/export/starexec/sandbox2/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


YES

Problem:
 active(and(true(),X)) -> mark(X)
 active(and(false(),Y)) -> mark(false())
 active(if(true(),X,Y)) -> mark(X)
 active(if(false(),X,Y)) -> mark(Y)
 active(add(0(),X)) -> mark(X)
 active(add(s(X),Y)) -> mark(s(add(X,Y)))
 active(first(0(),X)) -> mark(nil())
 active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(and(X1,X2)) -> and(active(X1),X2)
 active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
 active(add(X1,X2)) -> add(active(X1),X2)
 active(first(X1,X2)) -> first(active(X1),X2)
 active(first(X1,X2)) -> first(X1,active(X2))
 and(mark(X1),X2) -> mark(and(X1,X2))
 if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
 add(mark(X1),X2) -> mark(add(X1,X2))
 first(mark(X1),X2) -> mark(first(X1,X2))
 first(X1,mark(X2)) -> mark(first(X1,X2))
 proper(and(X1,X2)) -> and(proper(X1),proper(X2))
 proper(true()) -> ok(true())
 proper(false()) -> ok(false())
 proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
 proper(add(X1,X2)) -> add(proper(X1),proper(X2))
 proper(0()) -> ok(0())
 proper(s(X)) -> s(proper(X))
 proper(first(X1,X2)) -> first(proper(X1),proper(X2))
 proper(nil()) -> ok(nil())
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 and(ok(X1),ok(X2)) -> ok(and(X1,X2))
 if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
 add(ok(X1),ok(X2)) -> ok(add(X1,X2))
 s(ok(X)) -> ok(s(X))
 first(ok(X1),ok(X2)) -> ok(first(X1,X2))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 from(ok(X)) -> ok(from(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Proof:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
               [1 1 0]     [0]
   [top](x0) = [0 1 0]x0 + [0]
               [0 0 0]     [1],
   
              [1 0 1]  
   [ok](x0) = [0 1 0]x0
              [1 0 0]  ,
   
                    
   [proper](x0) = x0
                    ,
   
                [1 0 1]     [0]
   [from](x0) = [0 0 0]x0 + [1]
                [1 0 0]     [0],
   
                    [1 0 0]     [1 0 0]  
   [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1
                    [0 0 1]     [0 0 1]  ,
   
           [0]
   [nil] = [0]
           [0],
   
                               [0]
   [first](x0, x1) = x0 + x1 + [1]
                               [0],
   
             [1 0 0]  
   [s](x0) = [0 0 0]x0
             [0 0 1]  ,
   
                             [0]
   [add](x0, x1) = x0 + x1 + [1]
                             [0],
   
         [0]
   [0] = [0]
         [0],
   
                                     [0]
   [if](x0, x1, x2) = x0 + x1 + x2 + [1]
                                     [0],
   
             [0]
   [false] = [0]
             [0],
   
                [1 0 0]     [0]
   [mark](x0) = [0 1 0]x0 + [1]
                [0 0 0]     [0],
   
                  [1 0 1]  
   [active](x0) = [0 1 0]x0
                  [1 0 1]  ,
   
                             [0]
   [and](x0, x1) = x0 + x1 + [1]
                             [0],
   
            [0]
   [true] = [0]
            [0]
  orientation:
                           [1 0 1]    [0]    [1 0 0]    [0]          
   active(and(true(),X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = mark(X)
                           [1 0 1]    [0]    [0 0 0]    [0]          
   
                            [1 0 1]    [0]    [0]                
   active(and(false(),Y)) = [0 1 0]Y + [1] >= [1] = mark(false())
                            [1 0 1]    [0]    [0]                
   
                            [1 0 1]    [1 0 1]    [0]    [1 0 0]    [0]          
   active(if(true(),X,Y)) = [0 1 0]X + [0 1 0]Y + [1] >= [0 1 0]X + [1] = mark(X)
                            [1 0 1]    [1 0 1]    [0]    [0 0 0]    [0]          
   
                             [1 0 1]    [1 0 1]    [0]    [1 0 0]    [0]          
   active(if(false(),X,Y)) = [0 1 0]X + [0 1 0]Y + [1] >= [0 1 0]Y + [1] = mark(Y)
                             [1 0 1]    [1 0 1]    [0]    [0 0 0]    [0]          
   
                        [1 0 1]    [0]    [1 0 0]    [0]          
   active(add(0(),X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = mark(X)
                        [1 0 1]    [0]    [0 0 0]    [0]          
   
