/export/starexec/sandbox/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


YES

Problem:
 l(o(x)) -> o(l(x))
 a(l(x)) -> l(a(a(x)))
 o(x) -> a(l(x))
 C(0(),x) -> o(x)
 a(C(C(x,y),z)) -> a(f(C(x,y),z))
 a(f(0(),x)) -> x
 a(f(C(x,y),z)) -> h(a(f(x,y)),a(a(f(f(x,y),z))))
 a(h(x,y)) -> h(a(x),a(a(C(x,y))))
 h(x,y) -> o(y)
 a(f(x,y)) -> f(a(x),y)
 a(C(x,y)) -> C(a(x),a(y))
 a(x) -> x
 o(x) -> x

Proof:
 Embedding Processor:
  strict:
   l(o(x)) -> o(l(x))
   a(l(x)) -> l(a(a(x)))
   o(x) -> a(l(x))
   C(0(),x) -> o(x)
   a(C(C(x,y),z)) -> a(f(C(x,y),z))
   a(f(0(),x)) -> x
   a(f(C(x,y),z)) -> h(a(f(x,y)),a(a(f(f(x,y),z))))
   a(h(x,y)) -> h(a(x),a(a(C(x,y))))
   h(x,y) -> o(y)
   a(f(x,y)) -> f(a(x),y)
   a(C(x,y)) -> C(a(x),a(y))
   a(x) -> x
   o(x) -> x
  weak:
   l(x0) -> x0
   o(x0) -> x0
   a(x0) -> x0
   C(x0,x1) -> x0
   C(x0,x1) -> x1
   f(x0,x1) -> x0
   f(x0,x1) -> x1
   h(x0,x1) -> x0
   h(x0,x1) -> x1
  Higher Ordinal Interpretation Processor:
   degree: 2
   reverse arguments: false
   interpretation:
    l(x5) =  (+) x5
    o(x6) =  (+) x6 (+) 1
    a(x7) =  (+) x7
    C(x8,x9) = omega^( (+) x8) (+) x9 (+) 1
    0() =  (+) 1
    f(x10,x11) = omega^( (+) x10) (+) x11
    h(x12,x13) = x13 + omega^( (+) x12 (+) 1)
   problem:
    strict:
     l(o(x)) -> o(l(x))
     a(l(x)) -> l(a(a(x)))
     a(f(C(x,y),z)) -> h(a(f(x,y)),a(a(f(f(x,y),z))))
     a(h(x,y)) -> h(a(x),a(a(C(x,y))))
     a(f(x,y)) -> f(a(x),y)
     a(C(x,y)) -> C(a(x),a(y))
     a(x) -> x
    weak:
     l(x0) -> x0
     a(x0) -> x0
     f7(x12) -> h(x12,x13)
   Matrix Interpretation Processor: dim=2
    
    interpretation:
                [1 0]     [2]
     [f7](x0) = [1 2]x0 + [2],
     
                   [0]
     [h](x0, x1) = [0],
     
                   [0 1]     [0 0]     [2]
     [f](x0, x1) = [0 1]x0 + [0 2]x1 + [1],
     
                   [0 0]     [0 2]     [3]
     [C](x0, x1) = [1 0]x0 + [1 0]x1 + [1],
     
                 
     [a](x0) = x0,
     
               [1 1]  
     [l](x0) = [0 2]x0,
     
               [0 0]  
     [o](x0) = [0 1]x0
    orientation:
               [0 1]     [0 0]           
     l(o(x)) = [0 2]x >= [0 2]x = o(l(x))
     
               [1 1]     [1 1]              
     a(l(x)) = [0 2]x >= [0 2]x = l(a(a(x)))
     
                      [1 0]    [1 0]    [0 0]    [3]    [0]                                 
     a(f(C(x,y),z)) = [1 0]x + [1 0]y + [0 2]z + [2] >= [0] = h(a(f(x,y)),a(a(f(f(x,y),z))))
     
                 [0]    [0]                       
     a(h(x,y)) = [0] >= [0] = h(a(x),a(a(C(x,y))))
     
                 [0 1]    [0 0]    [2]    [0 1]    [0 0]    [2]            
     a(f(x,y)) = [0 1]x + [0 2]y + [1] >= [0 1]x + [0 2]y + [1] = f(a(x),y)
     
                 [0 0]    [0 2]    [3]    [0 0]    [0 2]    [3]               
     a(C(x,y)) = [1 0]x + [1 0]y + [1] >= [1 0]x + [1 0]y + [1] = C(a(x),a(y))
     
