NO

Problem:
 nats() -> adx(zeros())
 zeros() -> cons(n__0(),n__zeros())
 incr(cons(X,Y)) -> cons(n__s(activate(X)),n__incr(activate(Y)))
 adx(cons(X,Y)) -> incr(cons(activate(X),n__adx(activate(Y))))
 hd(cons(X,Y)) -> activate(X)
 tl(cons(X,Y)) -> activate(Y)
 0() -> n__0()
 zeros() -> n__zeros()
 s(X) -> n__s(X)
 incr(X) -> n__incr(X)
 adx(X) -> n__adx(X)
 activate(n__0()) -> 0()
 activate(n__zeros()) -> zeros()
 activate(n__s(X)) -> s(X)
 activate(n__incr(X)) -> incr(activate(X))
 activate(n__adx(X)) -> adx(activate(X))
 activate(X) -> X

Proof:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
                [0]
   [n__zeros] = [0]
                [1],
   
                [1 0 0]  
   [n__s](x0) = [0 0 0]x0
                [0 0 0]  ,
   
             [0]
   [zeros] = [1]
             [1],
   
            [0]
   [n__0] = [0]
            [1],
   
                    [1 0 0]     [1 1 0]  
   [cons](x0, x1) = [0 0 0]x0 + [0 0 1]x1
                    [0 1 1]     [0 0 0]  ,
   
                  [1 0 0]  
   [n__adx](x0) = [0 0 0]x0
                  [0 0 0]  ,
   
            [0]
   [nats] = [0]
            [0],
   
                         [0]
   [activate](x0) = x0 + [1]
                         [0],
   
             [1 0 0]  
   [s](x0) = [0 0 0]x0
             [0 0 0]  ,
   
              [1 0 0]     [0]
   [tl](x0) = [1 0 1]x0 + [1]
              [1 1 0]     [0],
   
              [1 0 0]     [1]
   [hd](x0) = [0 0 1]x0 + [1]
              [0 1 1]     [0],
   
                   [1 0 0]  
   [n__incr](x0) = [0 0 0]x0
                   [0 0 0]  ,
   
         [0]
   [0] = [1]
         [1],
   
                [1 0 0]  
   [incr](x0) = [0 0 0]x0
                [0 0 0]  ,
   
               [1 0 0]  
   [adx](x0) = [0 0 0]x0
               [0 0 0]  
  orientation:
            [0]    [0]               
   nats() = [0] >= [0] = adx(zeros())
            [0]    [0]               
   
             [0]    [0]                          
   zeros() = [1] >= [1] = cons(n__0(),n__zeros())
             [1]    [1]                          
   
                     [1 0 0]    [1 1 0]     [1 0 0]    [1 0 0]                                                
   incr(cons(X,Y)) = [0 0 0]X + [0 0 0]Y >= [0 0 0]X + [0 0 0]Y = cons(n__s(activate(X)),n__incr(activate(Y)))
                     [0 0 0]    [0 0 0]     [0 0 0]    [0 0 0]                                                
   
                    [1 0 0]    [1 1 0]     [1 0 0]    [1 0 0]                                               
   adx(cons(X,Y)) = [0 0 0]X + [0 0 0]Y >= [0 0 0]X + [0 0 0]Y = incr(cons(activate(X),n__adx(activate(Y))))
                    [0 0 0]    [0 0 0]     [0 0 0]    [0 0 0]                                               
   
                   [1 0 0]    [1 1 0]    [1]        [0]              
   hd(cons(X,Y)) = [0 1 1]X + [0 0 0]Y + [1] >= X + [1] = activate(X)
                   [0 1 1]    [0 0 1]    [0]        [0]              
   
                   [1 0 0]    [1 1 0]    [0]        [0]              
   tl(cons(X,Y)) = [1 1 1]X + [1 1 0]Y + [1] >= Y + [1] = activate(Y)
                   [1 0 0]    [1 1 1]    [0]        [0]              
   
         [0]    [0]         
   0() = [1] >= [0] = n__0()
         [1]    [1]         
   
             [0]    [0]             
   zeros() = [1] >= [0] = n__zeros()
             [1]    [1]             
   
          [1 0 0]     [1 0 0]           
   s(X) = [0 0 0]X >= [0 0 0]X = n__s(X)
          [0 0 0]     [0 0 0]           
   
             [1 0 0]     [1 0 0]              
   incr(X) = [0 0 0]X >= [0 0 0]X = n__incr(X)
             [0 0 0]     [0 0 0]              
   
            [1 0 0]     [1 0 0]             
   adx(X) = [0 0 0]X >= [0 0 0]X = n__adx(X)
            [0 0 0]     [0 0 0]             
   
                      [0]    [0]      
   activate(n__0()) = [1] >= [1] = 0()
                      [1]    [1]      
   
