Problem Transformed CSR 04 Ex4 7 56 Bor03 FR

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputTransformed CSR 04 Ex4 7 56 Bor03 FR

stdout:

MAYBE

Problem:
 from(X) -> cons(X,n__from(n__s(X)))
 after(0(),XS) -> XS
 after(s(N),cons(X,XS)) -> after(N,activate(XS))
 from(X) -> n__from(X)
 s(X) -> n__s(X)
 activate(n__from(X)) -> from(activate(X))
 activate(n__s(X)) -> s(activate(X))
 activate(X) -> X

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputTransformed CSR 04 Ex4 7 56 Bor03 FR

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
YES(?,O(n^2))
InputTransformed CSR 04 Ex4 7 56 Bor03 FR

stdout:

YES(?,O(n^2))

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           YES(?,O(n^2))
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  from(X) -> cons(X, n__from(n__s(X)))
     , after(0(), XS) -> XS
     , after(s(N), cons(X, XS)) -> after(N, activate(XS))
     , from(X) -> n__from(X)
     , s(X) -> n__s(X)
     , activate(n__from(X)) -> from(activate(X))
     , activate(n__s(X)) -> s(activate(X))
     , activate(X) -> X}

Proof Output:    
  'wdg' proved the best result:
  
  Details:
  --------
    'wdg' succeeded with the following output:
     'wdg'
     -----
     Answer:           YES(?,O(n^2))
     Input Problem:    innermost runtime-complexity with respect to
       Rules:
         {  from(X) -> cons(X, n__from(n__s(X)))
          , after(0(), XS) -> XS
          , after(s(N), cons(X, XS)) -> after(N, activate(XS))
          , from(X) -> n__from(X)
          , s(X) -> n__s(X)
          , activate(n__from(X)) -> from(activate(X))
          , activate(n__s(X)) -> s(activate(X))
          , activate(X) -> X}
     
     Proof Output:    
       Transformation Details:
       -----------------------
         We have computed the following set of weak (innermost) dependency pairs:
         
           {  1: from^#(X) -> c_0()
            , 2: after^#(0(), XS) -> c_1()
            , 3: after^#(s(N), cons(X, XS)) -> c_2(after^#(N, activate(XS)))
            , 4: from^#(X) -> c_3()
            , 5: s^#(X) -> c_4()
            , 6: activate^#(n__from(X)) -> c_5(from^#(activate(X)))
            , 7: activate^#(n__s(X)) -> c_6(s^#(activate(X)))
            , 8: activate^#(X) -> c_7()}
         
         Following Dependency Graph (modulo SCCs) was computed. (Answers to
         subproofs are indicated to the right.)
         
           ->{8}                                                       [    YES(?,O(1))     ]
           
           ->{7}                                                       [   YES(?,O(n^2))    ]
              |
              `->{5}                                                   [   YES(?,O(n^2))    ]
           
           ->{6}                                                       [   YES(?,O(n^2))    ]
              |
              |->{1}                                                   [   YES(?,O(n^2))    ]
              |
              `->{4}                                                   [   YES(?,O(n^2))    ]
           
           ->{3}                                                       [   YES(?,O(n^2))    ]
              |
              `->{2}                                                   [   YES(?,O(n^2))    ]
           
         
       
       Sub-problems:
       -------------
         * Path {3}: YES(?,O(n^2))
           -----------------------
           
           The usable rules for this path are:
           
             {  activate(n__from(X)) -> from(activate(X))
              , activate(n__s(X)) -> s(activate(X))
              , activate(X) -> X
              , from(X) -> cons(X, n__from(n__s(X)))
              , from(X) -> n__from(X)
              , s(X) -> n__s(X)}
           