                         [1 0 1]    [1 0 1]    [0]    [1 0 0]    [1 0 0]    [0]                    
   active(add(s(X),Y)) = [0 0 0]X + [0 1 0]Y + [1] >= [0 0 0]X + [0 0 0]Y + [1] = mark(s(add(X,Y)))
                         [1 0 1]    [1 0 1]    [0]    [0 0 0]    [0 0 0]    [0]                    
   
                          [1 0 1]    [0]    [0]              
   active(first(0(),X)) = [0 1 0]X + [1] >= [1] = mark(nil())
                          [1 0 1]    [0]    [0]              
   
                                   [1 0 1]    [1 0 1]    [1 0 1]    [0]    [1 0 0]    [1 0 0]    [1 0 0]    [0]                           
   active(first(s(X),cons(Y,Z))) = [0 0 0]X + [0 0 0]Y + [0 0 0]Z + [1] >= [0 0 0]X + [0 0 0]Y + [0 0 0]Z + [1] = mark(cons(Y,first(X,Z)))
                                   [1 0 1]    [1 0 1]    [1 0 1]    [0]    [0 0 0]    [0 0 0]    [0 0 0]    [0]                           
   
                     [2 0 1]    [0]    [2 0 1]    [0]                           
   active(from(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = mark(cons(X,from(s(X))))
                     [2 0 1]    [0]    [0 0 0]    [0]                           
   
                        [1 0 1]     [1 0 1]     [0]    [1 0 1]          [0]                     
   active(and(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + X2 + [1] = and(active(X1),X2)
                        [1 0 1]     [1 0 1]     [0]    [1 0 1]          [0]                     
   
                          [1 0 1]     [1 0 1]     [1 0 1]     [0]    [1 0 1]               [0]                       
   active(if(X1,X2,X3)) = [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] >= [0 1 0]X1 + X2 + X3 + [1] = if(active(X1),X2,X3)
                          [1 0 1]     [1 0 1]     [1 0 1]     [0]    [1 0 1]               [0]                       
   
                        [1 0 1]     [1 0 1]     [0]    [1 0 1]          [0]                     
   active(add(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + X2 + [1] = add(active(X1),X2)
                        [1 0 1]     [1 0 1]     [0]    [1 0 1]          [0]                     
   
                          [1 0 1]     [1 0 1]     [0]    [1 0 1]          [0]                       
   active(first(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + X2 + [1] = first(active(X1),X2)
                          [1 0 1]     [1 0 1]     [0]    [1 0 1]          [0]                       
   
                          [1 0 1]     [1 0 1]     [0]         [1 0 1]     [0]                       
   active(first(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= X1 + [0 1 0]X2 + [1] = first(X1,active(X2))
                          [1 0 1]     [1 0 1]     [0]         [1 0 1]     [0]                       
   
                      [1 0 0]          [0]    [1 0 0]     [1 0 0]     [0]                   
   and(mark(X1),X2) = [0 1 0]X1 + X2 + [2] >= [0 1 0]X1 + [0 1 0]X2 + [2] = mark(and(X1,X2))
                      [0 0 0]          [0]    [0 0 0]     [0 0 0]     [0]                   
   
                        [1 0 0]               [0]    [1 0 0]     [1 0 0]     [1 0 0]     [0]                     
   if(mark(X1),X2,X3) = [0 1 0]X1 + X2 + X3 + [2] >= [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [2] = mark(if(X1,X2,X3))
                        [0 0 0]               [0]    [0 0 0]     [0 0 0]     [0 0 0]     [0]                     
   
                      [1 0 0]          [0]    [1 0 0]     [1 0 0]     [0]                   
   add(mark(X1),X2) = [0 1 0]X1 + X2 + [2] >= [0 1 0]X1 + [0 1 0]X2 + [2] = mark(add(X1,X2))
                      [0 0 0]          [0]    [0 0 0]     [0 0 0]     [0]                   
   
                        [1 0 0]          [0]    [1 0 0]     [1 0 0]     [0]                     
   first(mark(X1),X2) = [0 1 0]X1 + X2 + [2] >= [0 1 0]X1 + [0 1 0]X2 + [2] = mark(first(X1,X2))
                        [0 0 0]          [0]    [0 0 0]     [0 0 0]     [0]                     
   
                             [1 0 0]     [0]    [1 0 0]     [1 0 0]     [0]                     
   first(X1,mark(X2)) = X1 + [0 1 0]X2 + [2] >= [0 1 0]X1 + [0 1 0]X2 + [2] = mark(first(X1,X2))
                             [0 0 0]     [0]    [0 0 0]     [0 0 0]     [0]                     
   
