Problem GTSSK07 cade17

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputGTSSK07 cade17

stdout:

MAYBE

Problem:
 log(x,s(s(y))) -> cond(le(x,s(s(y))),x,y)
 cond(true(),x,y) -> s(0())
 cond(false(),x,y) -> double(log(x,square(s(s(y)))))
 le(0(),v) -> true()
 le(s(u),0()) -> false()
 le(s(u),s(v)) -> le(u,v)
 double(0()) -> 0()
 double(s(x)) -> s(s(double(x)))
 square(0()) -> 0()
 square(s(x)) -> s(plus(square(x),double(x)))
 plus(n,0()) -> n
 plus(n,s(m)) -> s(plus(n,m))

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputGTSSK07 cade17

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputGTSSK07 cade17

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
     , cond(true(), x, y) -> s(0())
     , cond(false(), x, y) -> double(log(x, square(s(s(y)))))
     , le(0(), v) -> true()
     , le(s(u), 0()) -> false()
     , le(s(u), s(v)) -> le(u, v)
     , double(0()) -> 0()
     , double(s(x)) -> s(s(double(x)))
     , square(0()) -> 0()
     , square(s(x)) -> s(plus(square(x), double(x)))
     , plus(n, 0()) -> n
     , plus(n, s(m)) -> s(plus(n, m))}

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: log^#(x, s(s(y))) -> c_0(cond^#(le(x, s(s(y))), x, y))
              , 2: cond^#(true(), x, y) -> c_1()
              , 3: cond^#(false(), x, y) ->
                   c_2(double^#(log(x, square(s(s(y))))))
              , 4: le^#(0(), v) -> c_3()
              , 5: le^#(s(u), 0()) -> c_4()
              , 6: le^#(s(u), s(v)) -> c_5(le^#(u, v))
              , 7: double^#(0()) -> c_6()
              , 8: double^#(s(x)) -> c_7(double^#(x))
              , 9: square^#(0()) -> c_8()
              , 10: square^#(s(x)) -> c_9(plus^#(square(x), double(x)))
              , 11: plus^#(n, 0()) -> c_10()
              , 12: plus^#(n, s(m)) -> c_11(plus^#(n, m))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{10}                                                      [     inherited      ]
                |
                |->{11}                                                  [       MAYBE        ]
                |
                `->{12}                                                  [     inherited      ]
                    |
                    `->{11}                                              [         NA         ]
             
             ->{9}                                                       [    YES(?,O(1))     ]
             
             ->{6}                                                       [   YES(?,O(n^2))    ]
                |
                |->{4}                                                   [   YES(?,O(n^2))    ]
                |
                `->{5}                                                   [   YES(?,O(n^1))    ]
             
             ->{1}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    |->{7}                                               [         NA         ]
                    |
                    `->{8}                                               [     inherited      ]
                        |
                        `->{7}                                           [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
           
           * Path {1}->{2}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  le(0(), v) -> true()
                , le(s(u), 0()) -> false()
                , le(s(u), s(v)) -> le(u, v)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
                 Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
                 Uargs(log^#) = {}, Uargs(c_0) = {1}, Uargs(cond^#) = {1},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
                 Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(square^#) = {},
                 Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                s(x1) = [1 0] x1 + [2]
                        [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                le(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
                             [0 0]      [0 0]      [3]
                true() = [0]
                         [1]
                0() = [0]
                      [0]
                false() = [0]
                          [1]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                square(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                cond^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                square^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_10() = [0]
                         [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {1}->{3}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
           
           * Path {1}->{3}->{7}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  le(0(), v) -> true()
                , le(s(u), 0()) -> false()
                , le(s(u), s(v)) -> le(u, v)
                , log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
                , square(0()) -> 0()
                , square(s(x)) -> s(plus(square(x), double(x)))
                , cond(true(), x, y) -> s(0())
                , cond(false(), x, y) -> double(log(x, square(s(s(y)))))
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , plus(n, 0()) -> n
                , plus(n, s(m)) -> s(plus(n, m))}
             
             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}->{3}->{8}: inherited
             -----------------------------
             
             This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
           
           * Path {1}->{3}->{8}->{7}: NA
             ---------------------------
             
             The usable rules for this path are:
             