           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(from) = {1}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(after) = {}, Uargs(s) = {1},
               Uargs(activate) = {}, Uargs(from^#) = {}, Uargs(after^#) = {2},
               Uargs(c_2) = {1}, Uargs(s^#) = {}, Uargs(activate^#) = {},
               Uargs(c_5) = {}, Uargs(c_6) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [1 3] x1 + [1]
                         [0 1]      [3]
              cons(x1, x2) = [1 2] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [2]
              n__from(x1) = [1 3] x1 + [0]
                            [0 1]      [3]
              n__s(x1) = [1 2] x1 + [2]
                         [0 1]      [1]
              after(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
              0() = [0]
                    [0]
              s(x1) = [1 2] x1 + [3]
                      [0 1]      [1]
              activate(x1) = [1 2] x1 + [2]
                             [0 1]      [0]
              from^#(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
              c_0() = [0]
                      [0]
              after^#(x1, x2) = [0 0] x1 + [1 0] x2 + [0]
                                [3 3]      [3 3]      [0]
              c_1() = [0]
                      [0]
              c_2(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
              c_3() = [0]
                      [0]
              s^#(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_4() = [0]
                      [0]
              activate^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
              c_5(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_6(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_7() = [0]
                      [0]
           Complexity induced by the adequate RMI: YES(?,O(n^2))
           
           We apply the sub-processor on the resulting sub-problem:
           
           'matrix-interpretation of dimension 2'
           --------------------------------------
           Answer:           YES(?,O(n^1))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules:
               {after^#(s(N), cons(X, XS)) -> c_2(after^#(N, activate(XS)))}
             Weak Rules:
               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}
           
           Proof Output:    
             The following argument positions are usable:
               Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(s) = {}, Uargs(activate) = {},
               Uargs(after^#) = {}, Uargs(c_2) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0 0] x1 + [0]
                         [0 0]      [0]
              cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 1]      [0]
              n__from(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
              n__s(x1) = [1 0] x1 + [3]
                         [0 1]      [0]
              s(x1) = [1 0] x1 + [3]
                      [0 1]      [0]
              activate(x1) = [1 0] x1 + [0]
                             [0 1]      [0]
              after^#(x1, x2) = [1 0] x1 + [0 1] x2 + [1]
                                [0 0]      [0 1]      [0]
              c_2(x1) = [1 0] x1 + [1]
                        [0 0]      [0]
         
         * Path {3}->{2}: YES(?,O(n^2))
           ----------------------------
           
           The usable rules for this path are:
           
             {  activate(n__from(X)) -> from(activate(X))
              , activate(n__s(X)) -> s(activate(X))
              , activate(X) -> X
              , from(X) -> cons(X, n__from(n__s(X)))
              , from(X) -> n__from(X)
              , s(X) -> n__s(X)}
           
           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(from) = {1}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(after) = {}, Uargs(s) = {1},
               Uargs(activate) = {}, Uargs(from^#) = {}, Uargs(after^#) = {2},
               Uargs(c_2) = {1}, Uargs(s^#) = {}, Uargs(activate^#) = {},
               Uargs(c_5) = {}, Uargs(c_6) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [1 3] x1 + [1]
                         [0 1]      [2]
              cons(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
                             [0 0]      [0 1]      [0]
              n__from(x1) = [1 3] x1 + [0]
                            [0 1]      [2]
              n__s(x1) = [1 0] x1 + [2]
                         [0 1]      [0]
              after(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
              0() = [0]
                    [0]
              s(x1) = [1 0] x1 + [3]
                      [0 1]      [0]
              activate(x1) = [2 2] x1 + [2]
                             [0 1]      [0]
              from^#(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
              c_0() = [0]
                      [0]
              after^#(x1, x2) = [0 0] x1 + [3 0] x2 + [0]
                                [0 0]      [0 0]      [0]
              c_1() = [0]
                      [0]
              c_2(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
              c_3() = [0]
                      [0]
              s^#(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_4() = [0]
                      [0]
              activate^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
              c_5(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_6(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_7() = [0]
                      [0]
           Complexity induced by the adequate RMI: YES(?,O(n^2))
           
           We apply the sub-processor on the resulting sub-problem:
           
           'matrix-interpretation of dimension 2'
           --------------------------------------
           Answer:           YES(?,O(n^1))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {after^#(0(), XS) -> c_1()}
             Weak Rules:
               {  after^#(s(N), cons(X, XS)) -> c_2(after^#(N, activate(XS)))
                , activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}
           