                                  [0]              [0]                             
   proper(and(X1,X2)) = X1 + X2 + [1] >= X1 + X2 + [1] = and(proper(X1),proper(X2))
                                  [0]              [0]                             
   
                    [0]    [0]             
   proper(true()) = [0] >= [0] = ok(true())
                    [0]    [0]             
   
                     [0]    [0]              
   proper(false()) = [0] >= [0] = ok(false())
                     [0]    [0]              
   
                                         [0]                   [0]                                       
   proper(if(X1,X2,X3)) = X1 + X2 + X3 + [1] >= X1 + X2 + X3 + [1] = if(proper(X1),proper(X2),proper(X3))
                                         [0]                   [0]                                       
   
                                  [0]              [0]                             
   proper(add(X1,X2)) = X1 + X2 + [1] >= X1 + X2 + [1] = add(proper(X1),proper(X2))
                                  [0]              [0]                             
   
                 [0]    [0]          
   proper(0()) = [0] >= [0] = ok(0())
                 [0]    [0]          
   
                  [1 0 0]     [1 0 0]                
   proper(s(X)) = [0 0 0]X >= [0 0 0]X = s(proper(X))
                  [0 0 1]     [0 0 1]                
   
                                    [0]              [0]                               
   proper(first(X1,X2)) = X1 + X2 + [1] >= X1 + X2 + [1] = first(proper(X1),proper(X2))
                                    [0]              [0]                               
   
                   [0]    [0]            
   proper(nil()) = [0] >= [0] = ok(nil())
                   [0]    [0]            
   
                         [1 0 0]     [1 0 0]      [1 0 0]     [1 0 0]                                
   proper(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(proper(X1),proper(X2))
                         [0 0 1]     [0 0 1]      [0 0 1]     [0 0 1]                                
   
                     [1 0 1]    [0]    [1 0 1]    [0]                  
   proper(from(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = from(proper(X))
                     [1 0 0]    [0]    [1 0 0]    [0]                  
   
                        [1 0 1]     [1 0 1]     [0]    [1 0 1]     [1 0 1]     [0]                 
   and(ok(X1),ok(X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = ok(and(X1,X2))
                        [1 0 0]     [1 0 0]     [0]    [1 0 0]     [1 0 0]     [0]                 
   
                              [1 0 1]     [1 0 1]     [1 0 1]     [0]    [1 0 1]     [1 0 1]     [1 0 1]     [0]                   
   if(ok(X1),ok(X2),ok(X3)) = [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 + [1] = ok(if(X1,X2,X3))
                              [1 0 0]     [1 0 0]     [1 0 0]     [0]    [1 0 0]     [1 0 0]     [1 0 0]     [0]                   
   
                        [1 0 1]     [1 0 1]     [0]    [1 0 1]     [1 0 1]     [0]                 
   add(ok(X1),ok(X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = ok(add(X1,X2))
                        [1 0 0]     [1 0 0]     [0]    [1 0 0]     [1 0 0]     [0]                 
   
              [1 0 1]     [1 0 1]            
   s(ok(X)) = [0 0 0]X >= [0 0 0]X = ok(s(X))
              [1 0 0]     [1 0 0]            
   
                          [1 0 1]     [1 0 1]     [0]    [1 0 1]     [1 0 1]     [0]                   
   first(ok(X1),ok(X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = ok(first(X1,X2))
                          [1 0 0]     [1 0 0]     [0]    [1 0 0]     [1 0 0]     [0]                   
   
                         [1 0 1]     [1 0 1]      [1 0 1]     [1 0 1]                    
   cons(ok(X1),ok(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = ok(cons(X1,X2))
                         [1 0 0]     [1 0 0]      [1 0 0]     [1 0 0]                    
   
                 [2 0 1]    [0]    [2 0 1]    [0]              
   from(ok(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = ok(from(X))
                 [1 0 1]    [0]    [1 0 1]    [0]              
   
                  [1 1 0]    [1]    [1 1 0]    [0]                 
   top(mark(X)) = [0 1 0]X + [1] >= [0 1 0]X + [0] = top(proper(X))
                  [0 0 0]    [1]    [0 0 0]    [1]                 
   
