a()) -> b() f(x) -> h(x,x) b() -> a() h(x,y) -> h(g(x),g(y)) g(x) -> x a() -> b() Let C be the following subset of R: f(x) -> h(x,x) g(x) -> x h(x,y) -> h(g(x),g(y)) g(a()) -> b() a() -> b() b() -> a() 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) -> h(x,x) g(x) -> x h(x,y) -> h(g(x),g(y)) g(a()) -> b() a() -> b() b() -> a() Let C be the following subset of R: f(x) -> h(x,x) g(x) -> x h(x,y) -> h(g(x),g(y)) a() -> b() b() -> a() 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) -> h(x,x) g(x) -> x h(x,y) -> h(g(x),g(y)) a() -> b() b() -> a() Let C be the following subset of R: f(x) -> h(x,x) g(x) -> x h(x,y) -> h(g(x),g(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) -> h(x,x) g(x) -> x h(x,y) -> h(g(x),g(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.