           Proof Output:    
             The following argument positions are usable:
               Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(s) = {}, Uargs(activate) = {},
               Uargs(after^#) = {}, Uargs(c_2) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [1 0] x1 + [0]
                         [0 1]      [0]
              cons(x1, x2) = [1 0] x1 + [0 0] x2 + [0]
                             [0 1]      [0 0]      [0]
              n__from(x1) = [1 0] x1 + [0]
                            [0 1]      [0]
              n__s(x1) = [1 0] x1 + [0]
                         [0 0]      [0]
              0() = [2]
                    [0]
              s(x1) = [1 0] x1 + [0]
                      [0 0]      [0]
              activate(x1) = [1 0] x1 + [0]
                             [4 1]      [0]
              after^#(x1, x2) = [0 0] x1 + [0 0] x2 + [1]
                                [6 0]      [0 0]      [0]
              c_1() = [0]
                      [0]
              c_2(x1) = [1 0] x1 + [0]
                        [0 0]      [0]
         
         * Path {6}: YES(?,O(n^2))
           -----------------------
           
           The usable rules for this path are:
           
             {  activate(n__from(X)) -> from(activate(X))
              , activate(n__s(X)) -> s(activate(X))
              , activate(X) -> X
              , from(X) -> cons(X, n__from(n__s(X)))
              , from(X) -> n__from(X)
              , s(X) -> n__s(X)}
           
           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(from) = {1}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(after) = {}, Uargs(s) = {1},
               Uargs(activate) = {}, Uargs(from^#) = {1}, Uargs(after^#) = {},
               Uargs(c_2) = {}, Uargs(s^#) = {}, Uargs(activate^#) = {},
               Uargs(c_5) = {1}, Uargs(c_6) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [1 3] x1 + [3]
                         [0 1]      [0]
              cons(x1, x2) = [1 1] x1 + [0 0] x2 + [2]
                             [0 0]      [0 1]      [0]
              n__from(x1) = [1 3] x1 + [2]
                            [0 1]      [0]
              n__s(x1) = [1 0] x1 + [2]
                         [0 1]      [0]
              after(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
              0() = [0]
                    [0]
              s(x1) = [1 0] x1 + [3]
                      [0 1]      [0]
              activate(x1) = [2 1] x1 + [1]
                             [0 1]      [0]
              from^#(x1) = [1 0] x1 + [0]
                           [3 3]      [0]
              c_0() = [0]
                      [0]
              after^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
              c_1() = [0]
                      [0]
              c_2(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_3() = [0]
                      [0]
              s^#(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_4() = [0]
                      [0]
              activate^#(x1) = [2 2] x1 + [0]
                               [0 0]      [0]
              c_5(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
              c_6(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_7() = [0]
                      [0]
           Complexity induced by the adequate RMI: YES(?,O(n^2))
           
           We apply the sub-processor on the resulting sub-problem:
           
           'matrix-interpretation of dimension 2'
           --------------------------------------
           Answer:           YES(?,O(n^1))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {activate^#(n__from(X)) -> c_5(from^#(activate(X)))}
             Weak Rules:
               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}
           
           Proof Output:    
             The following argument positions are usable:
               Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(s) = {}, Uargs(activate) = {},
               Uargs(from^#) = {}, Uargs(activate^#) = {}, Uargs(c_5) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [1 1] x1 + [2]
                         [0 0]      [0]
              cons(x1, x2) = [1 1] x1 + [0 0] x2 + [2]
                             [0 0]      [0 0]      [0]
              n__from(x1) = [1 1] x1 + [2]
                            [0 0]      [0]
              n__s(x1) = [0 0] x1 + [0]
                         [0 0]      [0]
              s(x1) = [0 0] x1 + [0]
                      [0 0]      [0]
              activate(x1) = [1 0] x1 + [2]
                             [0 1]      [0]
              from^#(x1) = [2 0] x1 + [0]
                           [0 0]      [0]
              activate^#(x1) = [4 0] x1 + [7]
                               [4 0]      [7]
              c_5(x1) = [2 0] x1 + [3]
                        [1 0]      [6]
         
         * Path {6}->{1}: YES(?,O(n^2))
           ----------------------------
           
           The usable rules for this path are:
           
             {  activate(n__from(X)) -> from(activate(X))
              , activate(n__s(X)) -> s(activate(X))
              , activate(X) -> X
              , from(X) -> cons(X, n__from(n__s(X)))
              , from(X) -> n__from(X)
              , s(X) -> n__s(X)}
           
