YES

Proof:
This system is confluent.
By \cite{ALS94}, Theorem 4.1.
This system is of type 3 or smaller.
This system is strongly deterministic.
All 2 critical pairs are joinable.
x = y <= g(y) = s(x), g(g(y)) = x, g(x) = y, g(g(x)) = s(y):
This critical pair is unfeasible.
x = y <= g(y) = x, g(g(y)) = s(x), g(x) = s(y), g(g(x)) = y:
This critical pair is unfeasible.
By \cite{A14}, Theorem 11.5.9.
This system is of type 3 or smaller.
This system is deterministic.
System R transformed to V(R) + Emb.
Call external tool:
ttt2 - trs 30
Input:
  f(x, y) -> x
  f(x, y) -> g(g(x))
  f(x, y) -> g(x)
  f(x, y) -> y
  f(x, y) -> g(g(y))
  f(x, y) -> g(y)
  g(x) -> x
  f(x, y) -> x
  f(x, y) -> y

 DP Processor:
  DPs:
   f#(x,y) -> g#(x)
   f#(x,y) -> g#(g(x))
   f#(x,y) -> g#(y)
   f#(x,y) -> g#(g(y))
  TRS:
   f(x,y) -> x
   f(x,y) -> g(g(x))
   f(x,y) -> g(x)
   f(x,y) -> y
   f(x,y) -> g(g(y))
   f(x,y) -> g(y)
   g(x) -> x
  TDG Processor:
   DPs:
    f#(x,y) -> g#(x)
    f#(x,y) -> g#(g(x))
    f#(x,y) -> g#(y)
    f#(x,y) -> g#(g(y))
   TRS:
    f(x,y) -> x
    f(x,y) -> g(g(x))
    f(x,y) -> g(x)
    f(x,y) -> y
    f(x,y) -> g(g(y))
    f(x,y) -> g(y)
    g(x) -> x
   graph:
    
   Qed