                [1 1 1]    [0]    [1 1 1]    [0]                 
   top(ok(X)) = [0 1 0]X + [0] >= [0 1 0]X + [0] = top(active(X))
                [0 0 0]    [1]    [0 0 0]    [1]                 
  problem:
   active(and(true(),X)) -> mark(X)
   active(and(false(),Y)) -> mark(false())
   active(if(true(),X,Y)) -> mark(X)
   active(if(false(),X,Y)) -> mark(Y)
   active(add(0(),X)) -> mark(X)
   active(add(s(X),Y)) -> mark(s(add(X,Y)))
   active(first(0(),X)) -> mark(nil())
   active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
   active(from(X)) -> mark(cons(X,from(s(X))))
   active(and(X1,X2)) -> and(active(X1),X2)
   active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
   active(add(X1,X2)) -> add(active(X1),X2)
   active(first(X1,X2)) -> first(active(X1),X2)
   active(first(X1,X2)) -> first(X1,active(X2))
   and(mark(X1),X2) -> mark(and(X1,X2))
   if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
   add(mark(X1),X2) -> mark(add(X1,X2))
   first(mark(X1),X2) -> mark(first(X1,X2))
   first(X1,mark(X2)) -> mark(first(X1,X2))
   proper(and(X1,X2)) -> and(proper(X1),proper(X2))
   proper(true()) -> ok(true())
   proper(false()) -> ok(false())
   proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
   proper(add(X1,X2)) -> add(proper(X1),proper(X2))
   proper(0()) -> ok(0())
   proper(s(X)) -> s(proper(X))
   proper(first(X1,X2)) -> first(proper(X1),proper(X2))
   proper(nil()) -> ok(nil())
   proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
   proper(from(X)) -> from(proper(X))
   and(ok(X1),ok(X2)) -> ok(and(X1,X2))
   if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
   add(ok(X1),ok(X2)) -> ok(add(X1,X2))
   s(ok(X)) -> ok(s(X))
   first(ok(X1),ok(X2)) -> ok(first(X1,X2))
   cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
   from(ok(X)) -> ok(from(X))
   top(ok(X)) -> top(active(X))
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [top](x0) = x0,
    
    [ok](x0) = 2x0,
    
    [proper](x0) = 3x0,
    
    [from](x0) = x0,
    
    [cons](x0, x1) = x0 + x1,
    
    [nil] = 3,
    
    [first](x0, x1) = x0 + x1,
    
    [s](x0) = x0,
    
    [add](x0, x1) = 4x0 + 4x1,
    
    [0] = 2,
    
    [if](x0, x1, x2) = 2x0 + x1 + 2x2,
    
    [false] = 0,
    
    [mark](x0) = x0,
    
    [active](x0) = 2x0,
    
    [and](x0, x1) = 2x0 + x1,
    
    [true] = 7
   orientation:
    active(and(true(),X)) = 2X + 28 >= X = mark(X)
    
    active(and(false(),Y)) = 2Y >= 0 = mark(false())
    
    active(if(true(),X,Y)) = 2X + 4Y + 28 >= X = mark(X)
    
    active(if(false(),X,Y)) = 2X + 4Y >= Y = mark(Y)
    
    active(add(0(),X)) = 8X + 16 >= X = mark(X)
    
    active(add(s(X),Y)) = 8X + 8Y >= 4X + 4Y = mark(s(add(X,Y)))
    
    active(first(0(),X)) = 2X + 4 >= 3 = mark(nil())
    
    active(first(s(X),cons(Y,Z))) = 2X + 2Y + 2Z >= X + Y + Z = mark(cons(Y,first(X,Z)))
    
    active(from(X)) = 2X >= 2X = mark(cons(X,from(s(X))))
    
    active(and(X1,X2)) = 4X1 + 2X2 >= 4X1 + X2 = and(active(X1),X2)
    
    active(if(X1,X2,X3)) = 4X1 + 2X2 + 4X3 >= 4X1 + X2 + 2X3 = if(active(X1),X2,X3)
    
    active(add(X1,X2)) = 8X1 + 8X2 >= 8X1 + 4X2 = add(active(X1),X2)
    
    active(first(X1,X2)) = 2X1 + 2X2 >= 2X1 + X2 = first(active(X1),X2)
    
    active(first(X1,X2)) = 2X1 + 2X2 >= X1 + 2X2 = first(X1,active(X2))
    
    and(mark(X1),X2) = 2X1 + X2 >= 2X1 + X2 = mark(and(X1,X2))
    
    if(mark(X1),X2,X3) = 2X1 + X2 + 2X3 >= 2X1 + X2 + 2X3 = mark(if(X1,X2,X3))
    