           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(from) = {1}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(after) = {}, Uargs(s) = {1},
               Uargs(activate) = {}, Uargs(from^#) = {1}, Uargs(after^#) = {},
               Uargs(c_2) = {}, Uargs(s^#) = {}, Uargs(activate^#) = {},
               Uargs(c_5) = {1}, Uargs(c_6) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [1 3] x1 + [1]
                         [0 1]      [2]
              cons(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
                             [0 0]      [0 1]      [0]
              n__from(x1) = [1 3] x1 + [0]
                            [0 1]      [2]
              n__s(x1) = [1 0] x1 + [2]
                         [0 1]      [0]
              after(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
              0() = [0]
                    [0]
              s(x1) = [1 0] x1 + [3]
                      [0 1]      [0]
              activate(x1) = [2 2] x1 + [2]
                             [0 1]      [0]
              from^#(x1) = [3 0] x1 + [0]
                           [0 0]      [0]
              c_0() = [0]
                      [0]
              after^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
              c_1() = [0]
                      [0]
              c_2(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_3() = [0]
                      [0]
              s^#(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_4() = [0]
                      [0]
              activate^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
              c_5(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
              c_6(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_7() = [0]
                      [0]
           Complexity induced by the adequate RMI: YES(?,O(n^2))
           
           We apply the sub-processor on the resulting sub-problem:
           
           'matrix-interpretation of dimension 2'
           --------------------------------------
           Answer:           YES(?,O(n^1))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {from^#(X) -> c_0()}
             Weak Rules:
               {  activate^#(n__from(X)) -> c_5(from^#(activate(X)))
                , activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}
           
           Proof Output:    
             The following argument positions are usable:
               Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(s) = {}, Uargs(activate) = {},
               Uargs(from^#) = {}, Uargs(activate^#) = {}, Uargs(c_5) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [1 0] x1 + [2]
                         [0 0]      [4]
              cons(x1, x2) = [1 0] x1 + [0 0] x2 + [2]
                             [0 0]      [0 0]      [1]
              n__from(x1) = [1 0] x1 + [2]
                            [0 0]      [4]
              n__s(x1) = [0 0] x1 + [4]
                         [0 0]      [0]
              s(x1) = [0 0] x1 + [4]
                      [0 0]      [0]
              activate(x1) = [1 0] x1 + [0]
                             [2 1]      [0]
              from^#(x1) = [2 0] x1 + [1]
                           [0 0]      [0]
              c_0() = [0]
                      [0]
              activate^#(x1) = [6 0] x1 + [3]
                               [4 0]      [7]
              c_5(x1) = [2 0] x1 + [5]
                        [0 0]      [7]
         
         * Path {6}->{4}: YES(?,O(n^2))
           ----------------------------
           
           The usable rules for this path are:
           
             {  activate(n__from(X)) -> from(activate(X))
              , activate(n__s(X)) -> s(activate(X))
              , activate(X) -> X
              , from(X) -> cons(X, n__from(n__s(X)))
              , from(X) -> n__from(X)
              , s(X) -> n__s(X)}
           
           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(from) = {1}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(after) = {}, Uargs(s) = {1},
               Uargs(activate) = {}, Uargs(from^#) = {1}, Uargs(after^#) = {},
               Uargs(c_2) = {}, Uargs(s^#) = {}, Uargs(activate^#) = {},
               Uargs(c_5) = {1}, Uargs(c_6) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [1 3] x1 + [1]
                         [0 1]      [2]
              cons(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
                             [0 0]      [0 1]      [0]
              n__from(x1) = [1 3] x1 + [0]
                            [0 1]      [2]
              n__s(x1) = [1 0] x1 + [2]
                         [0 1]      [0]
              after(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
              0() = [0]
                    [0]
              s(x1) = [1 0] x1 + [3]
                      [0 1]      [0]
              activate(x1) = [2 2] x1 + [2]
                             [0 1]      [0]
              from^#(x1) = [3 0] x1 + [0]
                           [0 0]      [0]
              c_0() = [0]
                      [0]
              after^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
              c_1() = [0]
                      [0]
              c_2(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_3() = [0]
                      [0]
              s^#(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_4() = [0]
                      [0]
              activate^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
              c_5(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
              c_6(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_7() = [0]
                      [0]
           Complexity induced by the adequate RMI: YES(?,O(n^2))
           
           We apply the sub-processor on the resulting sub-problem:
           
           'matrix-interpretation of dimension 2'
           --------------------------------------
           Answer:           YES(?,O(n^1))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {from^#(X) -> c_3()}
             Weak Rules:
               {  activate^#(n__from(X)) -> c_5(from^#(activate(X)))
                , activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}
           
