Problem AG01 3.19

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputAG01 3.19

stdout:

MAYBE

Problem:
 minus(x,0()) -> x
 minus(s(x),s(y)) -> minus(x,y)
 double(0()) -> 0()
 double(s(x)) -> s(s(double(x)))
 plus(0(),y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(s(x),y) -> plus(x,s(y))
 plus(s(x),y) -> s(plus(minus(x,y),double(y)))
 plus(s(plus(x,y)),z) -> s(plus(plus(x,y),z))

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputAG01 3.19

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputAG01 3.19

stdout:

MAYBE

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

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: minus^#(x, 0()) -> c_0()
              , 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
              , 3: double^#(0()) -> c_2()
              , 4: double^#(s(x)) -> c_3(double^#(x))
              , 5: plus^#(0(), y) -> c_4()
              , 6: plus^#(s(x), y) -> c_5(plus^#(x, y))
              , 7: plus^#(s(x), y) -> c_6(plus^#(x, s(y)))
              , 8: plus^#(s(x), y) -> c_7(plus^#(minus(x, y), double(y)))
              , 9: plus^#(s(plus(x, y)), z) -> c_8(plus^#(plus(x, y), z))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6,9,8,7}                                                 [     inherited      ]
                |
                `->{5}                                                   [         NA         ]
             
             ->{4}                                                       [   YES(?,O(n^2))    ]
                |
                `->{3}                                                   [   YES(?,O(n^2))    ]
             
             ->{2}                                                       [   YES(?,O(n^1))    ]
                |
                `->{1}                                                   [   YES(?,O(n^3))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {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(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                double(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                plus(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]
                minus^#(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_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                double^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                plus^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(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: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 0] x1 + [2]
                        [0 0 2]      [2]
                        [0 0 0]      [0]
                minus^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
                                  [2 2 0]      [0 2 0]      [0]
                                  [4 0 0]      [0 2 0]      [0]
                c_1(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [7]
           
           * Path {2}->{1}: YES(?,O(n^3))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                double(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                plus(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]
                minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                double^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                plus^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(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^3))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {minus^#(x, 0()) -> c_0()}
               Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                s(x1) = [1 2 0] x1 + [2]
                        [0 1 3]      [2]
                        [0 0 1]      [2]
                minus^#(x1, x2) = [0 0 0] x1 + [0 2 2] x2 + [0]
                                  [0 0 2]      [2 2 0]      [0]
                                  [0 0 0]      [0 2 2]      [0]
                c_0() = [1]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [0]
                          [0 0 0]      [7]
           
           * 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(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(plus^#) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                double(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                plus(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]
                minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                double^#(x1) = [0 0 0] x1 + [0]
                               [3 3 3]      [0]
                               [3 3 3]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                plus^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(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: {double^#(s(x)) -> c_3(double^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 2] x1 + [2]
                        [0 1 2]      [2]
                        [0 0 0]      [0]
                double^#(x1) = [0 1 0] x1 + [2]
                               [6 0 0]      [0]
                               [2 3 0]      [2]
                c_3(x1) = [1 0 0] x1 + [1]
                          [2 0 2]      [0]
                          [0 0 0]      [0]
           
           * 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(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(plus^#) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                double(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                plus(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]
                minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                double^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                plus^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(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: {double^#(0()) -> c_2()}
               Weak Rules: {double^#(s(x)) -> c_3(double^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                s(x1) = [1 1 0] x1 + [0]
                        [0 1 1]      [1]
                        [0 0 0]      [0]
                double^#(x1) = [2 2 2] x1 + [0]
                               [0 6 0]      [0]
                               [0 0 2]      [0]
                c_2() = [1]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [2]
                          [0 0 0]      [3]
                          [0 0 0]      [0]
           
           * Path {6,9,8,7}: inherited
             -------------------------
             
             This path is subsumed by the proof of path {6,9,8,7}->{5}.
           