                      
     a(x) = x >= x = x
     
             [1 1]             
     l(x0) = [0 2]x0 >= x0 = x0
     
                          
     a(x0) = x0 >= x0 = x0
     
               [1 0]      [2]    [0]             
     f7(x12) = [1 2]x12 + [2] >= [0] = h(x12,x13)
    problem:
     strict:
      l(o(x)) -> o(l(x))
      a(l(x)) -> l(a(a(x)))
      a(h(x,y)) -> h(a(x),a(a(C(x,y))))
      a(f(x,y)) -> f(a(x),y)
      a(C(x,y)) -> C(a(x),a(y))
      a(x) -> x
     weak:
      l(x0) -> x0
      a(x0) -> x0
      f7(x12) -> h(x12,x13)
      f8() -> o(x14)
      f9() -> f(x15,x16)
      f10() -> C(x17,x18)
      f11() -> h(x19,x20)
    Matrix Interpretation Processor: dim=2
     
     interpretation:
              [0]
      [f11] = [3],
      
              [0]
      [f10] = [0],
      
             [0]
      [f9] = [0],
      
             [0]
      [f8] = [0],
      
                 [2 0]     [0]
      [f7](x0) = [0 0]x0 + [3],
      
                    [0]
      [h](x0, x1) = [3],
      
                    [0]
      [f](x0, x1) = [0],
      
                    [0]
      [C](x0, x1) = [0],
      
                [1 1]  
      [a](x0) = [0 1]x0,
      
                [1 0]     [0]
      [l](x0) = [0 2]x0 + [2],
      
                [0]
      [o](x0) = [0]
     orientation:
                [0]    [0]          
      l(o(x)) = [2] >= [0] = o(l(x))
      
                [1 2]    [2]    [1 2]    [0]             
      a(l(x)) = [0 2]x + [2] >= [0 2]x + [2] = l(a(a(x)))
      
                  [3]    [0]                       
      a(h(x,y)) = [3] >= [3] = h(a(x),a(a(C(x,y))))
      
                  [0]    [0]            
      a(f(x,y)) = [0] >= [0] = f(a(x),y)
      
                  [0]    [0]               
      a(C(x,y)) = [0] >= [0] = C(a(x),a(y))
      
             [1 1]          
      a(x) = [0 1]x >= x = x
      
              [1 0]     [0]           
      l(x0) = [0 2]x0 + [2] >= x0 = x0
      
              [1 1]             
      a(x0) = [0 1]x0 >= x0 = x0
      
                [2 0]      [0]    [0]             
      f7(x12) = [0 0]x12 + [3] >= [3] = h(x12,x13)
      
             [0]    [0]         
      f8() = [0] >= [0] = o(x14)
      
             [0]    [0]             
      f9() = [0] >= [0] = f(x15,x16)
      
              [0]    [0]             
      f10() = [0] >= [0] = C(x17,x18)
      
              [0]    [0]             
      f11() = [3] >= [3] = h(x19,x20)
     problem:
      strict:
       l(o(x)) -> o(l(x))
       a(f(x,y)) -> f(a(x),y)
       a(C(x,y)) -> C(a(x),a(y))
       a(x) -> x
      weak:
       l(x0) -> x0
       a(x0) -> x0
       f7(x12) -> h(x12,x13)
       f8() -> o(x14)
       f9() -> f(x15,x16)
       f10() -> C(x17,x18)
       f11() -> h(x19,x20)
       f12() -> o(x21)
       f13() -> h(x22,x23)
       f14() -> C(x24,x25)
       f15() -> f(x26,x27)
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [f15] = 0,
       
       [f14] = 0,
       
       [f13] = 0,
       
       [f12] = 0,
       
       [f11] = 0,
       
       [f10] = 0,
       
       [f9] = 0,
       
       [f8] = 0,
       
       [f7](x0) = 2x0,
       
       [h](x0, x1) = 0,
       
       [f](x0, x1) = 0,
       
       [C](x0, x1) = 0,
       
       [a](x0) = 2x0 + 3,
       
       [l](x0) = 2x0 + 3,
       
       [o](x0) = 0
      orientation:
       l(o(x)) = 3 >= 0 = o(l(x))
       
       a(f(x,y)) = 3 >= 0 = f(a(x),y)
       
       a(C(x,y)) = 3 >= 0 = C(a(x),a(y))
       
       a(x) = 2x + 3 >= x = x
       
       l(x0) = 2x0 + 3 >= x0 = x0
       
       a(x0) = 2x0 + 3 >= x0 = x0
       
       f7(x12) = 2x12 >= 0 = h(x12,x13)
       
       f8() = 0 >= 0 = o(x14)
       
       f9() = 0 >= 0 = f(x15,x16)
       
       f10() = 0 >= 0 = C(x17,x18)
       
       f11() = 0 >= 0 = h(x19,x20)
       
       f12() = 0 >= 0 = o(x21)
       
       f13() = 0 >= 0 = h(x22,x23)
       
       f14() = 0 >= 0 = C(x24,x25)
       
       f15() = 0 >= 0 = f(x26,x27)
      problem:
       strict:
        
       weak:
        l(x0) -> x0
        a(x0) -> x0
        f7(x12) -> h(x12,x13)
        f8() -> o(x14)
        f9() -> f(x15,x16)
        f10() -> C(x17,x18)
        f11() -> h(x19,x20)
        f12() -> o(x21)
        f13() -> h(x22,x23)
        f14() -> C(x24,x25)
        f15() -> f(x26,x27)
        f16() -> o(x28)
        f17() -> f(x29,x30)
        f18() -> C(x31,x32)
        f19() -> h(x33,x34)
      Qed