Problem AProVE 07 thiemann19

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputAProVE 07 thiemann19

stdout:

MAYBE

Problem:
 1024() -> 1024_1(0())
 1024_1(x) -> if(lt(x,10()),x)
 if(true(),x) -> double(1024_1(s(x)))
 if(false(),x) -> s(0())
 lt(0(),s(y)) -> true()
 lt(x,0()) -> false()
 lt(s(x),s(y)) -> lt(x,y)
 double(0()) -> 0()
 double(s(x)) -> s(s(double(x)))
 10() -> double(s(double(s(s(0())))))

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputAProVE 07 thiemann19

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputAProVE 07 thiemann19

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  1024() -> 1024_1(0())
     , 1024_1(x) -> if(lt(x, 10()), x)
     , if(true(), x) -> double(1024_1(s(x)))
     , if(false(), x) -> s(0())
     , lt(0(), s(y)) -> true()
     , lt(x, 0()) -> false()
     , lt(s(x), s(y)) -> lt(x, y)
     , double(0()) -> 0()
     , double(s(x)) -> s(s(double(x)))
     , 10() -> double(s(double(s(s(0())))))}

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: 1024^#() -> c_0(1024_1^#(0()))
              , 2: 1024_1^#(x) -> c_1(if^#(lt(x, 10()), x))
              , 3: if^#(true(), x) -> c_2(double^#(1024_1(s(x))))
              , 4: if^#(false(), x) -> c_3()
              , 5: lt^#(0(), s(y)) -> c_4()
              , 6: lt^#(x, 0()) -> c_5()
              , 7: lt^#(s(x), s(y)) -> c_6(lt^#(x, y))
              , 8: double^#(0()) -> c_7()
              , 9: double^#(s(x)) -> c_8(double^#(x))
              , 10: 10^#() -> c_9(double^#(s(double(s(s(0()))))))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{10}                                                      [         NA         ]
                |
                `->{9}                                                   [         NA         ]
                    |
                    `->{8}                                               [         NA         ]
             
             ->{7}                                                       [   YES(?,O(n^2))    ]
                |
                |->{5}                                                   [   YES(?,O(n^2))    ]
                |
                `->{6}                                                   [   YES(?,O(n^2))    ]
             
             ->{1}                                                       [     inherited      ]
                |
                `->{2}                                                   [     inherited      ]
                    |
                    |->{3}                                               [     inherited      ]
                    |   |
                    |   |->{8}                                           [         NA         ]
                    |   |
                    |   `->{9}                                           [     inherited      ]
                    |       |
                    |       `->{8}                                       [         NA         ]
                    |
                    `->{4}                                               [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
           
           * Path {1}->{2}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
           
           * Path {1}->{2}->{3}: inherited
             -----------------------------
             
             This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
           
           * Path {1}->{2}->{3}->{8}: NA
             ---------------------------
             
             The usable rules for this path are:
             
               {  lt(0(), s(y)) -> true()
                , lt(x, 0()) -> false()
                , lt(s(x), s(y)) -> lt(x, y)
                , 10() -> double(s(double(s(s(0())))))
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , 1024_1(x) -> if(lt(x, 10()), x)
                , if(true(), x) -> double(1024_1(s(x)))
                , if(false(), x) -> s(0())}
             
             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 {1}->{2}->{3}->{9}: inherited
             ----------------------------------
             
             This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
           
           * Path {1}->{2}->{3}->{9}->{8}: NA
             --------------------------------
             
             The usable rules for this path are:
             
               {  lt(0(), s(y)) -> true()
                , lt(x, 0()) -> false()
                , lt(s(x), s(y)) -> lt(x, y)
                , 10() -> double(s(double(s(s(0())))))
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , 1024_1(x) -> if(lt(x, 10()), x)
                , if(true(), x) -> double(1024_1(s(x)))
                , if(false(), x) -> s(0())}
             
             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 {1}->{2}->{4}: MAYBE
             -------------------------
             
             The usable rules for this path are:
             
