Problem GTSSK07 cade09

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputGTSSK07 cade09

stdout:

MAYBE

Problem:
 f(true(),x,y) -> f(gt(x,y),x,round(s(y)))
 round(0()) -> 0()
 round(s(0())) -> s(s(0()))
 round(s(s(x))) -> s(s(round(x)))
 gt(0(),v) -> false()
 gt(s(u),0()) -> true()
 gt(s(u),s(v)) -> gt(u,v)

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputGTSSK07 cade09

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputGTSSK07 cade09

stdout:

MAYBE

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

Proof Output:    
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(true(), x, y) -> c_0(f^#(gt(x, y), x, round(s(y))))
              , 2: round^#(0()) -> c_1()
              , 3: round^#(s(0())) -> c_2()
              , 4: round^#(s(s(x))) -> c_3(round^#(x))
              , 5: gt^#(0(), v) -> c_4()
              , 6: gt^#(s(u), 0()) -> c_5()
              , 7: gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{7}                                                       [   YES(?,O(n^1))    ]
                |
                |->{5}                                                   [   YES(?,O(n^1))    ]
                |
                `->{6}                                                   [   YES(?,O(n^2))    ]
             
             ->{4}                                                       [   YES(?,O(n^2))    ]
                |
                |->{2}                                                   [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^2))    ]
             
             ->{1}                                                       [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: NA
             ------------
             
             The usable rules for this path are:
             
               {  round(0()) -> 0()
                , round(s(0())) -> s(s(0()))
                , round(s(s(x))) -> s(s(round(x)))
                , gt(0(), v) -> false()
                , gt(s(u), 0()) -> true()
                , gt(s(u), s(v)) -> gt(u, v)}
             
             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 {4}: YES(?,O(n^2))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {1}, Uargs(gt^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                round(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                round^#(x1) = [0 0 0] x1 + [0]
                              [3 3 3]      [0]
                              [3 3 3]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                gt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {round^#(s(s(x))) -> c_3(round^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [0]
                        [0 0 1]      [0]
                        [0 0 1]      [2]
                round^#(x1) = [2 0 2] x1 + [0]
                              [4 0 2]      [0]
                              [0 0 0]      [2]
                c_3(x1) = [1 0 2] x1 + [3]
                          [2 0 0]      [0]
                          [0 0 0]      [2]
           
           * Path {4}->{2}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {1}, Uargs(gt^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                round(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                round^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                gt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {round^#(0()) -> c_1()}
               Weak Rules: {round^#(s(s(x))) -> c_3(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [0]
                        [0 0 0]      [2]
                        [0 0 0]      [2]
                0() = [2]
                      [2]
                      [2]
                round^#(x1) = [2 0 0] x1 + [0]
                              [0 0 2]      [4]
                              [1 2 2]      [0]
                c_1() = [1]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [7]
                          [0 0 0]      [6]
           
           * Path {4}->{3}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {1}, Uargs(gt^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                round(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                round^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                gt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {round^#(s(0())) -> c_2()}
               Weak Rules: {round^#(s(s(x))) -> c_3(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [2]
                        [0 1 2]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                round^#(x1) = [2 3 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 5 0]      [0]
                c_2() = [1]
                        [0]
                        [0]
                c_3(x1) = [1 0 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
           
           * Path {7}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                round(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                round^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                gt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [3 3 3]      [3 3 3]      [0]
                               [3 3 3]      [3 3 3]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 3] x1 + [0]
                        [0 0 1]      [4]
                        [0 0 0]      [2]
                gt^#(x1, x2) = [1 1 0] x1 + [1 0 2] x2 + [0]
                               [0 0 2]      [0 0 0]      [2]
                               [1 1 0]      [4 1 0]      [0]
                c_6(x1) = [1 2 0] x1 + [3]
                          [0 0 0]      [2]
                          [0 2 0]      [3]
           
