YES

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

Proof:
 Church Rosser Transformation Processor:
  strict:
   
  weak:
   
  critical peaks: 1
   f(g(g(a()))) <-1|0[]- f(g(a())) -0|[]-> f(h(a(),a()))
  Shift Processor lab=left (dd) (force):
   strict:
    f(g(a())) -> f(g(g(a())))
    f(g(a())) -> f(h(a(),a()))
    f(h(a(),a())) -> f(g(g(a())))
   weak:
    f(g(x)) -> f(h(x,x))
    g(a()) -> g(g(a()))
    h(a(),a()) -> g(g(a()))
   Matrix Interpretation Processor: dim=3
    
    interpretation:
               [1 0 1]  
     [f](x0) = [0 0 0]x0
               [0 1 0]  ,
     
           [0]
     [a] = [1]
           [0],
     
               [1 0 0]  
     [g](x0) = [1 0 0]x0
               [1 1 0]  ,
     
                   [1 0 0]     [1 0 0]  
     [h](x0, x1) = [0 0 0]x0 + [0 0 0]x1
                   [0 0 0]     [0 0 0]  
    orientation:
                 [1]    [0]               
     f(g(a())) = [0] >= [0] = f(g(g(a())))
                 [0]    [0]               
     
                 [1]    [0]                
     f(g(a())) = [0] >= [0] = f(h(a(),a()))
                 [0]    [0]                
     
                     [0]    [0]               
     f(h(a(),a())) = [0] >= [0] = f(g(g(a())))
                     [0]    [0]               
     
               [2 1 0]     [2 0 0]             
     f(g(x)) = [0 0 0]x >= [0 0 0]x = f(h(x,x))
               [1 0 0]     [0 0 0]             
     
              [0]    [0]            
     g(a()) = [0] >= [0] = g(g(a()))
              [1]    [0]            
     
                  [0]    [0]            
     h(a(),a()) = [0] >= [0] = g(g(a()))
                  [0]    [0]            
    problem:
     strict:
      f(h(a(),a())) -> f(g(g(a())))
     weak:
      f(g(x)) -> f(h(x,x))
      g(a()) -> g(g(a()))
      h(a(),a()) -> g(g(a()))
    Matrix Interpretation Processor: dim=3
     
     interpretation:
                [1 0 1]  
      [f](x0) = [0 0 0]x0
                [0 0 1]  ,
      
            [0]
      [a] = [1]
            [0],
      
                [1 0 0]  
      [g](x0) = [0 0 0]x0
                [1 1 0]  ,
      
                    [1 0 0]     [1 0 0]  
      [h](x0, x1) = [0 0 1]x0 + [0 0 0]x1
                    [0 0 0]     [0 1 0]  
     orientation:
                      [1]    [0]               
      f(h(a(),a())) = [0] >= [0] = f(g(g(a())))
                      [1]    [0]               
      
                [2 1 0]     [2 1 0]             
      f(g(x)) = [0 0 0]x >= [0 0 0]x = f(h(x,x))
                [1 1 0]     [0 1 0]             
      
               [0]    [0]            
      g(a()) = [0] >= [0] = g(g(a()))
               [1]    [0]            
      
                   [0]    [0]            
      h(a(),a()) = [0] >= [0] = g(g(a()))
                   [1]    [0]            
     problem:
      strict:
       
      weak:
       f(g(x)) -> f(h(x,x))
       g(a()) -> g(g(a()))
       h(a(),a()) -> g(g(a()))
     Shift Processor (ldh) lab=left (force):
      strict:
       
      weak:
       
      Rule Labeling Processor:
       strict:
        
       weak:
        
       rule labeling (right):
        f(g(x)) -> f(h(x,x)): 0
        g(a()) -> g(g(a())): 0
        h(a(),a()) -> g(g(a())): 0
       Rule Labeling Processor:
        strict:
         
        weak:
         
        rule labeling (left):
         f(g(x)) -> f(h(x,x)): 2
         g(a()) -> g(g(a())): 0
         h(a(),a()) -> g(g(a())): 0
        Decreasing Processor:
         The following diagrams are decreasing:
         peak:
          f(g(g(a()))) <-1|0[=>0,==1,==0]- f(g(a())) -0|[?=2,==1,==0]-> f(h(a(),a()))
         joins (1):
          
          f(h(a(),a())) -2|0[=>0,>>0,==0]-> f(g(g(a())))
         Qed