               {  lt(0(), s(y)) -> true()
                , lt(x, 0()) -> false()
                , lt(s(x), s(y)) -> lt(x, y)
                , 10() -> double(s(double(s(s(0())))))
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(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:
                 {  1024_1^#(x) -> c_1(if^#(lt(x, 10()), x))
                  , 1024^#() -> c_0(1024_1^#(0()))
                  , if^#(false(), x) -> c_3()
                  , lt(0(), s(y)) -> true()
                  , lt(x, 0()) -> false()
                  , lt(s(x), s(y)) -> lt(x, y)
                  , 10() -> double(s(double(s(s(0())))))
                  , double(0()) -> 0()
                  , double(s(x)) -> s(s(double(x)))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {7}: 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(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                         [0]
                1024_1(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                0() = [0]
                      [0]
                if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                10() = [0]
                       [0]
                true() = [0]
                         [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                1024^#() = [0]
                           [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                1024_1^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_3() = [0]
                        [0]
                lt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                               [3 3]      [3 3]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                10^#() = [0]
                         [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [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: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                lt^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                               [0 2]      [0 0]      [0]
                c_6(x1) = [1 2] x1 + [5]
                          [0 0]      [3]
           
           * Path {7}->{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(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                         [0]
                1024_1(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                0() = [0]
                      [0]
                if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                10() = [0]
                       [0]
                true() = [0]
                         [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                1024^#() = [0]
                           [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                1024_1^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_3() = [0]
                        [0]
                lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                10^#() = [0]
                         [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [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: {lt^#(0(), s(y)) -> c_4()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                s(x1) = [1 6] x1 + [2]
                        [0 1]      [2]
                lt^#(x1, x2) = [2 1] x1 + [0 1] x2 + [0]
                               [1 2]      [2 0]      [0]
                c_4() = [1]
                        [0]
                c_6(x1) = [1 0] x1 + [7]
                          [0 0]      [7]
           
           * Path {7}->{6}: 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(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                         [0]
                1024_1(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                0() = [0]
                      [0]
                if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                10() = [0]
                       [0]
                true() = [0]
                         [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                1024^#() = [0]
                           [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                1024_1^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_3() = [0]
                        [0]
                lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                10^#() = [0]
                         [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [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: {lt^#(x, 0()) -> c_5()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [0]
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [0]
                lt^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
                               [0 0]      [4 1]      [0]
                c_5() = [1]
                        [0]
                c_6(x1) = [1 0] x1 + [6]
                          [0 0]      [7]
           
           * Path {10}: NA
             -------------
             
             The usable rules for this path are:
             
               {  double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                         [0]
                1024_1(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                0() = [2]
                      [0]
                if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                10() = [0]
                       [0]
                true() = [0]
                         [0]
                double(x1) = [3 0] x1 + [0]
                             [0 0]      [0]
                s(x1) = [1 0] x1 + [2]
                        [0 0]      [0]
                false() = [0]
                          [0]
                1024^#() = [0]
                           [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                1024_1^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [3 0] x1 + [0]
                               [0 0]      [0]
                c_3() = [0]
                        [0]
                lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                10^#() = [0]
                         [0]
                c_9(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{9}: NA
             ------------------
             
             The usable rules for this path are:
             
               {  double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_9) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                         [0]
                1024_1(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                0() = [2]
                      [0]
                if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                10() = [0]
                       [0]
                true() = [0]
                         [0]
                double(x1) = [3 0] x1 + [0]
                             [0 0]      [0]
                s(x1) = [1 0] x1 + [3]
                        [0 0]      [0]
                false() = [0]
                          [0]
                1024^#() = [0]
                           [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                1024_1^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [3 0] x1 + [0]
                               [3 3]      [0]
                c_3() = [0]
                        [0]
                lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                10^#() = [0]
                         [0]
                c_9(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{9}->{8}: NA
             -----------------------
             
             The usable rules for this path are:
             