           * Path {7}->{5}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                round(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                round^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                gt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {gt^#(0(), v) -> c_4()}
               Weak Rules: {gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 4 2] x1 + [0]
                        [0 0 2]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [2]
                      [0]
                gt^#(x1, x2) = [2 2 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [2 0 0]      [0 4 0]      [0]
                c_4() = [1]
                        [0]
                        [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
           
           * Path {7}->{6}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                round(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                round^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                gt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {gt^#(s(u), 0()) -> c_5()}
               Weak Rules: {gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0 0] x1 + [0]
                        [0 1 2]      [2]
                        [0 0 0]      [2]
                0() = [0]
                      [0]
                      [0]
                gt^#(x1, x2) = [0 2 2] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [4 0 0]      [4]
                               [0 0 0]      [0 0 0]      [2]
                c_5() = [1]
                        [0]
                        [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 0 2]      [0]
                          [0 0 0]      [2]
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(true(), x, y) -> c_0(f^#(gt(x, y), x, round(s(y))))
              , 2: round^#(0()) -> c_1()
              , 3: round^#(s(0())) -> c_2()
              , 4: round^#(s(s(x))) -> c_3(round^#(x))
              , 5: gt^#(0(), v) -> c_4()
              , 6: gt^#(s(u), 0()) -> c_5()
              , 7: gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{7}                                                       [   YES(?,O(n^2))    ]
                |
                |->{5}                                                   [   YES(?,O(n^2))    ]
                |
                `->{6}                                                   [   YES(?,O(n^1))    ]
             
             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                |->{2}                                                   [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^2))    ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules for this path are:
             
               {  round(0()) -> 0()
                , round(s(0())) -> s(s(0()))
                , round(s(s(x))) -> s(s(round(x)))
                , gt(0(), v) -> false()
                , gt(s(u), 0()) -> true()
                , gt(s(u), s(v)) -> gt(u, v)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  f^#(true(), x, y) -> c_0(f^#(gt(x, y), x, round(s(y))))
                  , round(0()) -> 0()
                  , round(s(0())) -> s(s(0()))
                  , round(s(s(x))) -> s(s(round(x)))
                  , gt(0(), v) -> false()
                  , gt(s(u), 0()) -> true()
                  , gt(s(u), s(v)) -> gt(u, v)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {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(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {1}, Uargs(gt^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                round(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                round^#(x1) = [0 0] x1 + [0]
                              [3 3]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {round^#(s(s(x))) -> c_3(round^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [1]
                round^#(x1) = [2 2] x1 + [2]
                              [6 0]      [0]
                c_3(x1) = [1 0] x1 + [5]
                          [2 0]      [7]
           
           * Path {4}->{2}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {1}, Uargs(gt^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                round(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                round^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {round^#(0()) -> c_1()}
               Weak Rules: {round^#(s(s(x))) -> c_3(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [0]
                0() = [0]
                      [2]
                round^#(x1) = [2 2] x1 + [4]
                              [6 2]      [0]
                c_1() = [1]
                        [0]
                c_3(x1) = [1 0] x1 + [3]
                          [2 0]      [3]
           
           * Path {4}->{3}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {1}, Uargs(gt^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                round(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                round^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {round^#(s(0())) -> c_2()}
               Weak Rules: {round^#(s(s(x))) -> c_3(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1] x1 + [2]
                        [0 1]      [1]
                0() = [1]
                      [0]
                round^#(x1) = [2 1] x1 + [1]
                              [0 0]      [7]
                c_2() = [1]
                        [1]
                c_3(x1) = [1 1] x1 + [3]
                          [0 0]      [3]
           
           * Path {7}: YES(?,O(n^2))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                round(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                round^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                               [3 3]      [3 3]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             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: {gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 6] x1 + [2]
                        [0 1]      [0]
                gt^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
                               [0 0]      [0 0]      [4]
                c_6(x1) = [1 0] x1 + [1]
                          [0 0]      [3]
           
           * Path {7}->{5}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                round(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                round^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             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: {gt^#(0(), v) -> c_4()}
               Weak Rules: {gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [0]
                        [0 1]      [2]
                0() = [2]
                      [2]
                gt^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
                               [3 3]      [0 1]      [0]
                c_4() = [1]
                        [0]
                c_6(x1) = [1 0] x1 + [3]
                          [0 0]      [7]
           