           Proof Output:    
             The following argument positions are usable:
               Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(s) = {}, Uargs(activate) = {},
               Uargs(from^#) = {}, Uargs(activate^#) = {}, Uargs(c_5) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [1 0] x1 + [2]
                         [0 0]      [4]
              cons(x1, x2) = [1 0] x1 + [0 0] x2 + [2]
                             [0 0]      [0 0]      [1]
              n__from(x1) = [1 0] x1 + [2]
                            [0 0]      [4]
              n__s(x1) = [0 0] x1 + [4]
                         [0 0]      [0]
              s(x1) = [0 0] x1 + [4]
                      [0 0]      [0]
              activate(x1) = [1 0] x1 + [0]
                             [2 1]      [0]
              from^#(x1) = [2 0] x1 + [1]
                           [0 0]      [0]
              c_3() = [0]
                      [0]
              activate^#(x1) = [6 0] x1 + [3]
                               [4 0]      [7]
              c_5(x1) = [2 0] x1 + [5]
                        [0 0]      [7]
         
         * Path {7}: YES(?,O(n^2))
           -----------------------
           
           The usable rules for this path are:
           
             {  activate(n__from(X)) -> from(activate(X))
              , activate(n__s(X)) -> s(activate(X))
              , activate(X) -> X
              , from(X) -> cons(X, n__from(n__s(X)))
              , from(X) -> n__from(X)
              , s(X) -> n__s(X)}
           
           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(from) = {1}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(after) = {}, Uargs(s) = {1},
               Uargs(activate) = {}, Uargs(from^#) = {}, Uargs(after^#) = {},
               Uargs(c_2) = {}, Uargs(s^#) = {1}, Uargs(activate^#) = {},
               Uargs(c_5) = {}, Uargs(c_6) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [1 2] x1 + [2]
                         [0 0]      [1]
              cons(x1, x2) = [1 1] x1 + [0 0] x2 + [1]
                             [0 0]      [0 1]      [0]
              n__from(x1) = [1 2] x1 + [1]
                            [0 0]      [1]
              n__s(x1) = [1 2] x1 + [2]
                         [0 1]      [0]
              after(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
              0() = [0]
                    [0]
              s(x1) = [1 2] x1 + [3]
                      [0 1]      [0]
              activate(x1) = [2 1] x1 + [1]
                             [0 1]      [0]
              from^#(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
              c_0() = [0]
                      [0]
              after^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
              c_1() = [0]
                      [0]
              c_2(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_3() = [0]
                      [0]
              s^#(x1) = [1 0] x1 + [0]
                        [3 3]      [0]
              c_4() = [0]
                      [0]
              activate^#(x1) = [3 2] x1 + [0]
                               [0 0]      [0]
              c_5(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_6(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
              c_7() = [0]
                      [0]
           Complexity induced by the adequate RMI: YES(?,O(n^2))
           
           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: {activate^#(n__s(X)) -> c_6(s^#(activate(X)))}
             Weak Rules:
               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}
           
           Proof Output:    
             The following argument positions are usable:
               Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(s) = {}, Uargs(activate) = {},
               Uargs(s^#) = {}, Uargs(activate^#) = {}, Uargs(c_6) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0 1] x1 + [4]
                         [0 0]      [2]
              cons(x1, x2) = [0 1] x1 + [0 0] x2 + [4]
                             [0 0]      [0 0]      [2]
              n__from(x1) = [0 1] x1 + [0]
                            [0 0]      [2]
              n__s(x1) = [1 1] x1 + [0]
                         [0 1]      [2]
              s(x1) = [1 1] x1 + [0]
                      [0 1]      [2]
              activate(x1) = [1 2] x1 + [2]
                             [0 1]      [0]
              s^#(x1) = [2 2] x1 + [0]
                        [0 0]      [0]
              activate^#(x1) = [6 6] x1 + [3]
                               [6 2]      [7]
              c_6(x1) = [2 0] x1 + [3]
                        [1 0]      [3]
         
         * Path {7}->{5}: YES(?,O(n^2))
           ----------------------------
           
           The usable rules for this path are:
           
             {  activate(n__from(X)) -> from(activate(X))
              , activate(n__s(X)) -> s(activate(X))
              , activate(X) -> X
              , from(X) -> cons(X, n__from(n__s(X)))
              , from(X) -> n__from(X)
              , s(X) -> n__s(X)}
           