               {  double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_9) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                         [0]
                1024_1(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                0() = [2]
                      [0]
                if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                10() = [0]
                       [0]
                true() = [0]
                         [0]
                double(x1) = [3 0] x1 + [0]
                             [0 0]      [0]
                s(x1) = [1 0] x1 + [3]
                        [0 0]      [0]
                false() = [0]
                          [0]
                1024^#() = [0]
                           [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                1024_1^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [3 0] x1 + [0]
                               [0 0]      [0]
                c_3() = [0]
                        [0]
                lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                10^#() = [0]
                         [0]
                c_9(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: 1024^#() -> c_0(1024_1^#(0()))
              , 2: 1024_1^#(x) -> c_1(if^#(lt(x, 10()), x))
              , 3: if^#(true(), x) -> c_2(double^#(1024_1(s(x))))
              , 4: if^#(false(), x) -> c_3()
              , 5: lt^#(0(), s(y)) -> c_4()
              , 6: lt^#(x, 0()) -> c_5()
              , 7: lt^#(s(x), s(y)) -> c_6(lt^#(x, y))
              , 8: double^#(0()) -> c_7()
              , 9: double^#(s(x)) -> c_8(double^#(x))
              , 10: 10^#() -> c_9(double^#(s(double(s(s(0()))))))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{10}                                                      [   YES(?,O(n^1))    ]
                |
                `->{9}                                                   [         NA         ]
                    |
                    `->{8}                                               [         NA         ]
             
             ->{7}                                                       [   YES(?,O(n^1))    ]
                |
                |->{5}                                                   [   YES(?,O(n^1))    ]
                |
                `->{6}                                                   [   YES(?,O(n^1))    ]
             
             ->{1}                                                       [     inherited      ]
                |
                `->{2}                                                   [     inherited      ]
                    |
                    |->{3}                                               [     inherited      ]
                    |   |
                    |   |->{8}                                           [         NA         ]
                    |   |
                    |   `->{9}                                           [     inherited      ]
                    |       |
                    |       `->{8}                                       [         NA         ]
                    |
                    `->{4}                                               [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
           
           * Path {1}->{2}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
           
           * Path {1}->{2}->{3}: inherited
             -----------------------------
             
             This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
           
           * Path {1}->{2}->{3}->{8}: NA
             ---------------------------
             
             The usable rules for this path are:
             
               {  lt(0(), s(y)) -> true()
                , lt(x, 0()) -> false()
                , lt(s(x), s(y)) -> lt(x, y)
                , 10() -> double(s(double(s(s(0())))))
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , 1024_1(x) -> if(lt(x, 10()), x)
                , if(true(), x) -> double(1024_1(s(x)))
                , if(false(), x) -> s(0())}
             
             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 {1}->{2}->{3}->{9}: inherited
             ----------------------------------
             
             This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
           
           * Path {1}->{2}->{3}->{9}->{8}: NA
             --------------------------------
             
             The usable rules for this path are:
             
               {  lt(0(), s(y)) -> true()
                , lt(x, 0()) -> false()
                , lt(s(x), s(y)) -> lt(x, y)
                , 10() -> double(s(double(s(s(0())))))
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , 1024_1(x) -> if(lt(x, 10()), x)
                , if(true(), x) -> double(1024_1(s(x)))
                , if(false(), x) -> s(0())}
             
             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 {1}->{2}->{4}: MAYBE
             -------------------------
             
             The usable rules for this path are:
             
               {  lt(0(), s(y)) -> true()
                , lt(x, 0()) -> false()
                , lt(s(x), s(y)) -> lt(x, y)
                , 10() -> double(s(double(s(s(0())))))
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(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:
                 {  1024_1^#(x) -> c_1(if^#(lt(x, 10()), x))
                  , 1024^#() -> c_0(1024_1^#(0()))
                  , if^#(false(), x) -> c_3()
                  , lt(0(), s(y)) -> true()
                  , lt(x, 0()) -> false()
                  , lt(s(x), s(y)) -> lt(x, y)
                  , 10() -> double(s(double(s(s(0())))))
                  , double(0()) -> 0()
                  , double(s(x)) -> s(s(double(x)))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {7}: 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(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                1024_1(x1) = [0] x1 + [0]
                0() = [0]
                if(x1, x2) = [0] x1 + [0] x2 + [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                10() = [0]
                true() = [0]
                double(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                false() = [0]
                1024^#() = [0]
                c_0(x1) = [0] x1 + [0]
                1024_1^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                if^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                c_3() = [0]
                lt^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                10^#() = [0]
                c_9(x1) = [0] 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: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                lt^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_6(x1) = [1] x1 + [7]
           