           * Path {7}->{6}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                round(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                round^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             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: {gt^#(s(u), 0()) -> c_5()}
               Weak Rules: {gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [2]
                        [0 1]      [2]
                0() = [0]
                      [2]
                gt^#(x1, x2) = [2 0] x1 + [0 2] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_5() = [1]
                        [0]
                c_6(x1) = [1 0] x1 + [3]
                          [0 0]      [0]
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(true(), x, y) -> c_0(f^#(gt(x, y), x, round(s(y))))
              , 2: round^#(0()) -> c_1()
              , 3: round^#(s(0())) -> c_2()
              , 4: round^#(s(s(x))) -> c_3(round^#(x))
              , 5: gt^#(0(), v) -> c_4()
              , 6: gt^#(s(u), 0()) -> c_5()
              , 7: gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{7}                                                       [   YES(?,O(n^1))    ]
                |
                |->{5}                                                   [   YES(?,O(n^1))    ]
                |
                `->{6}                                                   [   YES(?,O(n^1))    ]
             
             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                |->{2}                                                   [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [    YES(?,O(1))     ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules for this path are:
             
               {  round(0()) -> 0()
                , round(s(0())) -> s(s(0()))
                , round(s(s(x))) -> s(s(round(x)))
                , gt(0(), v) -> false()
                , gt(s(u), 0()) -> true()
                , gt(s(u), s(v)) -> gt(u, v)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  f^#(true(), x, y) -> c_0(f^#(gt(x, y), x, round(s(y))))
                  , round(0()) -> 0()
                  , round(s(0())) -> s(s(0()))
                  , round(s(s(x))) -> s(s(round(x)))
                  , gt(0(), v) -> false()
                  , gt(s(u), 0()) -> true()
                  , gt(s(u), s(v)) -> gt(u, v)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {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(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {1}, Uargs(gt^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                gt(x1, x2) = [0] x1 + [0] x2 + [0]
                round(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                false() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                round^#(x1) = [3] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(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: {round^#(s(s(x))) -> c_3(round^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                round^#(x1) = [2] x1 + [0]
                c_3(x1) = [1] x1 + [7]
           
           * Path {4}->{2}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {1}, Uargs(gt^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                gt(x1, x2) = [0] x1 + [0] x2 + [0]
                round(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                round^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(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: {round^#(0()) -> c_1()}
               Weak Rules: {round^#(s(s(x))) -> c_3(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                0() = [2]
                round^#(x1) = [2] x1 + [4]
                c_1() = [1]
                c_3(x1) = [1] x1 + [0]
           
           * Path {4}->{3}: YES(?,O(1))
             --------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {1}, Uargs(gt^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                gt(x1, x2) = [0] x1 + [0] x2 + [0]
                round(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                round^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(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: {round^#(s(0())) -> c_2()}
               Weak Rules: {round^#(s(s(x))) -> c_3(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0] x1 + [0]
                0() = [0]
                round^#(x1) = [0] x1 + [1]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
           
           * Path {7}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                gt(x1, x2) = [0] x1 + [0] x2 + [0]
                round(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                false() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                round^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                gt^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_6(x1) = [1] x1 + [7]
           
           * Path {7}->{5}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                gt(x1, x2) = [0] x1 + [0] x2 + [0]
                round(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                round^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {gt^#(0(), v) -> c_4()}
               Weak Rules: {gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                gt^#(x1, x2) = [3] x1 + [3] x2 + [2]
                c_4() = [1]
                c_6(x1) = [1] x1 + [5]
           
           * Path {7}->{6}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                gt(x1, x2) = [0] x1 + [0] x2 + [0]
                round(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                round^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {gt^#(s(u), 0()) -> c_5()}
               Weak Rules: {gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_5() = [1]
                c_6(x1) = [1] x1 + [7]
    