    add(mark(X1),X2) = 4X1 + 4X2 >= 4X1 + 4X2 = mark(add(X1,X2))
    
    first(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(first(X1,X2))
    
    first(X1,mark(X2)) = X1 + X2 >= X1 + X2 = mark(first(X1,X2))
    
    proper(and(X1,X2)) = 6X1 + 3X2 >= 6X1 + 3X2 = and(proper(X1),proper(X2))
    
    proper(true()) = 21 >= 14 = ok(true())
    
    proper(false()) = 0 >= 0 = ok(false())
    
    proper(if(X1,X2,X3)) = 6X1 + 3X2 + 6X3 >= 6X1 + 3X2 + 6X3 = if(proper(X1),proper(X2),proper(X3))
    
    proper(add(X1,X2)) = 12X1 + 12X2 >= 12X1 + 12X2 = add(proper(X1),proper(X2))
    
    proper(0()) = 6 >= 4 = ok(0())
    
    proper(s(X)) = 3X >= 3X = s(proper(X))
    
    proper(first(X1,X2)) = 3X1 + 3X2 >= 3X1 + 3X2 = first(proper(X1),proper(X2))
    
    proper(nil()) = 9 >= 6 = ok(nil())
    
    proper(cons(X1,X2)) = 3X1 + 3X2 >= 3X1 + 3X2 = cons(proper(X1),proper(X2))
    
    proper(from(X)) = 3X >= 3X = from(proper(X))
    
    and(ok(X1),ok(X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = ok(and(X1,X2))
    
    if(ok(X1),ok(X2),ok(X3)) = 4X1 + 2X2 + 4X3 >= 4X1 + 2X2 + 4X3 = ok(if(X1,X2,X3))
    
    add(ok(X1),ok(X2)) = 8X1 + 8X2 >= 8X1 + 8X2 = ok(add(X1,X2))
    
    s(ok(X)) = 2X >= 2X = ok(s(X))
    
    first(ok(X1),ok(X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = ok(first(X1,X2))
    
    cons(ok(X1),ok(X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = ok(cons(X1,X2))
    
    from(ok(X)) = 2X >= 2X = ok(from(X))
    
    top(ok(X)) = 2X >= 2X = top(active(X))
   problem:
    active(and(false(),Y)) -> mark(false())
    active(if(false(),X,Y)) -> mark(Y)
    active(add(s(X),Y)) -> mark(s(add(X,Y)))
    active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
    active(from(X)) -> mark(cons(X,from(s(X))))
    active(and(X1,X2)) -> and(active(X1),X2)
    active(if(X1,X2,X3)) -> if(active(X1),X2,X3)
    active(add(X1,X2)) -> add(active(X1),X2)
    active(first(X1,X2)) -> first(active(X1),X2)
    active(first(X1,X2)) -> first(X1,active(X2))
    and(mark(X1),X2) -> mark(and(X1,X2))
    if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
    add(mark(X1),X2) -> mark(add(X1,X2))
    first(mark(X1),X2) -> mark(first(X1,X2))
    first(X1,mark(X2)) -> mark(first(X1,X2))
    proper(and(X1,X2)) -> and(proper(X1),proper(X2))
    proper(false()) -> ok(false())
    proper(if(X1,X2,X3)) -> if(proper(X1),proper(X2),proper(X3))
    proper(add(X1,X2)) -> add(proper(X1),proper(X2))
    proper(s(X)) -> s(proper(X))
    proper(first(X1,X2)) -> first(proper(X1),proper(X2))
    proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
    proper(from(X)) -> from(proper(X))
    and(ok(X1),ok(X2)) -> ok(and(X1,X2))
    if(ok(X1),ok(X2),ok(X3)) -> ok(if(X1,X2,X3))
    add(ok(X1),ok(X2)) -> ok(add(X1,X2))
    s(ok(X)) -> ok(s(X))
    first(ok(X1),ok(X2)) -> ok(first(X1,X2))
    cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
    from(ok(X)) -> ok(from(X))
    top(ok(X)) -> top(active(X))
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [top](x0) = 4x0 + 2,
     
