Problem:
 f(g(x)) -> h(g(x),g(x))
 f(s(x)) -> h(s(x),s(x))
 g(x) -> s(x)

Proof:
 Church Rosser Transformation Processor (star):
  
  strict:
   f(0)(s(0)(x)) -> h(0)(s(0)(x))
   f(0)(s(0)(x)) -> h(1)(s(0)(x))
   f(0)(g(0)(x)) -> h(0)(g(0)(x))
   f(0)(g(0)(x)) -> h(1)(g(0)(x))
  weak:
   g(0)(x) -> s(0)(x)
  critical peaks: 1
  f(s(x)) <-2|0[]- f(g(x)) -0|[]-> h(g(x),g(x))
  Matrix Interpretation Processor: dim=1, lab=right
   
   interpretation:
    [h(1)](x0) = 2x0 + 1,
    
    [h(0)](x0) = 2x0,
    
    [f(0)](x0) = 2x0 + 2,
    
    [s(0)](x0) = x0,
    
    [g(0)](x0) = x0
   orientation:
    f(0)(s(0)(x)) = 2x + 2 >= 2x = h(0)(s(0)(x))
    
    f(0)(s(0)(x)) = 2x + 2 >= 2x + 1 = h(1)(s(0)(x))
    
    f(0)(g(0)(x)) = 2x + 2 >= 2x = h(0)(g(0)(x))
    
    f(0)(g(0)(x)) = 2x + 2 >= 2x + 1 = h(1)(g(0)(x))
    
    g(0)(x) = x >= x = s(0)(x)
   problem:
    
    strict:
     
    weak:
     g(0)(x) -> s(0)(x)
   Matrix Interpretation Processor: dim=1, lab=right
    
    interpretation:
     [s(0)](x0) = x0,
     
     [g(0)](x0) = x0 + 1
    orientation:
     g(0)(x) = x + 1 >= x = s(0)(x)
    problem:
     
     strict:
      
     weak:
      
    Shift Processor lab=left (dd):
     
     strict:
      f(g(x)) -> f(s(x))
      f(g(x)) -> h(g(x),g(x))
      f(s(x)) -> h(s(x),s(x))
      h(g(x),g(x)) -> h(g(x),s(x))
      h(g(x),s(x)) -> h(s(x),s(x))
      f(s(x)) -> h(s(x),s(x))
      h(g(x),g(x)) -> h(s(x),g(x))
      h(s(x),g(x)) -> h(s(x),s(x))
     weak:
      f(g(x)) -> h(g(x),g(x))
      f(s(x)) -> h(s(x),s(x))
      g(x) -> s(x)
     LPO Processor:
      precedence:
       f > s ~ h ~ g
      problem:
       
       strict:
        f(g(x)) -> f(s(x))
        h(g(x),g(x)) -> h(g(x),s(x))
        h(g(x),s(x)) -> h(s(x),s(x))
        h(g(x),g(x)) -> h(s(x),g(x))
        h(s(x),g(x)) -> h(s(x),s(x))
       weak:
        g(x) -> s(x)
      LPO Processor:
       precedence:
        g > s ~ h ~ f
       problem:
        
        strict:
         
        weak:
         
       Shift Processor (ldh) lab=left (force):
        
        strict:
         
        weak:
         
        Decreasing Processor:
         The following diagrams are decreasing:
         peak:
          f(s(x)) <-2|0[1,0,0]- f(g(x)) -0|[1,0,0]-> h(g(x),g(x))
         joins:
          f(s(x)) -1|[0,0,0]-> h(s(x),s(x))
          h(g(x),g(x)) -2|1[0,0,0]-> h(g(x),s(x)) -2|0[0,0,0]-> h(s(x),s(x))
         Qed