Problem:
 f(x,x) -> a()
 c() -> h(c(),g(c()))
 h(g3(x),g1(g2(x))) -> f(x,h(x,g(x)))

Proof:
 Church Rosser Transformation Processor (to relative problem):
  strict:
   f(x,x) -> a()
   c() -> h(c(),g(c()))
   h(g3(x),g1(g2(x))) -> f(x,h(x,g(x)))
  weak:
   
  original problem:
   f(x,x) -> a()
   c() -> h(c(),g(c()))
   h(g3(x),g1(g2(x))) -> f(x,h(x,g(x)))
  critical peaks: 
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [g1](x0) = 2x0 + 6,
     
     [g2](x0) = 4x0 + 1,
     
     [g3](x0) = 2x0 + 3,
     
     [h](x0, x1) = 2x0 + x1,
     
     [g](x0) = 4x0,
     
     [c] = 0,
     
     [a] = 0,
     
     [f](x0, x1) = 2x0 + x1
    orientation:
     f(x,x) = 3x >= 0 = a()
     
     c() = 0 >= 0 = h(c(),g(c()))
     
     h(g3(x),g1(g2(x))) = 12x + 14 >= 8x = f(x,h(x,g(x)))
    problem:
     strict:
      f(x,x) -> a()
      c() -> h(c(),g(c()))
     weak:
      
     original problem:
      f(x,x) -> a()
      c() -> h(c(),g(c()))
      h(g3(x),g1(g2(x))) -> f(x,h(x,g(x)))
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [h](x0, x1) = x0 + x1,
      
      [g](x0) = 2x0,
      
      [c] = 0,
      
      [a] = 0,
      
      [f](x0, x1) = 6x0 + x1 + 2
     orientation:
      f(x,x) = 7x + 2 >= 0 = a()
      
      c() = 0 >= 0 = h(c(),g(c()))
     problem:
      strict:
       c() -> h(c(),g(c()))
      weak:
       
      original problem:
       f(x,x) -> a()
       c() -> h(c(),g(c()))
       h(g3(x),g1(g2(x))) -> f(x,h(x,g(x)))
     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(g3(x),g1(g2(x))) -> f(x,h(x,g(x)))
      
      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