               {  le(0(), v) -> true()
                , le(s(u), 0()) -> false()
                , le(s(u), s(v)) -> le(u, v)
                , log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
                , square(0()) -> 0()
                , square(s(x)) -> s(plus(square(x), double(x)))
                , cond(true(), x, y) -> s(0())
                , cond(false(), x, y) -> double(log(x, square(s(s(y)))))
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , plus(n, 0()) -> n
                , plus(n, s(m)) -> s(plus(n, m))}
             
             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}: 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(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
                 Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
                 Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
                 Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
                 Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                square(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                le^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                               [3 3]      [3 3]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                square^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_10() = [0]
                         [0]
                c_11(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: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 6] x1 + [2]
                        [0 1]      [0]
                le^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
                               [0 0]      [0 0]      [4]
                c_5(x1) = [1 0] x1 + [1]
                          [0 0]      [3]
           
           * Path {6}->{4}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
                 Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
                 Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
                 Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
                 Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                square(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                square^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_10() = [0]
                         [0]
                c_11(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: {le^#(0(), v) -> c_3()}
               Weak Rules: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [0]
                        [0 1]      [2]
                0() = [2]
                      [2]
                le^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
                               [3 3]      [0 1]      [0]
                c_3() = [1]
                        [0]
                c_5(x1) = [1 0] x1 + [3]
                          [0 0]      [7]
           
           * Path {6}->{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(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
                 Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
                 Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
                 Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
                 Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                square(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                square^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_10() = [0]
                         [0]
                c_11(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^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {le^#(s(u), 0()) -> c_4()}
               Weak Rules: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [2]
                        [0 1]      [2]
                0() = [0]
                      [2]
                le^#(x1, x2) = [2 0] x1 + [0 2] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [1]
                        [0]
                c_5(x1) = [1 0] x1 + [3]
                          [0 0]      [0]
           
           * Path {9}: 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(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
                 Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
                 Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
                 Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(square^#) = {},
                 Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                square(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                square^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_10() = [0]
                         [0]
                c_11(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(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {square^#(0()) -> c_8()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(square^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                square^#(x1) = [2 0] x1 + [7]
                               [2 2]      [7]
                c_8() = [0]
                        [1]
           
           * Path {10}: inherited
             --------------------
             
             This path is subsumed by the proof of path {10}->{12}->{11}.
           
           * Path {10}->{11}: MAYBE
             ----------------------
             
             The usable rules for this path are:
             
               {  double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , square(0()) -> 0()
                , square(s(x)) -> s(plus(square(x), double(x)))
                , plus(n, 0()) -> n
                , plus(n, s(m)) -> s(plus(n, m))}
             
             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:
                 {  square^#(s(x)) -> c_9(plus^#(square(x), double(x)))
                  , plus^#(n, 0()) -> c_10()
                  , double(0()) -> 0()
                  , double(s(x)) -> s(s(double(x)))
                  , square(0()) -> 0()
                  , square(s(x)) -> s(plus(square(x), double(x)))
                  , plus(n, 0()) -> n
                  , plus(n, s(m)) -> s(plus(n, m))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {10}->{12}: inherited
             --------------------------
             
             This path is subsumed by the proof of path {10}->{12}->{11}.
           
           * Path {10}->{12}->{11}: NA
             -------------------------
             
             The usable rules for this path are:
             
               {  double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , square(0()) -> 0()
                , square(s(x)) -> s(plus(square(x), double(x)))
                , plus(n, 0()) -> n
                , plus(n, s(m)) -> s(plus(n, m))}
             
             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.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: log^#(x, s(s(y))) -> c_0(cond^#(le(x, s(s(y))), x, y))
              , 2: cond^#(true(), x, y) -> c_1()
              , 3: cond^#(false(), x, y) ->
                   c_2(double^#(log(x, square(s(s(y))))))
              , 4: le^#(0(), v) -> c_3()
              , 5: le^#(s(u), 0()) -> c_4()
              , 6: le^#(s(u), s(v)) -> c_5(le^#(u, v))
              , 7: double^#(0()) -> c_6()
              , 8: double^#(s(x)) -> c_7(double^#(x))
              , 9: square^#(0()) -> c_8()
              , 10: square^#(s(x)) -> c_9(plus^#(square(x), double(x)))
              , 11: plus^#(n, 0()) -> c_10()
              , 12: plus^#(n, s(m)) -> c_11(plus^#(n, m))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{10}                                                      [     inherited      ]
                |
                |->{11}                                                  [       MAYBE        ]
                |
                `->{12}                                                  [     inherited      ]
                    |
                    `->{11}                                              [         NA         ]
             