           * Path {7}->{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(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                1024_1(x1) = [0] x1 + [0]
                0() = [0]
                if(x1, x2) = [0] x1 + [0] x2 + [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                10() = [0]
                true() = [0]
                double(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                false() = [0]
                1024^#() = [0]
                c_0(x1) = [0] x1 + [0]
                1024_1^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                if^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                c_3() = [0]
                lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                10^#() = [0]
                c_9(x1) = [0] 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: {lt^#(0(), s(y)) -> c_4()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                lt^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_4() = [1]
                c_6(x1) = [1] x1 + [7]
           
           * Path {7}->{6}: 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(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                1024_1(x1) = [0] x1 + [0]
                0() = [0]
                if(x1, x2) = [0] x1 + [0] x2 + [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                10() = [0]
                true() = [0]
                double(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                false() = [0]
                1024^#() = [0]
                c_0(x1) = [0] x1 + [0]
                1024_1^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                if^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                c_3() = [0]
                lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                10^#() = [0]
                c_9(x1) = [0] 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: {lt^#(x, 0()) -> c_5()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                lt^#(x1, x2) = [2] x1 + [2] x2 + [4]
                c_5() = [1]
                c_6(x1) = [1] x1 + [7]
           
           * Path {10}: YES(?,O(n^1))
             ------------------------
             
             The usable rules for this path are:
             
               {  double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                1024_1(x1) = [0] x1 + [0]
                0() = [1]
                if(x1, x2) = [0] x1 + [0] x2 + [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                10() = [0]
                true() = [0]
                double(x1) = [3] x1 + [0]
                s(x1) = [1] x1 + [1]
                false() = [0]
                1024^#() = [0]
                c_0(x1) = [0] x1 + [0]
                1024_1^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                if^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [3] x1 + [0]
                c_3() = [0]
                lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                10^#() = [0]
                c_9(x1) = [1] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             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: {10^#() -> c_9(double^#(s(double(s(s(0()))))))}
               Weak Rules:
                 {  double(0()) -> 0()
                  , double(s(x)) -> s(s(double(x)))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(double^#) = {},
                 Uargs(c_9) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                double(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                double^#(x1) = [2] x1 + [0]
                10^#() = [7]
                c_9(x1) = [2] x1 + [3]
           
           * Path {10}->{9}: NA
             ------------------
             
             The usable rules for this path are:
             
               {  double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_9) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                1024_1(x1) = [0] x1 + [0]
                0() = [1]
                if(x1, x2) = [0] x1 + [0] x2 + [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                10() = [0]
                true() = [0]
                double(x1) = [3] x1 + [0]
                s(x1) = [1] x1 + [1]
                false() = [0]
                1024^#() = [0]
                c_0(x1) = [0] x1 + [0]
                1024_1^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                if^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [3] x1 + [0]
                c_3() = [0]
                lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
                10^#() = [0]
                c_9(x1) = [1] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{9}->{8}: NA
             -----------------------
             
             The usable rules for this path are:
             
               {  double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_9) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                1024_1(x1) = [0] x1 + [0]
                0() = [1]
                if(x1, x2) = [0] x1 + [0] x2 + [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                10() = [0]
                true() = [0]
                double(x1) = [3] x1 + [0]
                s(x1) = [1] x1 + [1]
                false() = [0]
                1024^#() = [0]
                c_0(x1) = [0] x1 + [0]
                1024_1^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                if^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [3] x1 + [0]
                c_3() = [0]
                lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
                10^#() = [0]
                c_9(x1) = [1] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
    
    3) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    5) '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 07 thiemann19

