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

Proof:
 Church Rosser Transformation Processor (star):
  
  strict:
   
  weak:
   g(0)(x) -> h(0)(g(0)(x))
   h(0)(f(0)(x)) -> h(0)(g(0)(x))
   f(0)(x) -> h(0)(f(0)(x))
   f(0)(x) -> g(0)(x)
  critical peaks: 4
  g(x) <-0|[]- f(x) -1|[]-> h(f(x))
  h(g(x)) <-0|0[]- h(f(x)) -2|[]-> h(g(x))
  h(f(x)) <-1|[]- f(x) -0|[]-> g(x)
  h(h(f(x))) <-1|0[]- h(f(x)) -2|[]-> h(g(x))
  Matrix Interpretation Processor: dim=1, lab=right
   
   interpretation:
    [f(0)](x0) = 3x0 + 2,
    
    [h(0)](x0) = x0,
    
    [g(0)](x0) = x0
   orientation:
    g(0)(x) = x >= x = h(0)(g(0)(x))
    
    h(0)(f(0)(x)) = 3x + 2 >= x = h(0)(g(0)(x))
    
    f(0)(x) = 3x + 2 >= 3x + 2 = h(0)(f(0)(x))
    
    f(0)(x) = 3x + 2 >= x = g(0)(x)
   problem:
    
    strict:
     
    weak:
     g(0)(x) -> h(0)(g(0)(x))
     f(0)(x) -> h(0)(f(0)(x))
   Shift Processor lab=left (dd):
    
    strict:
     f(x) -> g(x)
     f(x) -> h(f(x))
     g(x) -> h(g(x))
     h(f(x)) -> h(g(x))
     h(f(x)) -> h(g(x))
     h(f(x)) -> h(g(x))
     f(x) -> h(f(x))
     f(x) -> g(x)
     h(f(x)) -> h(g(x))
     g(x) -> h(g(x))
     h(f(x)) -> h(h(f(x)))
     h(f(x)) -> h(g(x))
     h(h(f(x))) -> h(h(g(x)))
     h(g(x)) -> h(h(g(x)))
    weak:
     f(x) -> g(x)
     f(x) -> h(f(x))
     h(f(x)) -> h(g(x))
     g(x) -> h(g(x))
    Polynomial Interpretation Processor:
     dimension: 1
     interpretation:
      [h](x0) = x0,
      
      [g](x0) = x0x0,
      
      [f](x0) = x0x0 + 1
     orientation:
      f(x) = x*x + 1 >= x*x = g(x)
      
      f(x) = x*x + 1 >= x*x + 1 = h(f(x))
      
      g(x) = x*x >= x*x = h(g(x))
      
      h(f(x)) = x*x + 1 >= x*x = h(g(x))
      
      h(f(x)) = x*x + 1 >= x*x = h(g(x))
      
      h(f(x)) = x*x + 1 >= x*x = h(g(x))
      
      f(x) = x*x + 1 >= x*x + 1 = h(f(x))
      
      f(x) = x*x + 1 >= x*x = g(x)
      
      h(f(x)) = x*x + 1 >= x*x = h(g(x))
      
      g(x) = x*x >= x*x = h(g(x))
      
      h(f(x)) = x*x + 1 >= x*x + 1 = h(h(f(x)))
      
      h(f(x)) = x*x + 1 >= x*x = h(g(x))
      
      h(h(f(x))) = x*x + 1 >= x*x = h(h(g(x)))
      
      h(g(x)) = x*x >= x*x = h(h(g(x)))
     problem:
      
      strict:
       f(x) -> h(f(x))
       g(x) -> h(g(x))
       f(x) -> h(f(x))
       g(x) -> h(g(x))
       h(f(x)) -> h(h(f(x)))
       h(g(x)) -> h(h(g(x)))
      weak:
       f(x) -> h(f(x))
       g(x) -> h(g(x))
     Shift Processor (ldh) lab=left (force):
      
      strict:
       
      weak:
       
      Decreasing Processor:
       The following diagrams are decreasing:
       peak:
        g(x) <-0|[1,0]- f(x) -1|[1,0]-> h(f(x))
       joins:
        g(x) -3|[0,0]-> h(g(x))
        h(f(x)) -2|[1,0]-> h(g(x))
       peak:
        h(g(x)) <-0|0[1,0]- h(f(x)) -2|[1,0]-> h(g(x))
       joins:
        
        
       peak:
        h(f(x)) <-1|[1,0]- f(x) -0|[1,0]-> g(x)
       joins:
        h(f(x)) -2|[1,0]-> h(g(x))
        g(x) -3|[0,0]-> h(g(x))
       peak:
        h(h(f(x))) <-1|0[1,0]- h(f(x)) -2|[1,0]-> h(g(x))
       joins:
        h(h(f(x))) -2|0[1,0]-> h(h(g(x)))
        h(g(x)) -3|0[0,0]-> h(h(g(x)))
       Qed