    4) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    

Tool RC1

Execution TimeUnknown
Answer
MAYBE
InputGTSSK07 cade09

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputGTSSK07 cade09

stdout:

MAYBE

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

Proof Output:    
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(true(), x, y) -> c_0(f^#(gt(x, y), x, round(s(y))))
              , 2: round^#(0()) -> c_1()
              , 3: round^#(s(0())) -> c_2()
              , 4: round^#(s(s(x))) -> c_3(round^#(x))
              , 5: gt^#(0(), v) -> c_4()
              , 6: gt^#(s(u), 0()) -> c_5()
              , 7: gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{7}                                                       [   YES(?,O(n^1))    ]
                |
                |->{5}                                                   [   YES(?,O(n^1))    ]
                |
                `->{6}                                                   [   YES(?,O(n^2))    ]
             
             ->{4}                                                       [   YES(?,O(n^2))    ]
                |
                |->{2}                                                   [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^2))    ]
             
             ->{1}                                                       [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: NA
             ------------
             
             The usable rules for this path are:
             
               {  round(0()) -> 0()
                , round(s(0())) -> s(s(0()))
                , round(s(s(x))) -> s(s(round(x)))
                , gt(0(), v) -> false()
                , gt(s(u), 0()) -> true()
                , gt(s(u), s(v)) -> gt(u, v)}
             
             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 {4}: YES(?,O(n^2))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {1}, Uargs(gt^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                round(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                round^#(x1) = [0 0 0] x1 + [0]
                              [3 3 3]      [0]
                              [3 3 3]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                gt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {round^#(s(s(x))) -> c_3(round^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [0]
                        [0 0 1]      [0]
                        [0 0 1]      [2]
                round^#(x1) = [2 0 2] x1 + [0]
                              [4 0 2]      [0]
                              [0 0 0]      [2]
                c_3(x1) = [1 0 2] x1 + [3]
                          [2 0 0]      [0]
                          [0 0 0]      [2]
           
           * Path {4}->{2}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {1}, Uargs(gt^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                round(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                round^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                gt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {round^#(0()) -> c_1()}
               Weak Rules: {round^#(s(s(x))) -> c_3(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [0]
                        [0 0 0]      [2]
                        [0 0 0]      [2]
                0() = [2]
                      [2]
                      [2]
                round^#(x1) = [2 0 0] x1 + [0]
                              [0 0 2]      [4]
                              [1 2 2]      [0]
                c_1() = [1]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [7]
                          [0 0 0]      [6]
           
           * Path {4}->{3}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {1}, Uargs(gt^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                round(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                round^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                gt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {round^#(s(0())) -> c_2()}
               Weak Rules: {round^#(s(s(x))) -> c_3(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 0] x1 + [2]
                        [0 1 2]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                round^#(x1) = [2 3 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 5 0]      [0]
                c_2() = [1]
                        [0]
                        [0]
                c_3(x1) = [1 0 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
           
           * Path {7}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                round(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                round^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                gt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [3 3 3]      [3 3 3]      [0]
                               [3 3 3]      [3 3 3]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 3] x1 + [0]
                        [0 0 1]      [4]
                        [0 0 0]      [2]
                gt^#(x1, x2) = [1 1 0] x1 + [1 0 2] x2 + [0]
                               [0 0 2]      [0 0 0]      [2]
                               [1 1 0]      [4 1 0]      [0]
                c_6(x1) = [1 2 0] x1 + [3]
                          [0 0 0]      [2]
                          [0 2 0]      [3]
           
           * Path {7}->{5}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                round(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                round^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                gt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {gt^#(0(), v) -> c_4()}
               Weak Rules: {gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 4 2] x1 + [0]
                        [0 0 2]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [2]
                      [0]
                gt^#(x1, x2) = [2 2 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [2 0 0]      [0 4 0]      [0]
                c_4() = [1]
                        [0]
                        [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
           