           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(from) = {1}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(after) = {}, Uargs(s) = {1},
               Uargs(activate) = {}, Uargs(from^#) = {}, Uargs(after^#) = {},
               Uargs(c_2) = {}, Uargs(s^#) = {1}, Uargs(activate^#) = {},
               Uargs(c_5) = {}, Uargs(c_6) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [1 3] x1 + [1]
                         [0 1]      [2]
              cons(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
                             [0 0]      [0 1]      [0]
              n__from(x1) = [1 3] x1 + [0]
                            [0 1]      [2]
              n__s(x1) = [1 0] x1 + [2]
                         [0 1]      [0]
              after(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
              0() = [0]
                    [0]
              s(x1) = [1 0] x1 + [3]
                      [0 1]      [0]
              activate(x1) = [2 2] x1 + [2]
                             [0 1]      [0]
              from^#(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
              c_0() = [0]
                      [0]
              after^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
              c_1() = [0]
                      [0]
              c_2(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_3() = [0]
                      [0]
              s^#(x1) = [3 0] x1 + [0]
                        [0 0]      [0]
              c_4() = [0]
                      [0]
              activate^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
              c_5(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_6(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
              c_7() = [0]
                      [0]
           Complexity induced by the adequate RMI: YES(?,O(n^2))
           
           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: {s^#(X) -> c_4()}
             Weak Rules:
               {  activate^#(n__s(X)) -> c_6(s^#(activate(X)))
                , activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}
           
           Proof Output:    
             The following argument positions are usable:
               Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(s) = {}, Uargs(activate) = {},
               Uargs(s^#) = {}, Uargs(activate^#) = {}, Uargs(c_6) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0 2] x1 + [2]
                         [0 1]      [2]
              cons(x1, x2) = [0 2] x1 + [0 0] x2 + [1]
                             [0 1]      [0 0]      [2]
              n__from(x1) = [0 1] x1 + [2]
                            [0 1]      [2]
              n__s(x1) = [1 3] x1 + [0]
                         [0 1]      [4]
              s(x1) = [1 3] x1 + [0]
                      [0 1]      [4]
              activate(x1) = [1 2] x1 + [2]
                             [0 1]      [2]
              s^#(x1) = [2 0] x1 + [2]
                        [0 0]      [0]
              c_4() = [1]
                      [0]
              activate^#(x1) = [4 2] x1 + [7]
                               [2 2]      [7]
              c_6(x1) = [2 0] x1 + [3]
                        [0 0]      [7]
         
         * Path {8}: YES(?,O(1))
           ---------------------
           
           The usable rules of this path are empty.
           
           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
               Uargs(n__s) = {}, Uargs(after) = {}, Uargs(s) = {},
               Uargs(activate) = {}, Uargs(from^#) = {}, Uargs(after^#) = {},
               Uargs(c_2) = {}, Uargs(s^#) = {}, Uargs(activate^#) = {},
               Uargs(c_5) = {}, Uargs(c_6) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              from(x1) = [0 0] x1 + [0]
                         [0 0]      [0]
              cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
              n__from(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
              n__s(x1) = [0 0] x1 + [0]
                         [0 0]      [0]
              after(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
              0() = [0]
                    [0]
              s(x1) = [0 0] x1 + [0]
                      [0 0]      [0]
              activate(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
              from^#(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
              c_0() = [0]
                      [0]
              after^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
              c_1() = [0]
                      [0]
              c_2(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_3() = [0]
                      [0]
              s^#(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_4() = [0]
                      [0]
              activate^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
              c_5(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_6(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
              c_7() = [0]
                      [0]
           
           We apply the sub-processor on the resulting sub-problem:
           
           'matrix-interpretation of dimension 2'
           --------------------------------------
           Answer:           YES(?,O(1))
           Input Problem:    innermost DP runtime-complexity with respect to
             Strict Rules: {activate^#(X) -> c_7()}
             Weak Rules: {}
           
           Proof Output:    
             The following argument positions are usable:
               Uargs(activate^#) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              activate^#(x1) = [0 0] x1 + [7]
                               [0 0]      [7]
              c_7() = [0]
                      [1]

