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

Proof:
 Church Rosser Transformation Processor (to relative problem):
  strict:
   f(x,x) -> a()
   c() -> h(c(),g(c()))
   h(g(x),g(x)) -> f(x,h(x,g(c())))
  weak:
   
  original problem:
   f(x,x) -> a()
   c() -> h(c(),g(c()))
   h(g(x),g(x)) -> f(x,h(x,g(c())))
  critical peaks: 
   Matrix Interpretation Processor: dim=2
    
    interpretation:
                   [1 1]     [2 0]  
     [h](x0, x1) = [2 1]x0 + [0 0]x1,
     
               [1 1]     [0]
     [g](x0) = [0 0]x0 + [2],
     
           [0]
     [c] = [0],
     
           [0]
     [a] = [0],
     
                   [1 0]     [2 0]     [2]
     [f](x0, x1) = [2 0]x0 + [0 0]x1 + [0]
    orientation:
              [3 0]    [2]    [0]      
     f(x,x) = [2 0]x + [0] >= [0] = a()
     
           [0]    [0]                
     c() = [0] >= [0] = h(c(),g(c()))
     
                    [3 3]    [2]    [3 2]    [2]                   
     h(g(x),g(x)) = [2 2]x + [2] >= [2 0]x + [0] = f(x,h(x,g(c())))
    problem:
     strict:
      c() -> h(c(),g(c()))
      h(g(x),g(x)) -> f(x,h(x,g(c())))
     weak:
      
     original problem:
      f(x,x) -> a()
      c() -> h(c(),g(c()))
      h(g(x),g(x)) -> f(x,h(x,g(c())))
    Matrix Interpretation Processor: dim=2
     
     interpretation:
                    [1 2]     [2 0]  
      [h](x0, x1) = [1 0]x0 + [2 0]x1,
      
                [1 1]     [0]
      [g](x0) = [0 0]x0 + [1],
      
            [0]
      [c] = [0],
      
                    [1 1]     [1 0]     [1]
      [f](x0, x1) = [0 0]x0 + [1 1]x1 + [0]
     orientation:
            [0]    [0]                
      c() = [0] >= [0] = h(c(),g(c()))
      
                     [3 3]    [2]    [2 3]    [1]                   
      h(g(x),g(x)) = [3 3]x + [0] >= [2 2]x + [0] = f(x,h(x,g(c())))
     problem:
      strict:
       c() -> h(c(),g(c()))
      weak:
       
      original problem:
       f(x,x) -> a()
       c() -> h(c(),g(c()))
       h(g(x),g(x)) -> f(x,h(x,g(c())))
     KH confluence processor
      Split input TRS into two TRSs S and T:
      
      TRS S:
       c() -> h(c(),g(c()))
      
      TRS T:
       f(x,x) -> a()
       h(g(x),g(x)) -> f(x,h(x,g(c())))
      
      As established above, T/S is terminating.
      T is strongly non-overlapping on S and S is strongly non-overlapping on T
      
      Please install theorem prover 'Prover9' and 'Mace4' for handling more TRSs.
      
       All S-critical pairs are joinable.
      
      We have to check confluence of S.
      
      Church Rosser Transformation Processor (no redundant rules):
       strict:
        c() -> h(c(),g(c()))
       weak:
        
       critical peaks: 0
       Closedness Processor (*feeble*):
        
        Qed