             ->{9}                                                       [    YES(?,O(1))     ]
             
             ->{6}                                                       [   YES(?,O(n^1))    ]
                |
                |->{4}                                                   [   YES(?,O(n^1))    ]
                |
                `->{5}                                                   [   YES(?,O(n^1))    ]
             
             ->{1}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    |->{7}                                               [         NA         ]
                    |
                    `->{8}                                               [     inherited      ]
                        |
                        `->{7}                                           [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
           
           * Path {1}->{2}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  le(0(), v) -> true()
                , le(s(u), 0()) -> false()
                , le(s(u), s(v)) -> le(u, v)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
                 Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
                 Uargs(log^#) = {}, Uargs(c_0) = {1}, Uargs(cond^#) = {1},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
                 Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(square^#) = {},
                 Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [1] x1 + [2]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                le(x1, x2) = [0] x1 + [2] x2 + [2]
                true() = [1]
                0() = [0]
                false() = [1]
                double(x1) = [0] x1 + [0]
                square(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [1] x1 + [0]
                cond^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                square^#(x1) = [0] x1 + [0]
                c_8() = [0]
                c_9(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_10() = [0]
                c_11(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {1}->{3}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
           
           * Path {1}->{3}->{7}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  le(0(), v) -> true()
                , le(s(u), 0()) -> false()
                , le(s(u), s(v)) -> le(u, v)
                , log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
                , square(0()) -> 0()
                , square(s(x)) -> s(plus(square(x), double(x)))
                , cond(true(), x, y) -> s(0())
                , cond(false(), x, y) -> double(log(x, square(s(s(y)))))
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , plus(n, 0()) -> n
                , plus(n, s(m)) -> s(plus(n, m))}
             
             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}->{3}->{8}: inherited
             -----------------------------
             
             This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
           
           * Path {1}->{3}->{8}->{7}: NA
             ---------------------------
             
             The usable rules for this path are:
             
               {  le(0(), v) -> true()
                , le(s(u), 0()) -> false()
                , le(s(u), s(v)) -> le(u, v)
                , log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
                , square(0()) -> 0()
                , square(s(x)) -> s(plus(square(x), double(x)))
                , cond(true(), x, y) -> s(0())
                , cond(false(), x, y) -> double(log(x, square(s(s(y)))))
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , plus(n, 0()) -> n
                , plus(n, s(m)) -> s(plus(n, m))}
             
             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}: 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(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
                 Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
                 Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
                 Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
                 Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [1] x1 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                0() = [0]
                false() = [0]
                double(x1) = [0] x1 + [0]
                square(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                le^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [1] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                square^#(x1) = [0] x1 + [0]
                c_8() = [0]
                c_9(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_10() = [0]
                c_11(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: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                le^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_5(x1) = [1] x1 + [7]
           
           * Path {6}->{4}: 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(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
                 Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
                 Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
                 Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
                 Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                0() = [0]
                false() = [0]
                double(x1) = [0] x1 + [0]
                square(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [1] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                square^#(x1) = [0] x1 + [0]
                c_8() = [0]
                c_9(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_10() = [0]
                c_11(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: {le^#(0(), v) -> c_3()}
               Weak Rules: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                le^#(x1, x2) = [3] x1 + [3] x2 + [2]
                c_3() = [1]
                c_5(x1) = [1] x1 + [5]
           
           * Path {6}->{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(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
                 Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
                 Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
                 Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
                 Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                0() = [0]
                false() = [0]
                double(x1) = [0] x1 + [0]
                square(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [1] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                square^#(x1) = [0] x1 + [0]
                c_8() = [0]
                c_9(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_10() = [0]
                c_11(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: {le^#(s(u), 0()) -> c_4()}
               Weak Rules: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                le^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_4() = [1]
                c_5(x1) = [1] x1 + [7]
           