     [ok](x0) = 4x0 + 2,
     
     [proper](x0) = 6x0,
     
     [from](x0) = 4x0,
     
     [cons](x0, x1) = 3x0 + x1,
     
     [first](x0, x1) = x0 + 2x1,
     
     [s](x0) = x0,
     
     [add](x0, x1) = 4x0 + 3x1 + 4,
     
     [if](x0, x1, x2) = x0 + x1 + 5x2 + 3,
     
     [false] = 4,
     
     [mark](x0) = x0,
     
     [active](x0) = 4x0,
     
     [and](x0, x1) = x0 + x1
    orientation:
     active(and(false(),Y)) = 4Y + 16 >= 4 = mark(false())
     
     active(if(false(),X,Y)) = 4X + 20Y + 28 >= Y = mark(Y)
     
     active(add(s(X),Y)) = 16X + 12Y + 16 >= 4X + 3Y + 4 = mark(s(add(X,Y)))
     
     active(first(s(X),cons(Y,Z))) = 4X + 24Y + 8Z >= X + 3Y + 2Z = mark(cons(Y,first(X,Z)))
     
     active(from(X)) = 16X >= 7X = mark(cons(X,from(s(X))))
     
     active(and(X1,X2)) = 4X1 + 4X2 >= 4X1 + X2 = and(active(X1),X2)
     
     active(if(X1,X2,X3)) = 4X1 + 4X2 + 20X3 + 12 >= 4X1 + X2 + 5X3 + 3 = if(active(X1),X2,X3)
     
     active(add(X1,X2)) = 16X1 + 12X2 + 16 >= 16X1 + 3X2 + 4 = add(active(X1),X2)
     
     active(first(X1,X2)) = 4X1 + 8X2 >= 4X1 + 2X2 = first(active(X1),X2)
     
     active(first(X1,X2)) = 4X1 + 8X2 >= X1 + 8X2 = first(X1,active(X2))
     
     and(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(and(X1,X2))
     
     if(mark(X1),X2,X3) = X1 + X2 + 5X3 + 3 >= X1 + X2 + 5X3 + 3 = mark(if(X1,X2,X3))
     
     add(mark(X1),X2) = 4X1 + 3X2 + 4 >= 4X1 + 3X2 + 4 = mark(add(X1,X2))
     
     first(mark(X1),X2) = X1 + 2X2 >= X1 + 2X2 = mark(first(X1,X2))
     
     first(X1,mark(X2)) = X1 + 2X2 >= X1 + 2X2 = mark(first(X1,X2))
     
     proper(and(X1,X2)) = 6X1 + 6X2 >= 6X1 + 6X2 = and(proper(X1),proper(X2))
     
     proper(false()) = 24 >= 18 = ok(false())
     
     proper(if(X1,X2,X3)) = 6X1 + 6X2 + 30X3 + 18 >= 6X1 + 6X2 + 30X3 + 3 = if(proper(X1),proper(X2),proper(X3))
     
     proper(add(X1,X2)) = 24X1 + 18X2 + 24 >= 24X1 + 18X2 + 4 = add(proper(X1),proper(X2))
     
     proper(s(X)) = 6X >= 6X = s(proper(X))
     
     proper(first(X1,X2)) = 6X1 + 12X2 >= 6X1 + 12X2 = first(proper(X1),proper(X2))
     
     proper(cons(X1,X2)) = 18X1 + 6X2 >= 18X1 + 6X2 = cons(proper(X1),proper(X2))
     
     proper(from(X)) = 24X >= 24X = from(proper(X))
     
     and(ok(X1),ok(X2)) = 4X1 + 4X2 + 4 >= 4X1 + 4X2 + 2 = ok(and(X1,X2))
     
     if(ok(X1),ok(X2),ok(X3)) = 4X1 + 4X2 + 20X3 + 17 >= 4X1 + 4X2 + 20X3 + 14 = ok(if(X1,X2,X3))
     
     add(ok(X1),ok(X2)) = 16X1 + 12X2 + 18 >= 16X1 + 12X2 + 18 = ok(add(X1,X2))
     
     s(ok(X)) = 4X + 2 >= 4X + 2 = ok(s(X))
     
     first(ok(X1),ok(X2)) = 4X1 + 8X2 + 6 >= 4X1 + 8X2 + 2 = ok(first(X1,X2))
     
     cons(ok(X1),ok(X2)) = 12X1 + 4X2 + 8 >= 12X1 + 4X2 + 2 = ok(cons(X1,X2))
     
     from(ok(X)) = 16X + 8 >= 16X + 2 = ok(from(X))
     