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputAProVE 07 thiemann19

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  1024() -> 1024_1(0())
     , 1024_1(x) -> if(lt(x, 10()), x)
     , if(true(), x) -> double(1024_1(s(x)))
     , if(false(), x) -> s(0())
     , lt(0(), s(y)) -> true()
     , lt(x, 0()) -> false()
     , lt(s(x), s(y)) -> lt(x, y)
     , double(0()) -> 0()
     , double(s(x)) -> s(s(double(x)))
     , 10() -> double(s(double(s(s(0())))))}

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: 1024^#() -> c_0(1024_1^#(0()))
              , 2: 1024_1^#(x) -> c_1(if^#(lt(x, 10()), x))
              , 3: if^#(true(), x) -> c_2(double^#(1024_1(s(x))))
              , 4: if^#(false(), x) -> c_3()
              , 5: lt^#(0(), s(y)) -> c_4()
              , 6: lt^#(x, 0()) -> c_5()
              , 7: lt^#(s(x), s(y)) -> c_6(lt^#(x, y))
              , 8: double^#(0()) -> c_7()
              , 9: double^#(s(x)) -> c_8(double^#(x))
              , 10: 10^#() -> c_9(double^#(s(double(s(s(0()))))))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{10}                                                      [         NA         ]
                |
                `->{9}                                                   [         NA         ]
                    |
                    `->{8}                                               [         NA         ]
             
             ->{7}                                                       [   YES(?,O(n^2))    ]
                |
                |->{5}                                                   [   YES(?,O(n^2))    ]
                |
                `->{6}                                                   [   YES(?,O(n^2))    ]
             
             ->{1}                                                       [     inherited      ]
                |
                `->{2}                                                   [     inherited      ]
                    |
                    |->{3}                                               [     inherited      ]
                    |   |
                    |   |->{8}                                           [         NA         ]
                    |   |
                    |   `->{9}                                           [     inherited      ]
                    |       |
                    |       `->{8}                                       [         NA         ]
                    |
                    `->{4}                                               [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
           
           * Path {1}->{2}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
           
           * Path {1}->{2}->{3}: inherited
             -----------------------------
             
             This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
           
           * Path {1}->{2}->{3}->{8}: NA
             ---------------------------
             
             The usable rules for this path are:
             
               {  lt(0(), s(y)) -> true()
                , lt(x, 0()) -> false()
                , lt(s(x), s(y)) -> lt(x, y)
                , 10() -> double(s(double(s(s(0())))))
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , 1024_1(x) -> if(lt(x, 10()), x)
                , if(true(), x) -> double(1024_1(s(x)))
                , if(false(), x) -> s(0())}
             
             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 {1}->{2}->{3}->{9}: inherited
             ----------------------------------
             
             This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
           
           * Path {1}->{2}->{3}->{9}->{8}: NA
             --------------------------------
             
             The usable rules for this path are:
             
               {  lt(0(), s(y)) -> true()
                , lt(x, 0()) -> false()
                , lt(s(x), s(y)) -> lt(x, y)
                , 10() -> double(s(double(s(s(0())))))
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , 1024_1(x) -> if(lt(x, 10()), x)
                , if(true(), x) -> double(1024_1(s(x)))
                , if(false(), x) -> s(0())}
             
             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 {1}->{2}->{4}: MAYBE
             -------------------------
             
             The usable rules for this path are:
             
               {  lt(0(), s(y)) -> true()
                , lt(x, 0()) -> false()
                , lt(s(x), s(y)) -> lt(x, y)
                , 10() -> double(s(double(s(s(0())))))
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(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:
                 {  1024_1^#(x) -> c_1(if^#(lt(x, 10()), x))
                  , 1024^#() -> c_0(1024_1^#(0()))
                  , if^#(false(), x) -> c_3()
                  , lt(0(), s(y)) -> true()
                  , lt(x, 0()) -> false()
                  , lt(s(x), s(y)) -> lt(x, y)
                  , 10() -> double(s(double(s(s(0())))))
                  , double(0()) -> 0()
                  , double(s(x)) -> s(s(double(x)))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {7}: 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(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                         [0]
                1024_1(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                0() = [0]
                      [0]
                if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                10() = [0]
                       [0]
                true() = [0]
                         [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                1024^#() = [0]
                           [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                1024_1^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_3() = [0]
                        [0]
                lt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                               [3 3]      [3 3]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                10^#() = [0]
                         [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [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: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                lt^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                               [0 2]      [0 0]      [0]
                c_6(x1) = [1 2] x1 + [5]
                          [0 0]      [3]
           