           * Path {9}: 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(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
                 Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
                 Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
                 Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(square^#) = {},
                 Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                0() = [0]
                false() = [0]
                double(x1) = [0] x1 + [0]
                square(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                square^#(x1) = [0] x1 + [0]
                c_8() = [0]
                c_9(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_10() = [0]
                c_11(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {square^#(0()) -> c_8()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(square^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [7]
                square^#(x1) = [1] x1 + [7]
                c_8() = [1]
           
           * Path {10}: inherited
             --------------------
             
             This path is subsumed by the proof of path {10}->{12}->{11}.
           
           * Path {10}->{11}: MAYBE
             ----------------------
             
             The usable rules for this path are:
             
               {  double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , square(0()) -> 0()
                , square(s(x)) -> s(plus(square(x), double(x)))
                , plus(n, 0()) -> n
                , plus(n, s(m)) -> s(plus(n, m))}
             
             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:
                 {  square^#(s(x)) -> c_9(plus^#(square(x), double(x)))
                  , plus^#(n, 0()) -> c_10()
                  , double(0()) -> 0()
                  , double(s(x)) -> s(s(double(x)))
                  , square(0()) -> 0()
                  , square(s(x)) -> s(plus(square(x), double(x)))
                  , plus(n, 0()) -> n
                  , plus(n, s(m)) -> s(plus(n, m))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {10}->{12}: inherited
             --------------------------
             
             This path is subsumed by the proof of path {10}->{12}->{11}.
           
           * Path {10}->{12}->{11}: NA
             -------------------------
             
             The usable rules for this path are:
             
               {  double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , square(0()) -> 0()
                , square(s(x)) -> s(plus(square(x), double(x)))
                , plus(n, 0()) -> n
                , plus(n, s(m)) -> s(plus(n, m))}
             
             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.
    
    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
InputGTSSK07 cade17

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputGTSSK07 cade17

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
     , cond(true(), x, y) -> s(0())
     , cond(false(), x, y) -> double(log(x, square(s(s(y)))))
     , le(0(), v) -> true()
     , le(s(u), 0()) -> false()
     , le(s(u), s(v)) -> le(u, v)
     , double(0()) -> 0()
     , double(s(x)) -> s(s(double(x)))
     , square(0()) -> 0()
     , square(s(x)) -> s(plus(square(x), double(x)))
     , plus(n, 0()) -> n
     , plus(n, s(m)) -> s(plus(n, m))}

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: log^#(x, s(s(y))) -> c_0(cond^#(le(x, s(s(y))), x, y))
              , 2: cond^#(true(), x, y) -> c_1()
              , 3: cond^#(false(), x, y) ->
                   c_2(double^#(log(x, square(s(s(y))))))
              , 4: le^#(0(), v) -> c_3()
              , 5: le^#(s(u), 0()) -> c_4()
              , 6: le^#(s(u), s(v)) -> c_5(le^#(u, v))
              , 7: double^#(0()) -> c_6()
              , 8: double^#(s(x)) -> c_7(double^#(x))
              , 9: square^#(0()) -> c_8()
              , 10: square^#(s(x)) -> c_9(plus^#(square(x), double(x)))
              , 11: plus^#(n, 0()) -> c_10(n)
              , 12: plus^#(n, s(m)) -> c_11(plus^#(n, m))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{10}                                                      [     inherited      ]
                |
                |->{11}                                                  [       MAYBE        ]
                |
                `->{12}                                                  [     inherited      ]
                    |
                    `->{11}                                              [         NA         ]
             
             ->{9}                                                       [    YES(?,O(1))     ]
             
             ->{6}                                                       [   YES(?,O(n^2))    ]
                |
                |->{4}                                                   [   YES(?,O(n^2))    ]
                |
                `->{5}                                                   [   YES(?,O(n^1))    ]
             
             ->{1}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    |->{7}                                               [         NA         ]
                    |
                    `->{8}                                               [     inherited      ]
                        |
                        `->{7}                                           [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
           