           * Path {6,9,8,7}->{5}: NA
             -----------------------
             
             The usable rules for this path are:
             
               {  minus(x, 0()) -> x
                , minus(s(x), s(y)) -> minus(x, y)
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))
                , plus(s(x), y) -> plus(x, s(y))
                , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
                , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: minus^#(x, 0()) -> c_0()
              , 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
              , 3: double^#(0()) -> c_2()
              , 4: double^#(s(x)) -> c_3(double^#(x))
              , 5: plus^#(0(), y) -> c_4()
              , 6: plus^#(s(x), y) -> c_5(plus^#(x, y))
              , 7: plus^#(s(x), y) -> c_6(plus^#(x, s(y)))
              , 8: plus^#(s(x), y) -> c_7(plus^#(minus(x, y), double(y)))
              , 9: plus^#(s(plus(x, y)), z) -> c_8(plus^#(plus(x, y), z))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6,9,8,7}                                                 [     inherited      ]
                |
                `->{5}                                                   [       MAYBE        ]
             
             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^1))    ]
             
             ->{2}                                                       [   YES(?,O(n^2))    ]
                |
                `->{1}                                                   [   YES(?,O(n^2))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {2}: 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(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                minus^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                  [3 3]      [3 3]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(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: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                minus^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                                  [0 2]      [0 0]      [0]
                c_1(x1) = [1 2] x1 + [5]
                          [0 0]      [3]
           
           * Path {2}->{1}: 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(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(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: {minus^#(x, 0()) -> c_0()}
               Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [0]
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [0]
                minus^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
                                  [0 0]      [4 1]      [0]
                c_0() = [1]
                        [0]
                c_1(x1) = [1 0] x1 + [6]
                          [0 0]      [7]
           
           * 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(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(plus^#) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [3 3] x1 + [0]
                               [3 3]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(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: {double^#(s(x)) -> c_3(double^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [1]
                double^#(x1) = [0 1] x1 + [1]
                               [0 0]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
           
           * Path {4}->{3}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(plus^#) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(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: {double^#(0()) -> c_2()}
               Weak Rules: {double^#(s(x)) -> c_3(double^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [3]
                double^#(x1) = [1 2] x1 + [2]
                               [6 1]      [0]
                c_2() = [1]
                        [0]
                c_3(x1) = [1 0] x1 + [5]
                          [2 0]      [3]
           
           * Path {6,9,8,7}: inherited
             -------------------------
             
             This path is subsumed by the proof of path {6,9,8,7}->{5}.
           
           * Path {6,9,8,7}->{5}: MAYBE
             --------------------------
             
             The usable rules for this path are:
             
               {  minus(x, 0()) -> x
                , minus(s(x), s(y)) -> minus(x, y)
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))
                , plus(s(x), y) -> plus(x, s(y))
                , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
                , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}
             
             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:
                 {  plus^#(s(x), y) -> c_5(plus^#(x, y))
                  , plus^#(s(plus(x, y)), z) -> c_8(plus^#(plus(x, y), z))
                  , plus^#(s(x), y) -> c_7(plus^#(minus(x, y), double(y)))
                  , plus^#(s(x), y) -> c_6(plus^#(x, s(y)))
                  , plus^#(0(), y) -> c_4()
                  , minus(x, 0()) -> x
                  , minus(s(x), s(y)) -> minus(x, y)
                  , double(0()) -> 0()
                  , double(s(x)) -> s(s(double(x)))
                  , plus(0(), y) -> y
                  , plus(s(x), y) -> s(plus(x, y))
                  , plus(s(x), y) -> plus(x, s(y))
                  , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
                  , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}
             
