Problem AProVE 08 round

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputAProVE 08 round

stdout:

MAYBE

Problem:
 f(s(x),x) -> f(s(x),round(s(x)))
 round(0()) -> 0()
 round(0()) -> s(0())
 round(s(0())) -> s(0())
 round(s(s(x))) -> s(s(round(x)))

Proof:
 Complexity Transformation Processor:
  strict:
   f(s(x),x) -> f(s(x),round(s(x)))
   round(0()) -> 0()
   round(0()) -> s(0())
   round(s(0())) -> s(0())
   round(s(s(x))) -> s(s(round(x)))
  weak:
   
  Matrix Interpretation Processor:
   dimension: 1
   max_matrix:
    1
    interpretation:
     [0] = 8,
     
     [round](x0) = x0 + 102,
     
     [f](x0, x1) = x0 + x1 + 127,
     
     [s](x0) = x0 + 2
    orientation:
     f(s(x),x) = 2x + 129 >= 2x + 233 = f(s(x),round(s(x)))
     
     round(0()) = 110 >= 8 = 0()
     
     round(0()) = 110 >= 10 = s(0())
     
     round(s(0())) = 112 >= 10 = s(0())
     
     round(s(s(x))) = x + 106 >= x + 106 = s(s(round(x)))
    problem:
     strict:
      f(s(x),x) -> f(s(x),round(s(x)))
      round(s(s(x))) -> s(s(round(x)))
     weak:
      round(0()) -> 0()
      round(0()) -> s(0())
      round(s(0())) -> s(0())
    Open
 

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputAProVE 08 round

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputAProVE 08 round

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  f(s(x), x) -> f(s(x), round(s(x)))
     , round(0()) -> 0()
     , round(0()) -> s(0())
     , round(s(0())) -> s(0())
     , round(s(s(x))) -> s(s(round(x)))}

Proof Output:    
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(s(x), x) -> c_0(f^#(s(x), round(s(x))))
              , 2: round^#(0()) -> c_1()
              , 3: round^#(0()) -> c_2()
              , 4: round^#(s(0())) -> c_3()
              , 5: round^#(s(s(x))) -> c_4(round^#(x))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{5}                                                       [   YES(?,O(n^2))    ]
                |
                |->{2}                                                   [   YES(?,O(n^1))    ]
                |
                |->{3}                                                   [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [   YES(?,O(n^2))    ]
             
             ->{1}                                                       [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: NA
             ------------
             
             The usable rules for this path are:
             
               {  round(0()) -> 0()
                , round(0()) -> s(0())
                , round(s(0())) -> s(0())
                , round(s(s(x))) -> s(s(round(x)))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {5}: YES(?,O(n^2))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                round(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                round^#(x1) = [0 0 0] x1 + [0]
                              [3 3 3]      [0]
                              [3 3 3]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [0]
                        [0 0 1]      [0]
                        [0 0 1]      [2]
                round^#(x1) = [2 0 2] x1 + [0]
                              [4 0 2]      [0]
                              [0 0 0]      [2]
                c_4(x1) = [1 0 2] x1 + [3]
                          [2 0 0]      [0]
                          [0 0 0]      [2]
           
           * Path {5}->{2}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                round(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                round^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {round^#(0()) -> c_1()}
               Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [0]
                        [0 0 0]      [2]
                        [0 0 0]      [2]
                0() = [2]
                      [2]
                      [2]
                round^#(x1) = [2 0 0] x1 + [0]
                              [0 0 2]      [4]
                              [1 2 2]      [0]
                c_1() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [7]
                          [0 0 0]      [6]
           
           * Path {5}->{3}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                round(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                round^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {round^#(0()) -> c_2()}
               Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [0]
                        [0 0 0]      [2]
                        [0 0 0]      [2]
                0() = [2]
                      [2]
                      [2]
                round^#(x1) = [2 0 0] x1 + [0]
                              [0 0 2]      [4]
                              [1 2 2]      [0]
                c_2() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [7]
                          [0 0 0]      [6]
           