Tool RC1

Execution TimeUnknown
Answer
MAYBE
InputTransformed CSR 04 Ex4 7 56 Bor03 FR

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputTransformed CSR 04 Ex4 7 56 Bor03 FR

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  from(X) -> cons(X, n__from(n__s(X)))
     , after(0(), XS) -> XS
     , after(s(N), cons(X, XS)) -> after(N, activate(XS))
     , from(X) -> n__from(X)
     , s(X) -> n__s(X)
     , activate(n__from(X)) -> from(activate(X))
     , activate(n__s(X)) -> s(activate(X))
     , activate(X) -> X}

Proof Output:    
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: from^#(X) -> c_0(X, X)
              , 2: after^#(0(), XS) -> c_1(XS)
              , 3: after^#(s(N), cons(X, XS)) -> c_2(after^#(N, activate(XS)))
              , 4: from^#(X) -> c_3(X)
              , 5: s^#(X) -> c_4(X)
              , 6: activate^#(n__from(X)) -> c_5(from^#(activate(X)))
              , 7: activate^#(n__s(X)) -> c_6(s^#(activate(X)))
              , 8: activate^#(X) -> c_7(X)}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{8}                                                       [    YES(?,O(1))     ]
             
             ->{7}                                                       [     inherited      ]
                |
                `->{5}                                                   [         NA         ]
             
             ->{6}                                                       [     inherited      ]
                |
                |->{1}                                                   [       MAYBE        ]
                |
                `->{4}                                                   [         NA         ]
             
             ->{3}                                                       [     inherited      ]
                |
                `->{2}                                                   [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {3}: inherited
             -------------------
             
             This path is subsumed by the proof of path {3}->{2}.
           
           * Path {3}->{2}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {6}: inherited
             -------------------
             
             This path is subsumed by the proof of path {6}->{1}.
           
           * Path {6}->{1}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(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 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  activate^#(n__from(X)) -> c_5(from^#(activate(X)))
                  , from^#(X) -> c_0(X, X)
                  , activate(n__from(X)) -> from(activate(X))
                  , activate(n__s(X)) -> s(activate(X))
                  , activate(X) -> X
                  , from(X) -> cons(X, n__from(n__s(X)))
                  , from(X) -> n__from(X)
                  , s(X) -> n__s(X)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {6}->{4}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {7}: inherited
             -------------------
             
             This path is subsumed by the proof of path {7}->{5}.
           
           * Path {7}->{5}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {8}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
                 Uargs(n__s) = {}, Uargs(after) = {}, Uargs(s) = {},
                 Uargs(activate) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(after^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(c_3) = {}, Uargs(s^#) = {}, Uargs(c_4) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                cons(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                n__from(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                n__s(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                after(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                activate(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                from^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_0(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                after^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                s^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                activate^#(x1) = [3 3 3] x1 + [0]
                                 [0 0 0]      [0]
                                 [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7(x1) = [1 1 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {activate^#(X) -> c_7(X)}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                activate^#(x1) = [7 7 7] x1 + [7]
                                 [7 7 7]      [7]
                                 [7 7 7]      [7]
                c_7(x1) = [3 3 3] x1 + [0]
                          [3 1 3]      [1]
                          [1 1 1]      [1]
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: from^#(X) -> c_0(X, X)
              , 2: after^#(0(), XS) -> c_1(XS)
              , 3: after^#(s(N), cons(X, XS)) -> c_2(after^#(N, activate(XS)))
              , 4: from^#(X) -> c_3(X)
              , 5: s^#(X) -> c_4(X)
              , 6: activate^#(n__from(X)) -> c_5(from^#(activate(X)))
              , 7: activate^#(n__s(X)) -> c_6(s^#(activate(X)))
              , 8: activate^#(X) -> c_7(X)}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{8}                                                       [    YES(?,O(1))     ]
             
             ->{7}                                                       [     inherited      ]
                |
                `->{5}                                                   [         NA         ]
             
             ->{6}                                                       [     inherited      ]
                |
                |->{1}                                                   [         NA         ]
                |
                `->{4}                                                   [       MAYBE        ]
             
             ->{3}                                                       [     inherited      ]
                |
                `->{2}                                                   [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {3}: inherited
             -------------------
             
             This path is subsumed by the proof of path {3}->{2}.
           
           * Path {3}->{2}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {6}: inherited
             -------------------
             
             This path is subsumed by the proof of path {6}->{1}.
           