           * Path {7}->{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(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                         [0]
                1024_1(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                0() = [0]
                      [0]
                if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                10() = [0]
                       [0]
                true() = [0]
                         [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                1024^#() = [0]
                           [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                1024_1^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_3() = [0]
                        [0]
                lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                10^#() = [0]
                         [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [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: {lt^#(0(), s(y)) -> c_4()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                s(x1) = [1 6] x1 + [2]
                        [0 1]      [2]
                lt^#(x1, x2) = [2 1] x1 + [0 1] x2 + [0]
                               [1 2]      [2 0]      [0]
                c_4() = [1]
                        [0]
                c_6(x1) = [1 0] x1 + [7]
                          [0 0]      [7]
           
           * Path {7}->{6}: 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(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                         [0]
                1024_1(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                0() = [0]
                      [0]
                if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                10() = [0]
                       [0]
                true() = [0]
                         [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                false() = [0]
                          [0]
                1024^#() = [0]
                           [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                1024_1^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_3() = [0]
                        [0]
                lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                10^#() = [0]
                         [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [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: {lt^#(x, 0()) -> c_5()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [0]
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [0]
                lt^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
                               [0 0]      [4 1]      [0]
                c_5() = [1]
                        [0]
                c_6(x1) = [1 0] x1 + [6]
                          [0 0]      [7]
           
           * Path {10}: NA
             -------------
             
             The usable rules for this path are:
             
               {  double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                         [0]
                1024_1(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                0() = [2]
                      [0]
                if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                10() = [0]
                       [0]
                true() = [0]
                         [0]
                double(x1) = [3 0] x1 + [0]
                             [0 0]      [0]
                s(x1) = [1 0] x1 + [2]
                        [0 0]      [0]
                false() = [0]
                          [0]
                1024^#() = [0]
                           [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                1024_1^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [3 0] x1 + [0]
                               [0 0]      [0]
                c_3() = [0]
                        [0]
                lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                10^#() = [0]
                         [0]
                c_9(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{9}: NA
             ------------------
             
             The usable rules for this path are:
             
               {  double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_9) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                         [0]
                1024_1(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                0() = [2]
                      [0]
                if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                10() = [0]
                       [0]
                true() = [0]
                         [0]
                double(x1) = [3 0] x1 + [0]
                             [0 0]      [0]
                s(x1) = [1 0] x1 + [3]
                        [0 0]      [0]
                false() = [0]
                          [0]
                1024^#() = [0]
                           [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                1024_1^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [3 0] x1 + [0]
                               [3 3]      [0]
                c_3() = [0]
                        [0]
                lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                10^#() = [0]
                         [0]
                c_9(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{9}->{8}: NA
             -----------------------
             
             The usable rules for this path are:
             
               {  double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_9) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                         [0]
                1024_1(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                0() = [2]
                      [0]
                if(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                10() = [0]
                       [0]
                true() = [0]
                         [0]
                double(x1) = [3 0] x1 + [0]
                             [0 0]      [0]
                s(x1) = [1 0] x1 + [3]
                        [0 0]      [0]
                false() = [0]
                          [0]
                1024^#() = [0]
                           [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                1024_1^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [3 0] x1 + [0]
                               [0 0]      [0]
                c_3() = [0]
                        [0]
                lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                10^#() = [0]
                         [0]
                c_9(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: 1024^#() -> c_0(1024_1^#(0()))
              , 2: 1024_1^#(x) -> c_1(if^#(lt(x, 10()), x))
              , 3: if^#(true(), x) -> c_2(double^#(1024_1(s(x))))
              , 4: if^#(false(), x) -> c_3()
              , 5: lt^#(0(), s(y)) -> c_4()
              , 6: lt^#(x, 0()) -> c_5()
              , 7: lt^#(s(x), s(y)) -> c_6(lt^#(x, y))
              , 8: double^#(0()) -> c_7()
              , 9: double^#(s(x)) -> c_8(double^#(x))
              , 10: 10^#() -> c_9(double^#(s(double(s(s(0()))))))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{10}                                                      [   YES(?,O(n^1))    ]
                |
                `->{9}                                                   [         NA         ]
                    |
                    `->{8}                                               [         NA         ]
             