             Proof Output:    
               The input cannot be shown compatible
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: minus^#(x, 0()) -> c_0()
              , 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
              , 3: double^#(0()) -> c_2()
              , 4: double^#(s(x)) -> c_3(double^#(x))
              , 5: plus^#(0(), y) -> c_4()
              , 6: plus^#(s(x), y) -> c_5(plus^#(x, y))
              , 7: plus^#(s(x), y) -> c_6(plus^#(x, s(y)))
              , 8: plus^#(s(x), y) -> c_7(plus^#(minus(x, y), double(y)))
              , 9: plus^#(s(plus(x, y)), z) -> c_8(plus^#(plus(x, y), z))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6,9,8,7}                                                 [     inherited      ]
                |
                `->{5}                                                   [       MAYBE        ]
             
             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^1))    ]
             
             ->{2}                                                       [   YES(?,O(n^1))    ]
                |
                `->{1}                                                   [   YES(?,O(n^1))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {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(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                double(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                minus^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_0() = [0]
                c_1(x1) = [1] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(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: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                minus^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_1(x1) = [1] x1 + [7]
           
           * Path {2}->{1}: 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(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(plus^#) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                double(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [1] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(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: {minus^#(x, 0()) -> c_0()}
               Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                minus^#(x1, x2) = [2] x1 + [2] x2 + [4]
                c_0() = [1]
                c_1(x1) = [1] x1 + [7]
           
           * 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(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(plus^#) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                double(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                double^#(x1) = [3] x1 + [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(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: {double^#(s(x)) -> c_3(double^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [4]
                double^#(x1) = [2] x1 + [0]
                c_3(x1) = [1] x1 + [7]
           
           * Path {4}->{3}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(plus^#) = {},
                 Uargs(c_5) = {}, Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                double(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(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: {double^#(0()) -> c_2()}
               Weak Rules: {double^#(s(x)) -> c_3(double^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [0]
                double^#(x1) = [2] x1 + [0]
                c_2() = [1]
                c_3(x1) = [1] x1 + [0]
           
           * Path {6,9,8,7}: inherited
             -------------------------
             
             This path is subsumed by the proof of path {6,9,8,7}->{5}.
           
           * Path {6,9,8,7}->{5}: MAYBE
             --------------------------
             
             The usable rules for this path are:
             
               {  minus(x, 0()) -> x
                , minus(s(x), s(y)) -> minus(x, y)
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))
                , plus(s(x), y) -> plus(x, s(y))
                , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
                , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}
             
             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:
                 {  plus^#(s(x), y) -> c_5(plus^#(x, y))
                  , plus^#(s(plus(x, y)), z) -> c_8(plus^#(plus(x, y), z))
                  , plus^#(s(x), y) -> c_7(plus^#(minus(x, y), double(y)))
                  , plus^#(s(x), y) -> c_6(plus^#(x, s(y)))
                  , plus^#(0(), y) -> c_4()
                  , minus(x, 0()) -> x
                  , minus(s(x), s(y)) -> minus(x, y)
                  , double(0()) -> 0()
                  , double(s(x)) -> s(s(double(x)))
                  , plus(0(), y) -> y
                  , plus(s(x), y) -> s(plus(x, y))
                  , plus(s(x), y) -> plus(x, s(y))
                  , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
                  , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}
             
             Proof Output:    
               The input cannot be shown compatible
    
    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
InputAG01 3.19

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputAG01 3.19

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  minus(x, 0()) -> x
     , minus(s(x), s(y)) -> minus(x, y)
     , double(0()) -> 0()
     , double(s(x)) -> s(s(double(x)))
     , plus(0(), y) -> y
     , plus(s(x), y) -> s(plus(x, y))
     , plus(s(x), y) -> plus(x, s(y))
     , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
     , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}

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: minus^#(x, 0()) -> c_0(x)
              , 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
              , 3: double^#(0()) -> c_2()
              , 4: double^#(s(x)) -> c_3(double^#(x))
              , 5: plus^#(0(), y) -> c_4(y)
              , 6: plus^#(s(x), y) -> c_5(plus^#(x, y))
              , 7: plus^#(s(x), y) -> c_6(plus^#(x, s(y)))
              , 8: plus^#(s(x), y) -> c_7(plus^#(minus(x, y), double(y)))
              , 9: plus^#(s(plus(x, y)), z) -> c_8(plus^#(plus(x, y), z))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6,9,8,7}                                                 [     inherited      ]
                |
                `->{5}                                                   [         NA         ]
             