           * Path {6}->{1}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {6}->{4}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(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:
                 {  activate^#(n__from(X)) -> c_5(from^#(activate(X)))
                  , from^#(X) -> c_3(X)
                  , activate(n__from(X)) -> from(activate(X))
                  , activate(n__s(X)) -> s(activate(X))
                  , activate(X) -> X
                  , from(X) -> cons(X, n__from(n__s(X)))
                  , from(X) -> n__from(X)
                  , s(X) -> n__s(X)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {7}: inherited
             -------------------
             
             This path is subsumed by the proof of path {7}->{5}.
           
           * Path {7}->{5}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {8}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
                 Uargs(n__s) = {}, Uargs(after) = {}, Uargs(s) = {},
                 Uargs(activate) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(after^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(c_3) = {}, Uargs(s^#) = {}, Uargs(c_4) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                n__from(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                n__s(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                after(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                activate(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_0(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                after^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                s^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                activate^#(x1) = [3 3] x1 + [0]
                                 [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {activate^#(X) -> c_7(X)}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                activate^#(x1) = [7 7] x1 + [7]
                                 [7 7]      [7]
                c_7(x1) = [1 3] x1 + [0]
                          [3 1]      [3]
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: from^#(X) -> c_0(X, X)
              , 2: after^#(0(), XS) -> c_1(XS)
              , 3: after^#(s(N), cons(X, XS)) -> c_2(after^#(N, activate(XS)))
              , 4: from^#(X) -> c_3(X)
              , 5: s^#(X) -> c_4(X)
              , 6: activate^#(n__from(X)) -> c_5(from^#(activate(X)))
              , 7: activate^#(n__s(X)) -> c_6(s^#(activate(X)))
              , 8: activate^#(X) -> c_7(X)}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{8}                                                       [    YES(?,O(1))     ]
             
             ->{7}                                                       [     inherited      ]
                |
                `->{5}                                                   [       MAYBE        ]
             
             ->{6}                                                       [     inherited      ]
                |
                |->{1}                                                   [         NA         ]
                |
                `->{4}                                                   [         NA         ]
             
             ->{3}                                                       [     inherited      ]
                |
                `->{2}                                                   [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {3}: inherited
             -------------------
             
             This path is subsumed by the proof of path {3}->{2}.
           
           * Path {3}->{2}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {6}: inherited
             -------------------
             
             This path is subsumed by the proof of path {6}->{1}.
           
           * Path {6}->{1}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {6}->{4}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(X)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {7}: inherited
             -------------------
             
             This path is subsumed by the proof of path {7}->{5}.
           
           * Path {7}->{5}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  activate(n__from(X)) -> from(activate(X))
                , activate(n__s(X)) -> s(activate(X))
                , activate(X) -> X
                , from(X) -> cons(X, n__from(n__s(X)))
                , from(X) -> n__from(X)
                , s(X) -> n__s(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:
                 {  activate^#(n__s(X)) -> c_6(s^#(activate(X)))
                  , s^#(X) -> c_4(X)
                  , activate(n__from(X)) -> from(activate(X))
                  , activate(n__s(X)) -> s(activate(X))
                  , activate(X) -> X
                  , from(X) -> cons(X, n__from(n__s(X)))
                  , from(X) -> n__from(X)
                  , s(X) -> n__s(X)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {8}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(from) = {}, Uargs(cons) = {}, Uargs(n__from) = {},
                 Uargs(n__s) = {}, Uargs(after) = {}, Uargs(s) = {},
                 Uargs(activate) = {}, Uargs(from^#) = {}, Uargs(c_0) = {},
                 Uargs(after^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {},
                 Uargs(c_3) = {}, Uargs(s^#) = {}, Uargs(c_4) = {},
                 Uargs(activate^#) = {}, Uargs(c_5) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                from(x1) = [0] x1 + [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                n__from(x1) = [0] x1 + [0]
                n__s(x1) = [0] x1 + [0]
                after(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                activate(x1) = [0] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_0(x1, x2) = [0] x1 + [0] x2 + [0]
                after^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                s^#(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                activate^#(x1) = [3] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {activate^#(X) -> c_7(X)}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(activate^#) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                activate^#(x1) = [7] x1 + [7]
                c_7(x1) = [1] x1 + [0]
    
    4) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.