             ->{7}                                                       [   YES(?,O(n^1))    ]
                |
                |->{5}                                                   [   YES(?,O(n^1))    ]
                |
                `->{6}                                                   [   YES(?,O(n^1))    ]
             
             ->{1}                                                       [     inherited      ]
                |
                `->{2}                                                   [     inherited      ]
                    |
                    |->{3}                                               [     inherited      ]
                    |   |
                    |   |->{8}                                           [         NA         ]
                    |   |
                    |   `->{9}                                           [     inherited      ]
                    |       |
                    |       `->{8}                                       [         NA         ]
                    |
                    `->{4}                                               [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
           
           * Path {1}->{2}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
           
           * Path {1}->{2}->{3}: inherited
             -----------------------------
             
             This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
           
           * Path {1}->{2}->{3}->{8}: NA
             ---------------------------
             
             The usable rules for this path are:
             
               {  lt(0(), s(y)) -> true()
                , lt(x, 0()) -> false()
                , lt(s(x), s(y)) -> lt(x, y)
                , 10() -> double(s(double(s(s(0())))))
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , 1024_1(x) -> if(lt(x, 10()), x)
                , if(true(), x) -> double(1024_1(s(x)))
                , if(false(), x) -> s(0())}
             
             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 {1}->{2}->{3}->{9}: inherited
             ----------------------------------
             
             This path is subsumed by the proof of path {1}->{2}->{3}->{9}->{8}.
           
           * Path {1}->{2}->{3}->{9}->{8}: NA
             --------------------------------
             
             The usable rules for this path are:
             
               {  lt(0(), s(y)) -> true()
                , lt(x, 0()) -> false()
                , lt(s(x), s(y)) -> lt(x, y)
                , 10() -> double(s(double(s(s(0())))))
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , 1024_1(x) -> if(lt(x, 10()), x)
                , if(true(), x) -> double(1024_1(s(x)))
                , if(false(), x) -> s(0())}
             
             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 {1}->{2}->{4}: MAYBE
             -------------------------
             
             The usable rules for this path are:
             
               {  lt(0(), s(y)) -> true()
                , lt(x, 0()) -> false()
                , lt(s(x), s(y)) -> lt(x, y)
                , 10() -> double(s(double(s(s(0())))))
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(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:
                 {  1024_1^#(x) -> c_1(if^#(lt(x, 10()), x))
                  , 1024^#() -> c_0(1024_1^#(0()))
                  , if^#(false(), x) -> c_3()
                  , lt(0(), s(y)) -> true()
                  , lt(x, 0()) -> false()
                  , lt(s(x), s(y)) -> lt(x, y)
                  , 10() -> double(s(double(s(s(0())))))
                  , double(0()) -> 0()
                  , double(s(x)) -> s(s(double(x)))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {7}: 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(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                1024_1(x1) = [0] x1 + [0]
                0() = [0]
                if(x1, x2) = [0] x1 + [0] x2 + [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                10() = [0]
                true() = [0]
                double(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                false() = [0]
                1024^#() = [0]
                c_0(x1) = [0] x1 + [0]
                1024_1^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                if^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                c_3() = [0]
                lt^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                10^#() = [0]
                c_9(x1) = [0] 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: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                lt^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_6(x1) = [1] x1 + [7]
           
