Problem:
 f(a(),a(),b(),b()) -> f(c(),c(),c(),c())
 a() -> b()
 a() -> c()
 b() -> a()
 b() -> c()

Proof:
 AT confluence processor
  Complete TRS T' of input TRS:
  f(a(),a(),b(),b()) -> f(c(),c(),c(),c())
  a() -> c()
  b() -> c()
  a() -> b()
  b() -> a()
  
   T' = (P union S) with
  
   TRS P:a() -> b()
         b() -> a()
  
   TRS S:f(a(),a(),b(),b()) -> f(c(),c(),c(),c())
         a() -> c()
         b() -> c()
  
  S is linear and P is reversible.
  
   CP(S,S) = 
  f(c(),a(),b(),b()) = f(c(),c(),c(),c()), f(a(),c(),b(),b()) = f(c(),c(),c(),c()), 
  f(a(),a(),c(),b()) = f(c(),c(),c(),c()), f(a(),a(),b(),c()) = f(c(),c(),c(),c())
  
   CP(S,P union P^-1) = 
  c() = b(), c() = a()
  
   CP(P union P^-1,S) = 
  f(b(),a(),b(),b()) = f(c(),c(),c(),c()), f(a(),b(),b(),b()) = f(c(),c(),c(),c()), 
  b() = c(), f(a(),a(),a(),b()) = f(c(),c(),c(),c()), f(a(),a(),b(),a()) = 
  f(c(),c(),c(),c()), a() = c()
  
  
  We have to check termination of S:
  
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [c] = 0,
    
    [f](x0, x1, x2, x3) = x0 + 2x1 + 2x2 + 2x3,
    
    [b] = 0,
    
    [a] = 1
   orientation:
    f(a(),a(),b(),b()) = 3 >= 0 = f(c(),c(),c(),c())
    
    a() = 1 >= 0 = c()
    
    b() = 0 >= 0 = c()
   problem:
    b() -> c()
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [c] = 0,
     
     [b] = 1
    orientation:
     b() = 1 >= 0 = c()
    problem:
     
    Qed