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

Proof:
 Church Rosser Transformation Processor (critical pair closing system, Thm 2.11):
  f(h(g(x80))) -> b()
  a() -> b()
  f(g(h(x81))) -> b()
  f(h(h(x82))) -> c()
  b() -> c()
  critical peaks: joinable
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [c] = 0,
    
    [b] = 4,
    
    [h](x0) = 4x0 + 4,
    
    [a] = 4,
    
    [f](x0) = x0,
    
    [g](x0) = 4x0
   orientation:
    f(h(g(x80))) = 16x80 + 4 >= 4 = b()
    
    a() = 4 >= 4 = b()
    
    f(g(h(x81))) = 16x81 + 16 >= 4 = b()
    
    f(h(h(x82))) = 16x82 + 20 >= 0 = c()
    
    b() = 4 >= 0 = c()
   problem:
    f(h(g(x80))) -> b()
    a() -> b()
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [b] = 0,
     
     [h](x0) = 4x0 + 1,
     
     [a] = 1,
     
     [f](x0) = 2x0 + 6,
     
     [g](x0) = 2x0
    orientation:
     f(h(g(x80))) = 16x80 + 8 >= 0 = b()
     
     a() = 1 >= 0 = b()
    problem:
     
    Qed