           * Path {5}->{4}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                round(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                round^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {round^#(s(0())) -> c_3()}
               Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [2]
                        [0 1 2]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                round^#(x1) = [2 3 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 5 0]      [0]
                c_3() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(s(x), x) -> c_0(f^#(s(x), round(s(x))))
              , 2: round^#(0()) -> c_1()
              , 3: round^#(0()) -> c_2()
              , 4: round^#(s(0())) -> c_3()
              , 5: round^#(s(s(x))) -> c_4(round^#(x))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                |->{2}                                                   [   YES(?,O(n^1))    ]
                |
                |->{3}                                                   [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [   YES(?,O(n^2))    ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules for this path are:
             
               {  round(0()) -> 0()
                , round(0()) -> s(0())
                , round(s(0())) -> s(0())
                , round(s(s(x))) -> s(s(round(x)))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  f^#(s(x), x) -> c_0(f^#(s(x), round(s(x))))
                  , round(0()) -> 0()
                  , round(0()) -> s(0())
                  , round(s(0())) -> s(0())
                  , round(s(s(x))) -> s(s(round(x)))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {5}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 0]      [0]
                round(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                0() = [0]
                      [0]
                f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                round^#(x1) = [0 0] x1 + [0]
                              [3 3]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [1]
                round^#(x1) = [2 2] x1 + [2]
                              [6 0]      [0]
                c_4(x1) = [1 0] x1 + [5]
                          [2 0]      [7]
           
           * Path {5}->{2}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                round(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                0() = [0]
                      [0]
                f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                round^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {round^#(0()) -> c_1()}
               Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [0]
                0() = [0]
                      [2]
                round^#(x1) = [2 2] x1 + [4]
                              [6 2]      [0]
                c_1() = [1]
                        [0]
                c_4(x1) = [1 0] x1 + [3]
                          [2 0]      [3]
           
           * Path {5}->{3}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                round(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                0() = [0]
                      [0]
                f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                round^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {round^#(0()) -> c_2()}
               Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [0]
                0() = [0]
                      [2]
                round^#(x1) = [2 2] x1 + [4]
                              [6 2]      [0]
                c_2() = [1]
                        [0]
                c_4(x1) = [1 0] x1 + [3]
                          [2 0]      [3]
           
           * Path {5}->{4}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                round(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                0() = [0]
                      [0]
                f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                round^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {round^#(s(0())) -> c_3()}
               Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1] x1 + [2]
                        [0 1]      [1]
                0() = [1]
                      [0]
                round^#(x1) = [2 1] x1 + [1]
                              [0 0]      [7]
                c_3() = [1]
                        [1]
                c_4(x1) = [1 1] x1 + [3]
                          [0 0]      [3]
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(s(x), x) -> c_0(f^#(s(x), round(s(x))))
              , 2: round^#(0()) -> c_1()
              , 3: round^#(0()) -> c_2()
              , 4: round^#(s(0())) -> c_3()
              , 5: round^#(s(s(x))) -> c_4(round^#(x))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                |->{2}                                                   [   YES(?,O(n^1))    ]
                |
                |->{3}                                                   [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [    YES(?,O(1))     ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules for this path are:
             
               {  round(0()) -> 0()
                , round(0()) -> s(0())
                , round(s(0())) -> s(0())
                , round(s(s(x))) -> s(s(round(x)))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  f^#(s(x), x) -> c_0(f^#(s(x), round(s(x))))
                  , round(0()) -> 0()
                  , round(0()) -> s(0())
                  , round(s(0())) -> s(0())
                  , round(s(s(x))) -> s(s(round(x)))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {5}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [1] x1 + [0]
                round(x1) = [0] x1 + [0]
                0() = [0]
                f^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                round^#(x1) = [3] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                round^#(x1) = [2] x1 + [0]
                c_4(x1) = [1] x1 + [7]
           
           * Path {5}->{2}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                round(x1) = [0] x1 + [0]
                0() = [0]
                f^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                round^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {round^#(0()) -> c_1()}
               Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                0() = [2]
                round^#(x1) = [2] x1 + [4]
                c_1() = [1]
                c_4(x1) = [1] x1 + [0]
           