             ->{4}                                                       [   YES(?,O(n^2))    ]
                |
                `->{3}                                                   [   YES(?,O(n^2))    ]
             
             ->{2}                                                       [   YES(?,O(n^1))    ]
                |
                `->{1}                                                   [   YES(?,O(n^2))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {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(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {1}, Uargs(double^#) = {}, Uargs(c_3) = {},
                 Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                double(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                plus(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]
                minus^#(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_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                double^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                plus^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(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: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 0] x1 + [2]
                        [0 0 2]      [2]
                        [0 0 0]      [0]
                minus^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
                                  [2 2 0]      [0 2 0]      [0]
                                  [4 0 0]      [0 2 0]      [0]
                c_1(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [7]
           
           * Path {2}->{1}: 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(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {1}, Uargs(double^#) = {}, Uargs(c_3) = {},
                 Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                double(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                plus(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]
                minus^#(x1, x2) = [3 3 3] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [1 1 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                double^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                plus^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(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: {minus^#(x, 0()) -> c_0(x)}
               Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                s(x1) = [1 2 1] x1 + [0]
                        [0 0 4]      [2]
                        [0 0 1]      [2]
                minus^#(x1, x2) = [2 0 2] x1 + [2 0 2] x2 + [0]
                                  [0 2 0]      [0 3 2]      [0]
                                  [4 2 2]      [5 2 0]      [0]
                c_0(x1) = [0 0 0] x1 + [1]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1(x1) = [1 0 0] x1 + [6]
                          [0 0 0]      [3]
                          [2 3 0]      [7]
           
           * 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(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1},
                 Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 0]      [0]
                double(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                plus(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]
                minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                double^#(x1) = [0 0 0] x1 + [0]
                               [3 3 3]      [0]
                               [3 3 3]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                plus^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(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: {double^#(s(x)) -> c_3(double^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 2] x1 + [2]
                        [0 1 2]      [2]
                        [0 0 0]      [0]
                double^#(x1) = [0 1 0] x1 + [2]
                               [6 0 0]      [0]
                               [2 3 0]      [2]
                c_3(x1) = [1 0 0] x1 + [1]
                          [2 0 2]      [0]
                          [0 0 0]      [0]
           
           * 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(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1},
                 Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                double(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                plus(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]
                minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                double^#(x1) = [0 0 0] x1 + [0]
                               [0 0 0]      [0]
                               [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                plus^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_8(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: {double^#(0()) -> c_2()}
               Weak Rules: {double^#(s(x)) -> c_3(double^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                s(x1) = [1 1 0] x1 + [0]
                        [0 1 1]      [1]
                        [0 0 0]      [0]
                double^#(x1) = [2 2 2] x1 + [0]
                               [0 6 0]      [0]
                               [0 0 2]      [0]
                c_2() = [1]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [2]
                          [0 0 0]      [3]
                          [0 0 0]      [0]
           
           * Path {6,9,8,7}: inherited
             -------------------------
             
             This path is subsumed by the proof of path {6,9,8,7}->{5}.
           
           * Path {6,9,8,7}->{5}: NA
             -----------------------
             
             The usable rules for this path are:
             
               {  minus(x, 0()) -> x
                , minus(s(x), s(y)) -> minus(x, y)
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))
                , plus(s(x), y) -> plus(x, s(y))
                , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
                , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: minus^#(x, 0()) -> c_0(x)
              , 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
              , 3: double^#(0()) -> c_2()
              , 4: double^#(s(x)) -> c_3(double^#(x))
              , 5: plus^#(0(), y) -> c_4(y)
              , 6: plus^#(s(x), y) -> c_5(plus^#(x, y))
              , 7: plus^#(s(x), y) -> c_6(plus^#(x, s(y)))
              , 8: plus^#(s(x), y) -> c_7(plus^#(minus(x, y), double(y)))
              , 9: plus^#(s(plus(x, y)), z) -> c_8(plus^#(plus(x, y), z))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6,9,8,7}                                                 [     inherited      ]
                |
                `->{5}                                                   [       MAYBE        ]
             