           * Path {1}->{2}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  le(0(), v) -> true()
                , le(s(u), 0()) -> false()
                , le(s(u), s(v)) -> le(u, v)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
                 Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
                 Uargs(log^#) = {}, Uargs(c_0) = {1}, Uargs(cond^#) = {1},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
                 Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(square^#) = {},
                 Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_10) = {},
                 Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                s(x1) = [1 0] x1 + [2]
                        [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                le(x1, x2) = [0 0] x1 + [1 0] x2 + [1]
                             [0 0]      [0 0]      [3]
                true() = [0]
                         [1]
                0() = [0]
                      [0]
                false() = [0]
                          [1]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                square(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                cond^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                square^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {1}->{3}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
           
           * Path {1}->{3}->{7}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  le(0(), v) -> true()
                , le(s(u), 0()) -> false()
                , le(s(u), s(v)) -> le(u, v)
                , log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
                , square(0()) -> 0()
                , square(s(x)) -> s(plus(square(x), double(x)))
                , cond(true(), x, y) -> s(0())
                , cond(false(), x, y) -> double(log(x, square(s(s(y)))))
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , plus(n, 0()) -> n
                , plus(n, s(m)) -> s(plus(n, m))}
             
             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}->{3}->{8}: inherited
             -----------------------------
             
             This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
           
           * Path {1}->{3}->{8}->{7}: NA
             ---------------------------
             
             The usable rules for this path are:
             
               {  le(0(), v) -> true()
                , le(s(u), 0()) -> false()
                , le(s(u), s(v)) -> le(u, v)
                , log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
                , square(0()) -> 0()
                , square(s(x)) -> s(plus(square(x), double(x)))
                , cond(true(), x, y) -> s(0())
                , cond(false(), x, y) -> double(log(x, square(s(s(y)))))
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , plus(n, 0()) -> n
                , plus(n, s(m)) -> s(plus(n, m))}
             
             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}: 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(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
                 Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
                 Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
                 Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
                 Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_10) = {},
                 Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                square(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                le^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                               [3 3]      [3 3]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                square^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11(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: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 6] x1 + [2]
                        [0 1]      [0]
                le^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
                               [0 0]      [0 0]      [4]
                c_5(x1) = [1 0] x1 + [1]
                          [0 0]      [3]
           
           * Path {6}->{4}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
                 Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
                 Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
                 Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
                 Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_10) = {},
                 Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                square(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                square^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11(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: {le^#(0(), v) -> c_3()}
               Weak Rules: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [0]
                        [0 1]      [2]
                0() = [2]
                      [2]
                le^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
                               [3 3]      [0 1]      [0]
                c_3() = [1]
                        [0]
                c_5(x1) = [1 0] x1 + [3]
                          [0 0]      [7]
           
           * Path {6}->{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(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
                 Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
                 Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
                 Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
                 Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_10) = {},
                 Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                square(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                square^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11(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^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {le^#(s(u), 0()) -> c_4()}
               Weak Rules: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [2]
                        [0 1]      [2]
                0() = [0]
                      [2]
                le^#(x1, x2) = [2 0] x1 + [0 2] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [1]
                        [0]
                c_5(x1) = [1 0] x1 + [3]
                          [0 0]      [0]
           
           * Path {9}: 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(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
                 Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
                 Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
                 Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(square^#) = {},
                 Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_10) = {},
                 Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                log(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                square(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                log^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                le^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                square^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11(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(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {square^#(0()) -> c_8()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(square^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                square^#(x1) = [2 0] x1 + [7]
                               [2 2]      [7]
                c_8() = [0]
                        [1]
           
           * Path {10}: inherited
             --------------------
             
             This path is subsumed by the proof of path {10}->{12}->{11}.
           
           * Path {10}->{11}: MAYBE
             ----------------------
             
             The usable rules for this path are:
             
               {  double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , square(0()) -> 0()
                , square(s(x)) -> s(plus(square(x), double(x)))
                , plus(n, 0()) -> n
                , plus(n, s(m)) -> s(plus(n, m))}
             
             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:
                 {  square^#(s(x)) -> c_9(plus^#(square(x), double(x)))
                  , plus^#(n, 0()) -> c_10(n)
                  , double(0()) -> 0()
                  , double(s(x)) -> s(s(double(x)))
                  , square(0()) -> 0()
                  , square(s(x)) -> s(plus(square(x), double(x)))
                  , plus(n, 0()) -> n
                  , plus(n, s(m)) -> s(plus(n, m))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {10}->{12}: inherited
             --------------------------
             
             This path is subsumed by the proof of path {10}->{12}->{11}.
           