           * Path {5}->{3}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                round(x1) = [0] x1 + [0]
                0() = [0]
                f^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                round^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {round^#(0()) -> c_2()}
               Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                0() = [2]
                round^#(x1) = [2] x1 + [4]
                c_2() = [1]
                c_4(x1) = [1] x1 + [0]
           
           * Path {5}->{4}: YES(?,O(1))
             --------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                round(x1) = [0] x1 + [0]
                0() = [0]
                f^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                round^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {round^#(s(0())) -> c_3()}
               Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0] x1 + [0]
                0() = [0]
                round^#(x1) = [0] x1 + [1]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
    
    4) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    

Tool RC1

Execution TimeUnknown
Answer
MAYBE
InputAProVE 08 round

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputAProVE 08 round

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  f(s(x), x) -> f(s(x), round(s(x)))
     , round(0()) -> 0()
     , round(0()) -> s(0())
     , round(s(0())) -> s(0())
     , round(s(s(x))) -> s(s(round(x)))}

Proof Output:    
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(s(x), x) -> c_0(f^#(s(x), round(s(x))))
              , 2: round^#(0()) -> c_1()
              , 3: round^#(0()) -> c_2()
              , 4: round^#(s(0())) -> c_3()
              , 5: round^#(s(s(x))) -> c_4(round^#(x))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{5}                                                       [   YES(?,O(n^2))    ]
                |
                |->{2}                                                   [   YES(?,O(n^1))    ]
                |
                |->{3}                                                   [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [   YES(?,O(n^2))    ]
             
             ->{1}                                                       [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: NA
             ------------
             
             The usable rules for this path are:
             
               {  round(0()) -> 0()
                , round(0()) -> s(0())
                , round(s(0())) -> s(0())
                , round(s(s(x))) -> s(s(round(x)))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {5}: YES(?,O(n^2))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                round(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                round^#(x1) = [0 0 0] x1 + [0]
                              [3 3 3]      [0]
                              [3 3 3]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [0]
                        [0 0 1]      [0]
                        [0 0 1]      [2]
                round^#(x1) = [2 0 2] x1 + [0]
                              [4 0 2]      [0]
                              [0 0 0]      [2]
                c_4(x1) = [1 0 2] x1 + [3]
                          [2 0 0]      [0]
                          [0 0 0]      [2]
           
           * Path {5}->{2}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                round(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                round^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {round^#(0()) -> c_1()}
               Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [0]
                        [0 0 0]      [2]
                        [0 0 0]      [2]
                0() = [2]
                      [2]
                      [2]
                round^#(x1) = [2 0 0] x1 + [0]
                              [0 0 2]      [4]
                              [1 2 2]      [0]
                c_1() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [7]
                          [0 0 0]      [6]
           
           * Path {5}->{3}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                round(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                round^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {round^#(0()) -> c_2()}
               Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [0]
                        [0 0 0]      [2]
                        [0 0 0]      [2]
                0() = [2]
                      [2]
                      [2]
                round^#(x1) = [2 0 0] x1 + [0]
                              [0 0 2]      [4]
                              [1 2 2]      [0]
                c_2() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [7]
                          [0 0 0]      [6]
           
           * Path {5}->{4}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                round(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                round^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {round^#(s(0())) -> c_3()}
               Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [2]
                        [0 1 2]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                round^#(x1) = [2 3 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 5 0]      [0]
                c_3() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(s(x), x) -> c_0(f^#(s(x), round(s(x))))
              , 2: round^#(0()) -> c_1()
              , 3: round^#(0()) -> c_2()
              , 4: round^#(s(0())) -> c_3()
              , 5: round^#(s(s(x))) -> c_4(round^#(x))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                |->{2}                                                   [   YES(?,O(n^1))    ]
                |
                |->{3}                                                   [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [   YES(?,O(n^2))    ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules for this path are:
             