                          [0]    [0]          
   activate(n__zeros()) = [1] >= [1] = zeros()
                          [1]    [1]          
   
                       [1 0 0]    [0]    [1 0 0]        
   activate(n__s(X)) = [0 0 0]X + [1] >= [0 0 0]X = s(X)
                       [0 0 0]    [0]    [0 0 0]        
   
                          [1 0 0]    [0]    [1 0 0]                     
   activate(n__incr(X)) = [0 0 0]X + [1] >= [0 0 0]X = incr(activate(X))
                          [0 0 0]    [0]    [0 0 0]                     
   
                         [1 0 0]    [0]    [1 0 0]                    
   activate(n__adx(X)) = [0 0 0]X + [1] >= [0 0 0]X = adx(activate(X))
                         [0 0 0]    [0]    [0 0 0]                    
   
                     [0]         
   activate(X) = X + [1] >= X = X
                     [0]         
  problem:
   nats() -> adx(zeros())
   zeros() -> cons(n__0(),n__zeros())
   incr(cons(X,Y)) -> cons(n__s(activate(X)),n__incr(activate(Y)))
   adx(cons(X,Y)) -> incr(cons(activate(X),n__adx(activate(Y))))
   tl(cons(X,Y)) -> activate(Y)
   0() -> n__0()
   zeros() -> n__zeros()
   s(X) -> n__s(X)
   incr(X) -> n__incr(X)
   adx(X) -> n__adx(X)
   activate(n__0()) -> 0()
   activate(n__zeros()) -> zeros()
   activate(n__s(X)) -> s(X)
   activate(n__incr(X)) -> incr(activate(X))
   activate(n__adx(X)) -> adx(activate(X))
   activate(X) -> X
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [n__zeros] = 0,
    
    [n__s](x0) = x0,
    
    [zeros] = 0,
    
    [n__0] = 0,
    
    [cons](x0, x1) = 2x0 + 2x1,
    
    [n__adx](x0) = x0,
    
    [nats] = 4,
    
    [activate](x0) = x0,
    
    [s](x0) = x0,
    
    [tl](x0) = 2x0 + 6,
    
    [n__incr](x0) = x0,
    
    [0] = 0,
    
    [incr](x0) = x0,
    
    [adx](x0) = x0
   orientation:
    nats() = 4 >= 0 = adx(zeros())
    
    zeros() = 0 >= 0 = cons(n__0(),n__zeros())
    
    incr(cons(X,Y)) = 2X + 2Y >= 2X + 2Y = cons(n__s(activate(X)),n__incr(activate(Y)))
    
    adx(cons(X,Y)) = 2X + 2Y >= 2X + 2Y = incr(cons(activate(X),n__adx(activate(Y))))
    
    tl(cons(X,Y)) = 4X + 4Y + 6 >= Y = activate(Y)
    
    0() = 0 >= 0 = n__0()
    
    zeros() = 0 >= 0 = n__zeros()
    
    s(X) = X >= X = n__s(X)
    
    incr(X) = X >= X = n__incr(X)
    
    adx(X) = X >= X = n__adx(X)
    
    activate(n__0()) = 0 >= 0 = 0()
    
    activate(n__zeros()) = 0 >= 0 = zeros()
    
    activate(n__s(X)) = X >= X = s(X)
    
    activate(n__incr(X)) = X >= X = incr(activate(X))
    
    activate(n__adx(X)) = X >= X = adx(activate(X))
    
    activate(X) = X >= X = X
   problem:
    zeros() -> cons(n__0(),n__zeros())
    incr(cons(X,Y)) -> cons(n__s(activate(X)),n__incr(activate(Y)))
    adx(cons(X,Y)) -> incr(cons(activate(X),n__adx(activate(Y))))
    0() -> n__0()
    zeros() -> n__zeros()
    s(X) -> n__s(X)
    incr(X) -> n__incr(X)
    adx(X) -> n__adx(X)
    activate(n__0()) -> 0()
    activate(n__zeros()) -> zeros()
    activate(n__s(X)) -> s(X)
    activate(n__incr(X)) -> incr(activate(X))
    activate(n__adx(X)) -> adx(activate(X))
    activate(X) -> X
   Unfolding Processor:
    loop length: 6
    terms:
     incr(cons(X,n__adx(n__zeros())))
     cons(n__s(activate(X)),n__incr(activate(n__adx(n__zeros()))))
     cons(n__s(activate(X)),n__incr(adx(activate(n__zeros()))))
     cons(n__s(activate(X)),n__incr(adx(zeros())))
     cons(n__s(activate(X)),n__incr(adx(cons(n__0(),n__zeros()))))
     cons(n__s(activate(X)),n__incr(incr(cons(activate(n__0()),n__adx(activate(n__zeros()))))))
    context: cons(n__s(activate(X)),n__incr([]))
    substitution:
     X -> activate(n__0())
    Qed