           * Path {10}->{12}->{11}: NA
             -------------------------
             
             The usable rules for this path are:
             
               {  double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , square(0()) -> 0()
                , square(s(x)) -> s(plus(square(x), double(x)))
                , plus(n, 0()) -> n
                , plus(n, s(m)) -> s(plus(n, m))}
             
             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.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: log^#(x, s(s(y))) -> c_0(cond^#(le(x, s(s(y))), x, y))
              , 2: cond^#(true(), x, y) -> c_1()
              , 3: cond^#(false(), x, y) ->
                   c_2(double^#(log(x, square(s(s(y))))))
              , 4: le^#(0(), v) -> c_3()
              , 5: le^#(s(u), 0()) -> c_4()
              , 6: le^#(s(u), s(v)) -> c_5(le^#(u, v))
              , 7: double^#(0()) -> c_6()
              , 8: double^#(s(x)) -> c_7(double^#(x))
              , 9: square^#(0()) -> c_8()
              , 10: square^#(s(x)) -> c_9(plus^#(square(x), double(x)))
              , 11: plus^#(n, 0()) -> c_10(n)
              , 12: plus^#(n, s(m)) -> c_11(plus^#(n, m))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{10}                                                      [     inherited      ]
                |
                |->{11}                                                  [       MAYBE        ]
                |
                `->{12}                                                  [     inherited      ]
                    |
                    `->{11}                                              [         NA         ]
             
             ->{9}                                                       [    YES(?,O(1))     ]
             
             ->{6}                                                       [   YES(?,O(n^1))    ]
                |
                |->{4}                                                   [   YES(?,O(n^1))    ]
                |
                `->{5}                                                   [   YES(?,O(n^1))    ]
             
             ->{1}                                                       [     inherited      ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    |->{7}                                               [         NA         ]
                    |
                    `->{8}                                               [     inherited      ]
                        |
                        `->{7}                                           [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
           
           * Path {1}->{2}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  le(0(), v) -> true()
                , le(s(u), 0()) -> false()
                , le(s(u), s(v)) -> le(u, v)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
                 Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
                 Uargs(log^#) = {}, Uargs(c_0) = {1}, Uargs(cond^#) = {1},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
                 Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(square^#) = {},
                 Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_10) = {},
                 Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [1] x1 + [2]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                le(x1, x2) = [0] x1 + [2] x2 + [2]
                true() = [1]
                0() = [0]
                false() = [1]
                double(x1) = [0] x1 + [0]
                square(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [1] x1 + [0]
                cond^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                square^#(x1) = [0] x1 + [0]
                c_8() = [0]
                c_9(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_10(x1) = [0] x1 + [0]
                c_11(x1) = [0] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {1}->{3}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
           
           * Path {1}->{3}->{7}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  le(0(), v) -> true()
                , le(s(u), 0()) -> false()
                , le(s(u), s(v)) -> le(u, v)
                , log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
                , square(0()) -> 0()
                , square(s(x)) -> s(plus(square(x), double(x)))
                , cond(true(), x, y) -> s(0())
                , cond(false(), x, y) -> double(log(x, square(s(s(y)))))
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , plus(n, 0()) -> n
                , plus(n, s(m)) -> s(plus(n, m))}
             
             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}->{3}->{8}: inherited
             -----------------------------
             
             This path is subsumed by the proof of path {1}->{3}->{8}->{7}.
           