           * Path {7}->{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(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                1024_1(x1) = [0] x1 + [0]
                0() = [0]
                if(x1, x2) = [0] x1 + [0] x2 + [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                10() = [0]
                true() = [0]
                double(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                false() = [0]
                1024^#() = [0]
                c_0(x1) = [0] x1 + [0]
                1024_1^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                if^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                c_3() = [0]
                lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                10^#() = [0]
                c_9(x1) = [0] 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: {lt^#(0(), s(y)) -> c_4()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                lt^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_4() = [1]
                c_6(x1) = [1] x1 + [7]
           
           * Path {7}->{6}: 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(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(lt^#) = {},
                 Uargs(c_6) = {1}, Uargs(c_8) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                1024_1(x1) = [0] x1 + [0]
                0() = [0]
                if(x1, x2) = [0] x1 + [0] x2 + [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                10() = [0]
                true() = [0]
                double(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                false() = [0]
                1024^#() = [0]
                c_0(x1) = [0] x1 + [0]
                1024_1^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                if^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                c_3() = [0]
                lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                10^#() = [0]
                c_9(x1) = [0] 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: {lt^#(x, 0()) -> c_5()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_6(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                lt^#(x1, x2) = [2] x1 + [2] x2 + [4]
                c_5() = [1]
                c_6(x1) = [1] x1 + [7]
           
           * Path {10}: YES(?,O(n^1))
             ------------------------
             
             The usable rules for this path are:
             
               {  double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {}, Uargs(c_9) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                1024_1(x1) = [0] x1 + [0]
                0() = [1]
                if(x1, x2) = [0] x1 + [0] x2 + [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                10() = [0]
                true() = [0]
                double(x1) = [3] x1 + [0]
                s(x1) = [1] x1 + [1]
                false() = [0]
                1024^#() = [0]
                c_0(x1) = [0] x1 + [0]
                1024_1^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                if^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [3] x1 + [0]
                c_3() = [0]
                lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
                10^#() = [0]
                c_9(x1) = [1] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             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: {10^#() -> c_9(double^#(s(double(s(s(0()))))))}
               Weak Rules:
                 {  double(0()) -> 0()
                  , double(s(x)) -> s(s(double(x)))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(double^#) = {},
                 Uargs(c_9) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                double(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                double^#(x1) = [2] x1 + [0]
                10^#() = [7]
                c_9(x1) = [2] x1 + [3]
           
           * Path {10}->{9}: NA
             ------------------
             
             The usable rules for this path are:
             
               {  double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_9) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                1024_1(x1) = [0] x1 + [0]
                0() = [1]
                if(x1, x2) = [0] x1 + [0] x2 + [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                10() = [0]
                true() = [0]
                double(x1) = [3] x1 + [0]
                s(x1) = [1] x1 + [1]
                false() = [0]
                1024^#() = [0]
                c_0(x1) = [0] x1 + [0]
                1024_1^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                if^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [3] x1 + [0]
                c_3() = [0]
                lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
                10^#() = [0]
                c_9(x1) = [1] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {10}->{9}->{8}: NA
             -----------------------
             
             The usable rules for this path are:
             
               {  double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(1024_1) = {}, Uargs(if) = {}, Uargs(lt) = {},
                 Uargs(double) = {}, Uargs(s) = {1}, Uargs(c_0) = {},
                 Uargs(1024_1^#) = {}, Uargs(c_1) = {}, Uargs(if^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {1}, Uargs(lt^#) = {},
                 Uargs(c_6) = {}, Uargs(c_8) = {1}, Uargs(c_9) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                1024() = [0]
                1024_1(x1) = [0] x1 + [0]
                0() = [1]
                if(x1, x2) = [0] x1 + [0] x2 + [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                10() = [0]
                true() = [0]
                double(x1) = [3] x1 + [0]
                s(x1) = [1] x1 + [1]
                false() = [0]
                1024^#() = [0]
                c_0(x1) = [0] x1 + [0]
                1024_1^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                if^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [3] x1 + [0]
                c_3() = [0]
                lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
                10^#() = [0]
                c_9(x1) = [1] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
    
    3) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.