     top(ok(X)) = 16X + 10 >= 16X + 2 = top(active(X))
    problem:
     active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
     active(from(X)) -> mark(cons(X,from(s(X))))
     active(and(X1,X2)) -> and(active(X1),X2)
     active(first(X1,X2)) -> first(active(X1),X2)
     active(first(X1,X2)) -> first(X1,active(X2))
     and(mark(X1),X2) -> mark(and(X1,X2))
     if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
     add(mark(X1),X2) -> mark(add(X1,X2))
     first(mark(X1),X2) -> mark(first(X1,X2))
     first(X1,mark(X2)) -> mark(first(X1,X2))
     proper(and(X1,X2)) -> and(proper(X1),proper(X2))
     proper(s(X)) -> s(proper(X))
     proper(first(X1,X2)) -> first(proper(X1),proper(X2))
     proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
     proper(from(X)) -> from(proper(X))
     add(ok(X1),ok(X2)) -> ok(add(X1,X2))
     s(ok(X)) -> ok(s(X))
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [ok](x0) = x0 + 5,
      
      [proper](x0) = x0,
      
      [from](x0) = 2x0 + 3,
      
      [cons](x0, x1) = x0 + x1 + 1,
      
      [first](x0, x1) = 2x0 + 4x1,
      
      [s](x0) = 2x0,
      
      [add](x0, x1) = 2x0 + 2x1 + 5,
      
      [if](x0, x1, x2) = x0 + 2x1 + 4x2 + 6,
      
      [mark](x0) = x0 + 5,
      
      [active](x0) = 3x0,
      
      [and](x0, x1) = 5x0 + x1 + 1
     orientation:
      active(first(s(X),cons(Y,Z))) = 12X + 12Y + 12Z + 12 >= 2X + Y + 4Z + 6 = mark(cons(Y,first(X,Z)))
      
      active(from(X)) = 6X + 9 >= 5X + 9 = mark(cons(X,from(s(X))))
      
      active(and(X1,X2)) = 15X1 + 3X2 + 3 >= 15X1 + X2 + 1 = and(active(X1),X2)
      
      active(first(X1,X2)) = 6X1 + 12X2 >= 6X1 + 4X2 = first(active(X1),X2)
      
      active(first(X1,X2)) = 6X1 + 12X2 >= 2X1 + 12X2 = first(X1,active(X2))
      
      and(mark(X1),X2) = 5X1 + X2 + 26 >= 5X1 + X2 + 6 = mark(and(X1,X2))
      
      if(mark(X1),X2,X3) = X1 + 2X2 + 4X3 + 11 >= X1 + 2X2 + 4X3 + 11 = mark(if(X1,X2,X3))
      
      add(mark(X1),X2) = 2X1 + 2X2 + 15 >= 2X1 + 2X2 + 10 = mark(add(X1,X2))
      
      first(mark(X1),X2) = 2X1 + 4X2 + 10 >= 2X1 + 4X2 + 5 = mark(first(X1,X2))
      
      first(X1,mark(X2)) = 2X1 + 4X2 + 20 >= 2X1 + 4X2 + 5 = mark(first(X1,X2))
      
      proper(and(X1,X2)) = 5X1 + X2 + 1 >= 5X1 + X2 + 1 = and(proper(X1),proper(X2))
      
      proper(s(X)) = 2X >= 2X = s(proper(X))
      
      proper(first(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = first(proper(X1),proper(X2))
      
      proper(cons(X1,X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = cons(proper(X1),proper(X2))
      
      proper(from(X)) = 2X + 3 >= 2X + 3 = from(proper(X))
      
      add(ok(X1),ok(X2)) = 2X1 + 2X2 + 25 >= 2X1 + 2X2 + 10 = ok(add(X1,X2))
      
      s(ok(X)) = 2X + 10 >= 2X + 5 = ok(s(X))
     problem:
      active(from(X)) -> mark(cons(X,from(s(X))))
      active(first(X1,X2)) -> first(active(X1),X2)
      active(first(X1,X2)) -> first(X1,active(X2))
      if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
      proper(and(X1,X2)) -> and(proper(X1),proper(X2))
      proper(s(X)) -> s(proper(X))
      proper(first(X1,X2)) -> first(proper(X1),proper(X2))
      proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
      proper(from(X)) -> from(proper(X))
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [proper](x0) = 3x0,
       