           * Path {7}->{6}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                gt(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                round(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                f^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                round^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                gt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {gt^#(s(u), 0()) -> c_5()}
               Weak Rules: {gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0 0] x1 + [0]
                        [0 1 2]      [2]
                        [0 0 0]      [2]
                0() = [0]
                      [0]
                      [0]
                gt^#(x1, x2) = [0 2 2] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [4 0 0]      [4]
                               [0 0 0]      [0 0 0]      [2]
                c_5() = [1]
                        [0]
                        [0]
                c_6(x1) = [1 0 0] x1 + [0]
                          [0 0 2]      [0]
                          [0 0 0]      [2]
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(true(), x, y) -> c_0(f^#(gt(x, y), x, round(s(y))))
              , 2: round^#(0()) -> c_1()
              , 3: round^#(s(0())) -> c_2()
              , 4: round^#(s(s(x))) -> c_3(round^#(x))
              , 5: gt^#(0(), v) -> c_4()
              , 6: gt^#(s(u), 0()) -> c_5()
              , 7: gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{7}                                                       [   YES(?,O(n^2))    ]
                |
                |->{5}                                                   [   YES(?,O(n^2))    ]
                |
                `->{6}                                                   [   YES(?,O(n^1))    ]
             
             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                |->{2}                                                   [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^2))    ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules for this path are:
             
               {  round(0()) -> 0()
                , round(s(0())) -> s(s(0()))
                , round(s(s(x))) -> s(s(round(x)))
                , gt(0(), v) -> false()
                , gt(s(u), 0()) -> true()
                , gt(s(u), s(v)) -> gt(u, v)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  f^#(true(), x, y) -> c_0(f^#(gt(x, y), x, round(s(y))))
                  , round(0()) -> 0()
                  , round(s(0())) -> s(s(0()))
                  , round(s(s(x))) -> s(s(round(x)))
                  , gt(0(), v) -> false()
                  , gt(s(u), 0()) -> true()
                  , gt(s(u), s(v)) -> gt(u, v)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {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(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {1}, Uargs(gt^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                round(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                round^#(x1) = [0 0] x1 + [0]
                              [3 3]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {round^#(s(s(x))) -> c_3(round^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [1]
                round^#(x1) = [2 2] x1 + [2]
                              [6 0]      [0]
                c_3(x1) = [1 0] x1 + [5]
                          [2 0]      [7]
           
           * Path {4}->{2}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {1}, Uargs(gt^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                round(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                round^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {round^#(0()) -> c_1()}
               Weak Rules: {round^#(s(s(x))) -> c_3(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [0]
                0() = [0]
                      [2]
                round^#(x1) = [2 2] x1 + [4]
                              [6 2]      [0]
                c_1() = [1]
                        [0]
                c_3(x1) = [1 0] x1 + [3]
                          [2 0]      [3]
           
           * Path {4}->{3}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {1}, Uargs(gt^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                round(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                round^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {round^#(s(0())) -> c_2()}
               Weak Rules: {round^#(s(s(x))) -> c_3(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1] x1 + [2]
                        [0 1]      [1]
                0() = [1]
                      [0]
                round^#(x1) = [2 1] x1 + [1]
                              [0 0]      [7]
                c_2() = [1]
                        [1]
                c_3(x1) = [1 1] x1 + [3]
                          [0 0]      [3]
           
           * Path {7}: YES(?,O(n^2))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                round(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                round^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                               [3 3]      [3 3]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             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: {gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 6] x1 + [2]
                        [0 1]      [0]
                gt^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
                               [0 0]      [0 0]      [4]
                c_6(x1) = [1 0] x1 + [1]
                          [0 0]      [3]
           
           * Path {7}->{5}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                round(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                round^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             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: {gt^#(0(), v) -> c_4()}
               Weak Rules: {gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [0]
                        [0 1]      [2]
                0() = [2]
                      [2]
                gt^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
                               [3 3]      [0 1]      [0]
                c_4() = [1]
                        [0]
                c_6(x1) = [1 0] x1 + [3]
                          [0 0]      [7]
           
