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

Proof:
 Church Rosser Transformation Processor (kb):
  f(g(g(x))) -> a()
  f(g(h(x))) -> b()
  f(h(g(x))) -> b()
  f(h(h(x))) -> c()
  g(x) -> h(x)
  a() -> b()
  b() -> c()
  critical peaks: joinable
  Matrix Interpretation Processor: dim=3
   
   interpretation:
          [0]
    [c] = [0]
          [0],
    
          [1]
    [b] = [0]
          [0],
    
              [1 1 0]     [0]
    [h](x0) = [0 0 0]x0 + [1]
              [0 0 0]     [0],
    
          [1]
    [a] = [1]
          [0],
    
              [1 0 0]  
    [f](x0) = [0 1 0]x0
              [0 0 0]  ,
    
              [1 1 0]     [0]
    [g](x0) = [0 0 0]x0 + [1]
              [0 0 0]     [0]
   orientation:
                 [1 1 0]    [1]    [1]      
    f(g(g(x))) = [0 0 0]x + [1] >= [1] = a()
                 [0 0 0]    [0]    [0]      
    
                 [1 1 0]    [1]    [1]      
    f(g(h(x))) = [0 0 0]x + [1] >= [0] = b()
                 [0 0 0]    [0]    [0]      
    
                 [1 1 0]    [1]    [1]      
    f(h(g(x))) = [0 0 0]x + [1] >= [0] = b()
                 [0 0 0]    [0]    [0]      
    
                 [1 1 0]    [1]    [0]      
    f(h(h(x))) = [0 0 0]x + [1] >= [0] = c()
                 [0 0 0]    [0]    [0]      
    
           [1 1 0]    [0]    [1 1 0]    [0]       
    g(x) = [0 0 0]x + [1] >= [0 0 0]x + [1] = h(x)
           [0 0 0]    [0]    [0 0 0]    [0]       
    
          [1]    [1]      
    a() = [1] >= [0] = b()
          [0]    [0]      
    
          [1]    [0]      
    b() = [0] >= [0] = c()
          [0]    [0]      
   problem:
    f(g(g(x))) -> a()
    f(g(h(x))) -> b()
    f(h(g(x))) -> b()
    g(x) -> h(x)
    a() -> b()
   Matrix Interpretation Processor: dim=3
    
    interpretation:
           [0]
     [b] = [0]
           [0],
     
               [1 0 0]  
     [h](x0) = [0 0 0]x0
               [0 0 0]  ,
     
           [1]
     [a] = [0]
           [0],
     
               [1 0 0]     [0]
     [f](x0) = [0 0 0]x0 + [1]
               [1 0 0]     [0],
     
               [1 1 0]     [0]
     [g](x0) = [0 0 0]x0 + [1]
               [0 0 0]     [0]
    orientation:
                  [1 1 0]    [1]    [1]      
     f(g(g(x))) = [0 0 0]x + [1] >= [0] = a()
                  [1 1 0]    [1]    [0]      
     
                  [1 0 0]    [0]    [0]      
     f(g(h(x))) = [0 0 0]x + [1] >= [0] = b()
                  [1 0 0]    [0]    [0]      
     
                  [1 1 0]    [0]    [0]      
     f(h(g(x))) = [0 0 0]x + [1] >= [0] = b()
                  [1 1 0]    [0]    [0]      
     
            [1 1 0]    [0]    [1 0 0]        
     g(x) = [0 0 0]x + [1] >= [0 0 0]x = h(x)
            [0 0 0]    [0]    [0 0 0]        
     
           [1]    [0]      
     a() = [0] >= [0] = b()
           [0]    [0]      
    problem:
     f(g(g(x))) -> a()
     f(g(h(x))) -> b()
     f(h(g(x))) -> b()
     g(x) -> h(x)
    Matrix Interpretation Processor: dim=3
     
     interpretation:
            [0]
      [b] = [0]
            [0],
      
                [1 1 0]  
      [h](x0) = [0 0 0]x0
                [0 0 0]  ,
      
            [0]
      [a] = [0]
            [0],
      
                [1 0 0]  
      [f](x0) = [0 0 0]x0
                [0 0 0]  ,
      
                [1 1 0]     [0]
      [g](x0) = [0 0 0]x0 + [1]
                [0 0 0]     [0]
     orientation:
                   [1 1 0]    [1]    [0]      
      f(g(g(x))) = [0 0 0]x + [0] >= [0] = a()
                   [0 0 0]    [0]    [0]      
      
                   [1 1 0]     [0]      
      f(g(h(x))) = [0 0 0]x >= [0] = b()
                   [0 0 0]     [0]      
      
                   [1 1 0]    [1]    [0]      
      f(h(g(x))) = [0 0 0]x + [0] >= [0] = b()
                   [0 0 0]    [0]    [0]      
      
             [1 1 0]    [0]    [1 1 0]        
      g(x) = [0 0 0]x + [1] >= [0 0 0]x = h(x)
             [0 0 0]    [0]    [0 0 0]        
     problem:
      f(g(h(x))) -> b()
      g(x) -> h(x)
     Matrix Interpretation Processor: dim=3
      
      interpretation:
             [0]
       [b] = [0]
             [0],
       
                 [1 0 0]  
       [h](x0) = [0 0 0]x0
                 [0 0 0]  ,
       
                 [1 0 0]  
       [f](x0) = [0 0 0]x0
                 [0 0 0]  ,
       
                 [1 0 0]     [1]
       [g](x0) = [0 0 0]x0 + [0]
                 [0 0 0]     [0]
      orientation:
                    [1 0 0]    [1]    [0]      
       f(g(h(x))) = [0 0 0]x + [0] >= [0] = b()
                    [0 0 0]    [0]    [0]      
       
              [1 0 0]    [1]    [1 0 0]        
       g(x) = [0 0 0]x + [0] >= [0 0 0]x = h(x)
              [0 0 0]    [0]    [0 0 0]        
      problem:
       
      Qed