           * Path {1}->{3}->{8}->{7}: NA
             ---------------------------
             
             The usable rules for this path are:
             
               {  le(0(), v) -> true()
                , le(s(u), 0()) -> false()
                , le(s(u), s(v)) -> le(u, v)
                , log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y)
                , square(0()) -> 0()
                , square(s(x)) -> s(plus(square(x), double(x)))
                , cond(true(), x, y) -> s(0())
                , cond(false(), x, y) -> double(log(x, square(s(s(y)))))
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , plus(n, 0()) -> n
                , plus(n, s(m)) -> s(plus(n, m))}
             
             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}: 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(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
                 Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
                 Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
                 Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
                 Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_10) = {},
                 Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [1] x1 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                0() = [0]
                false() = [0]
                double(x1) = [0] x1 + [0]
                square(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                le^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [1] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                square^#(x1) = [0] x1 + [0]
                c_8() = [0]
                c_9(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_10(x1) = [0] x1 + [0]
                c_11(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: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                le^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_5(x1) = [1] x1 + [7]
           
           * Path {6}->{4}: 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(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
                 Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
                 Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
                 Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
                 Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_10) = {},
                 Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                0() = [0]
                false() = [0]
                double(x1) = [0] x1 + [0]
                square(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [1] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                square^#(x1) = [0] x1 + [0]
                c_8() = [0]
                c_9(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_10(x1) = [0] x1 + [0]
                c_11(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: {le^#(0(), v) -> c_3()}
               Weak Rules: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                le^#(x1, x2) = [3] x1 + [3] x2 + [2]
                c_3() = [1]
                c_5(x1) = [1] x1 + [5]
           
           * Path {6}->{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(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
                 Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
                 Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
                 Uargs(c_5) = {1}, Uargs(c_7) = {}, Uargs(square^#) = {},
                 Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_10) = {},
                 Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                0() = [0]
                false() = [0]
                double(x1) = [0] x1 + [0]
                square(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [1] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                square^#(x1) = [0] x1 + [0]
                c_8() = [0]
                c_9(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_10(x1) = [0] x1 + [0]
                c_11(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: {le^#(s(u), 0()) -> c_4()}
               Weak Rules: {le^#(s(u), s(v)) -> c_5(le^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                le^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_4() = [1]
                c_5(x1) = [1] x1 + [7]
           
           * Path {9}: 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(log) = {}, Uargs(s) = {}, Uargs(cond) = {}, Uargs(le) = {},
                 Uargs(double) = {}, Uargs(square) = {}, Uargs(plus) = {},
                 Uargs(log^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_2) = {}, Uargs(double^#) = {}, Uargs(le^#) = {},
                 Uargs(c_5) = {}, Uargs(c_7) = {}, Uargs(square^#) = {},
                 Uargs(c_9) = {}, Uargs(plus^#) = {}, Uargs(c_10) = {},
                 Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                log(x1, x2) = [0] x1 + [0] x2 + [0]
                s(x1) = [0] x1 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                0() = [0]
                false() = [0]
                double(x1) = [0] x1 + [0]
                square(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                log^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                square^#(x1) = [0] x1 + [0]
                c_8() = [0]
                c_9(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_10(x1) = [0] x1 + [0]
                c_11(x1) = [0] 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: {square^#(0()) -> c_8()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(square^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [7]
                square^#(x1) = [1] x1 + [7]
                c_8() = [1]
           
           * Path {10}: inherited
             --------------------
             
             This path is subsumed by the proof of path {10}->{12}->{11}.
           
           * Path {10}->{11}: MAYBE
             ----------------------
             
             The usable rules for this path are:
             
               {  double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , square(0()) -> 0()
                , square(s(x)) -> s(plus(square(x), double(x)))
                , plus(n, 0()) -> n
                , plus(n, s(m)) -> s(plus(n, m))}
             
             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:
                 {  square^#(s(x)) -> c_9(plus^#(square(x), double(x)))
                  , plus^#(n, 0()) -> c_10(n)
                  , double(0()) -> 0()
                  , double(s(x)) -> s(s(double(x)))
                  , square(0()) -> 0()
                  , square(s(x)) -> s(plus(square(x), double(x)))
                  , plus(n, 0()) -> n
                  , plus(n, s(m)) -> s(plus(n, m))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {10}->{12}: inherited
             --------------------------
             
             This path is subsumed by the proof of path {10}->{12}->{11}.
           
           * Path {10}->{12}->{11}: NA
             -------------------------
             
             The usable rules for this path are:
             
               {  double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , square(0()) -> 0()
                , square(s(x)) -> s(plus(square(x), double(x)))
                , plus(n, 0()) -> n
                , plus(n, s(m)) -> s(plus(n, m))}
             
             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.
    
    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.