YES

Problem:
 g(a()) -> f(g(a()))
 g(b()) -> c()
 a() -> b()
 f(x) -> h(x,x)
 h(x,y) -> c()

Proof:
 Church Rosser Transformation Processor:
  strict:
   
  weak:
   
  critical peaks: 1
   g(b()) <-2|0[]- g(a()) -0|[]-> f(g(a()))
  Shift Processor lab=left (dd) (force):
   strict:
    g(a()) -> g(b())
    g(a()) -> f(g(a()))
    g(b()) -> c()
    f(g(a())) -> h(g(a()),g(a()))
    h(g(a()),g(a())) -> c()
   weak:
    g(a()) -> f(g(a()))
    g(b()) -> c()
    a() -> b()
    f(x) -> h(x,x)
    h(x,y) -> c()
   Shift Processor (ldh) lab=left (force):
    strict:
     
    weak:
     
    Rule Labeling Processor:
     strict:
      
     weak:
      
     rule labeling (right):
      g(a()) -> f(g(a())): 1
      g(b()) -> c(): 0
      a() -> b(): 1
      f(x) -> h(x,x): 0
      h(x,y) -> c(): 0
     Shift Processor lab=left (par):
      strict:
       
      weak:
       
      Decreasing Processor (par):
       The following diagrams are decreasing:
       peak:
        g(b()) <-2|0[==1,==1]- g(a()) -0|[==1,==1]-> f(g(a()))
       joins (1):
        g(b()) -1|[==1,>>0]-> c()
        f(g(a())) -3|[==1,>>0]-> h(g(a()),g(a())) -4|[==1,>>0]-> c()
       Qed