       [from](x0) = 4x0 + 2,
       
       [cons](x0, x1) = 2x0 + 2x1,
       
       [first](x0, x1) = x0 + 2x1 + 1,
       
       [s](x0) = 2x0 + 2,
       
       [if](x0, x1, x2) = x0 + 4x1 + 4x2 + 2,
       
       [mark](x0) = x0,
       
       [active](x0) = 7x0 + 6,
       
       [and](x0, x1) = x0 + x1
      orientation:
       active(from(X)) = 28X + 20 >= 18X + 20 = mark(cons(X,from(s(X))))
       
       active(first(X1,X2)) = 7X1 + 14X2 + 13 >= 7X1 + 2X2 + 7 = first(active(X1),X2)
       
       active(first(X1,X2)) = 7X1 + 14X2 + 13 >= X1 + 14X2 + 13 = first(X1,active(X2))
       
       if(mark(X1),X2,X3) = X1 + 4X2 + 4X3 + 2 >= X1 + 4X2 + 4X3 + 2 = mark(if(X1,X2,X3))
       
       proper(and(X1,X2)) = 3X1 + 3X2 >= 3X1 + 3X2 = and(proper(X1),proper(X2))
       
       proper(s(X)) = 6X + 6 >= 6X + 2 = s(proper(X))
       
       proper(first(X1,X2)) = 3X1 + 6X2 + 3 >= 3X1 + 6X2 + 1 = first(proper(X1),proper(X2))
       
       proper(cons(X1,X2)) = 6X1 + 6X2 >= 6X1 + 6X2 = cons(proper(X1),proper(X2))
       
       proper(from(X)) = 12X + 6 >= 12X + 2 = from(proper(X))
      problem:
       active(from(X)) -> mark(cons(X,from(s(X))))
       active(first(X1,X2)) -> first(X1,active(X2))
       if(mark(X1),X2,X3) -> mark(if(X1,X2,X3))
       proper(and(X1,X2)) -> and(proper(X1),proper(X2))
       proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
      Matrix Interpretation Processor: dim=1
       
       interpretation:
        [proper](x0) = 4x0,
        
        [from](x0) = 2x0 + 2,
        
        [cons](x0, x1) = 4x0 + x1 + 2,
        
        [first](x0, x1) = 4x0 + 2x1,
        
        [s](x0) = x0 + 1,
        
        [if](x0, x1, x2) = 2x0 + x1 + x2,
        
        [mark](x0) = x0 + 2,
        
        [active](x0) = 4x0,
        
        [and](x0, x1) = x0 + x1 + 4
       orientation:
        active(from(X)) = 8X + 8 >= 6X + 8 = mark(cons(X,from(s(X))))
        
        active(first(X1,X2)) = 16X1 + 8X2 >= 4X1 + 8X2 = first(X1,active(X2))
        
        if(mark(X1),X2,X3) = 2X1 + X2 + X3 + 4 >= 2X1 + X2 + X3 + 2 = mark(if(X1,X2,X3))
        
        proper(and(X1,X2)) = 4X1 + 4X2 + 16 >= 4X1 + 4X2 + 4 = and(proper(X1),proper(X2))
        
        proper(cons(X1,X2)) = 16X1 + 4X2 + 8 >= 16X1 + 4X2 + 2 = cons(proper(X1),proper(X2))
       problem:
        active(from(X)) -> mark(cons(X,from(s(X))))
        active(first(X1,X2)) -> first(X1,active(X2))
       Matrix Interpretation Processor: dim=1
        
        interpretation:
         [from](x0) = 4x0,
         
         [cons](x0, x1) = x0 + 3x1,
         
         [first](x0, x1) = x0 + 2x1 + 1,
         
         [s](x0) = x0,
         
         [mark](x0) = x0 + 1,
         
         [active](x0) = 4x0 + 3
        orientation:
         active(from(X)) = 16X + 3 >= 13X + 1 = mark(cons(X,from(s(X))))
         
         active(first(X1,X2)) = 4X1 + 8X2 + 7 >= X1 + 8X2 + 7 = first(X1,active(X2))
        problem:
         active(first(X1,X2)) -> first(X1,active(X2))
        Matrix Interpretation Processor: dim=3
         
         interpretation:
                            [1 0 0]     [1 0 0]     [0]
          [first](x0, x1) = [0 0 0]x0 + [0 0 1]x1 + [0]
                            [0 0 0]     [0 0 1]     [1],
          
                         [1 0 1]  
          [active](x0) = [0 1 0]x0
                         [0 0 1]  
         orientation:
                                 [1 0 0]     [1 0 1]     [1]    [1 0 0]     [1 0 1]     [0]                       
          active(first(X1,X2)) = [0 0 0]X1 + [0 0 1]X2 + [0] >= [0 0 0]X1 + [0 0 1]X2 + [0] = first(X1,active(X2))
                                 [0 0 0]     [0 0 1]     [1]    [0 0 0]     [0 0 1]     [1]                       
         problem:
          
         Qed