           * Path {7}->{6}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                gt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                round(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                f^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                round^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             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: {gt^#(s(u), 0()) -> c_5()}
               Weak Rules: {gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [2]
                        [0 1]      [2]
                0() = [0]
                      [2]
                gt^#(x1, x2) = [2 0] x1 + [0 2] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_5() = [1]
                        [0]
                c_6(x1) = [1 0] x1 + [3]
                          [0 0]      [0]
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(true(), x, y) -> c_0(f^#(gt(x, y), x, round(s(y))))
              , 2: round^#(0()) -> c_1()
              , 3: round^#(s(0())) -> c_2()
              , 4: round^#(s(s(x))) -> c_3(round^#(x))
              , 5: gt^#(0(), v) -> c_4()
              , 6: gt^#(s(u), 0()) -> c_5()
              , 7: gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{7}                                                       [   YES(?,O(n^1))    ]
                |
                |->{5}                                                   [   YES(?,O(n^1))    ]
                |
                `->{6}                                                   [   YES(?,O(n^1))    ]
             
             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                |->{2}                                                   [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [    YES(?,O(1))     ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules for this path are:
             
               {  round(0()) -> 0()
                , round(s(0())) -> s(s(0()))
                , round(s(s(x))) -> s(s(round(x)))
                , gt(0(), v) -> false()
                , gt(s(u), 0()) -> true()
                , gt(s(u), s(v)) -> gt(u, v)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  f^#(true(), x, y) -> c_0(f^#(gt(x, y), x, round(s(y))))
                  , round(0()) -> 0()
                  , round(s(0())) -> s(s(0()))
                  , round(s(s(x))) -> s(s(round(x)))
                  , gt(0(), v) -> false()
                  , gt(s(u), 0()) -> true()
                  , gt(s(u), s(v)) -> gt(u, v)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {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(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {1}, Uargs(gt^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                gt(x1, x2) = [0] x1 + [0] x2 + [0]
                round(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                false() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                round^#(x1) = [3] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(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: {round^#(s(s(x))) -> c_3(round^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                round^#(x1) = [2] x1 + [0]
                c_3(x1) = [1] x1 + [7]
           
           * Path {4}->{2}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {1}, Uargs(gt^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                gt(x1, x2) = [0] x1 + [0] x2 + [0]
                round(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                round^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(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: {round^#(0()) -> c_1()}
               Weak Rules: {round^#(s(s(x))) -> c_3(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                0() = [2]
                round^#(x1) = [2] x1 + [4]
                c_1() = [1]
                c_3(x1) = [1] x1 + [0]
           
           * Path {4}->{3}: YES(?,O(1))
             --------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {1}, Uargs(gt^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                gt(x1, x2) = [0] x1 + [0] x2 + [0]
                round(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                round^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(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: {round^#(s(0())) -> c_2()}
               Weak Rules: {round^#(s(s(x))) -> c_3(round^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(round^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0] x1 + [0]
                0() = [0]
                round^#(x1) = [0] x1 + [1]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
           
           * Path {7}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                gt(x1, x2) = [0] x1 + [0] x2 + [0]
                round(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                false() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                round^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                gt^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_6(x1) = [1] x1 + [7]
           
           * Path {7}->{5}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                gt(x1, x2) = [0] x1 + [0] x2 + [0]
                round(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                round^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {gt^#(0(), v) -> c_4()}
               Weak Rules: {gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                gt^#(x1, x2) = [3] x1 + [3] x2 + [2]
                c_4() = [1]
                c_6(x1) = [1] x1 + [5]
           
           * Path {7}->{6}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(gt) = {}, Uargs(round) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(round^#) = {},
                 Uargs(c_3) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                gt(x1, x2) = [0] x1 + [0] x2 + [0]
                round(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                round^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                gt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5() = [0]
                c_6(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {gt^#(s(u), 0()) -> c_5()}
               Weak Rules: {gt^#(s(u), s(v)) -> c_6(gt^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gt^#) = {}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                gt^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_5() = [1]
                c_6(x1) = [1] x1 + [7]
    
    4) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.