               {  round(0()) -> 0()
                , round(0()) -> s(0())
                , round(s(0())) -> s(0())
                , round(s(s(x))) -> s(s(round(x)))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  f^#(s(x), x) -> c_0(f^#(s(x), round(s(x))))
                  , round(0()) -> 0()
                  , round(0()) -> s(0())
                  , round(s(0())) -> s(0())
                  , round(s(s(x))) -> s(s(round(x)))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {5}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 0]      [0]
                round(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                0() = [0]
                      [0]
                f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                round^#(x1) = [0 0] x1 + [0]
                              [3 3]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [1]
                round^#(x1) = [2 2] x1 + [2]
                              [6 0]      [0]
                c_4(x1) = [1 0] x1 + [5]
                          [2 0]      [7]
           
           * Path {5}->{2}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                round(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                0() = [0]
                      [0]
                f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                round^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {round^#(0()) -> c_1()}
               Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [0]
                0() = [0]
                      [2]
                round^#(x1) = [2 2] x1 + [4]
                              [6 2]      [0]
                c_1() = [1]
                        [0]
                c_4(x1) = [1 0] x1 + [3]
                          [2 0]      [3]
           
           * Path {5}->{3}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                round(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                0() = [0]
                      [0]
                f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                round^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {round^#(0()) -> c_2()}
               Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [0]
                0() = [0]
                      [2]
                round^#(x1) = [2 2] x1 + [4]
                              [6 2]      [0]
                c_2() = [1]
                        [0]
                c_4(x1) = [1 0] x1 + [3]
                          [2 0]      [3]
           
           * Path {5}->{4}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                round(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                0() = [0]
                      [0]
                f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                round^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {round^#(s(0())) -> c_3()}
               Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1] x1 + [2]
                        [0 1]      [1]
                0() = [1]
                      [0]
                round^#(x1) = [2 1] x1 + [1]
                              [0 0]      [7]
                c_3() = [1]
                        [1]
                c_4(x1) = [1 1] x1 + [3]
                          [0 0]      [3]
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(s(x), x) -> c_0(f^#(s(x), round(s(x))))
              , 2: round^#(0()) -> c_1()
              , 3: round^#(0()) -> c_2()
              , 4: round^#(s(0())) -> c_3()
              , 5: round^#(s(s(x))) -> c_4(round^#(x))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                |->{2}                                                   [   YES(?,O(n^1))    ]
                |
                |->{3}                                                   [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [    YES(?,O(1))     ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules for this path are:
             
               {  round(0()) -> 0()
                , round(0()) -> s(0())
                , round(s(0())) -> s(0())
                , round(s(s(x))) -> s(s(round(x)))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  f^#(s(x), x) -> c_0(f^#(s(x), round(s(x))))
                  , round(0()) -> 0()
                  , round(0()) -> s(0())
                  , round(s(0())) -> s(0())
                  , round(s(s(x))) -> s(s(round(x)))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {5}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [1] x1 + [0]
                round(x1) = [0] x1 + [0]
                0() = [0]
                f^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                round^#(x1) = [3] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                round^#(x1) = [2] x1 + [0]
                c_4(x1) = [1] x1 + [7]
           
           * Path {5}->{2}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                round(x1) = [0] x1 + [0]
                0() = [0]
                f^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                round^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {round^#(0()) -> c_1()}
               Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                0() = [2]
                round^#(x1) = [2] x1 + [4]
                c_1() = [1]
                c_4(x1) = [1] x1 + [0]
           
           * Path {5}->{3}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                round(x1) = [0] x1 + [0]
                0() = [0]
                f^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                round^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {round^#(0()) -> c_2()}
               Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                0() = [2]
                round^#(x1) = [2] x1 + [4]
                c_2() = [1]
                c_4(x1) = [1] x1 + [0]
           
           * Path {5}->{4}: YES(?,O(1))
             --------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(s) = {}, Uargs(round) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                round(x1) = [0] x1 + [0]
                0() = [0]
                f^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                round^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {round^#(s(0())) -> c_3()}
               Weak Rules: {round^#(s(s(x))) -> c_4(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0] x1 + [0]
                0() = [0]
                round^#(x1) = [0] x1 + [1]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
    
    4) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.