             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^1))    ]
             
             ->{2}                                                       [   YES(?,O(n^2))    ]
                |
                `->{1}                                                   [   YES(?,O(n^2))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {2}: 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(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {1}, Uargs(double^#) = {}, Uargs(c_3) = {},
                 Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                minus^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                  [3 3]      [3 3]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(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: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                minus^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                                  [0 2]      [0 0]      [0]
                c_1(x1) = [1 2] x1 + [5]
                          [0 0]      [3]
           
           * Path {2}->{1}: 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(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {1}, Uargs(double^#) = {}, Uargs(c_3) = {},
                 Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                minus^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(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: {minus^#(x, 0()) -> c_0(x)}
               Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [2]
                s(x1) = [1 2] x1 + [0]
                        [0 1]      [0]
                minus^#(x1, x2) = [2 2] x1 + [0 2] x2 + [0]
                                  [4 1]      [3 2]      [0]
                c_0(x1) = [0 0] x1 + [1]
                          [0 1]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [2 0]      [0]
           
           * 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(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1},
                 Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [3 3] x1 + [0]
                               [3 3]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(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: {double^#(s(x)) -> c_3(double^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [1]
                double^#(x1) = [0 1] x1 + [1]
                               [0 0]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
           
           * Path {4}->{3}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1},
                 Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                plus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_8(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: {double^#(0()) -> c_2()}
               Weak Rules: {double^#(s(x)) -> c_3(double^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [3]
                double^#(x1) = [1 2] x1 + [2]
                               [6 1]      [0]
                c_2() = [1]
                        [0]
                c_3(x1) = [1 0] x1 + [5]
                          [2 0]      [3]
           
           * Path {6,9,8,7}: inherited
             -------------------------
             
             This path is subsumed by the proof of path {6,9,8,7}->{5}.
           
           * Path {6,9,8,7}->{5}: MAYBE
             --------------------------
             
             The usable rules for this path are:
             
               {  minus(x, 0()) -> x
                , minus(s(x), s(y)) -> minus(x, y)
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))
                , plus(s(x), y) -> plus(x, s(y))
                , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
                , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}
             
             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:
                 {  plus^#(s(x), y) -> c_5(plus^#(x, y))
                  , plus^#(s(plus(x, y)), z) -> c_8(plus^#(plus(x, y), z))
                  , plus^#(s(x), y) -> c_7(plus^#(minus(x, y), double(y)))
                  , plus^#(s(x), y) -> c_6(plus^#(x, s(y)))
                  , plus^#(0(), y) -> c_4(y)
                  , minus(x, 0()) -> x
                  , minus(s(x), s(y)) -> minus(x, y)
                  , double(0()) -> 0()
                  , double(s(x)) -> s(s(double(x)))
                  , plus(0(), y) -> y
                  , plus(s(x), y) -> s(plus(x, y))
                  , plus(s(x), y) -> plus(x, s(y))
                  , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
                  , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}
             
             Proof Output:    
               The input cannot be shown compatible
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: minus^#(x, 0()) -> c_0(x)
              , 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
              , 3: double^#(0()) -> c_2()
              , 4: double^#(s(x)) -> c_3(double^#(x))
              , 5: plus^#(0(), y) -> c_4(y)
              , 6: plus^#(s(x), y) -> c_5(plus^#(x, y))
              , 7: plus^#(s(x), y) -> c_6(plus^#(x, s(y)))
              , 8: plus^#(s(x), y) -> c_7(plus^#(minus(x, y), double(y)))
              , 9: plus^#(s(plus(x, y)), z) -> c_8(plus^#(plus(x, y), z))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6,9,8,7}                                                 [     inherited      ]
                |
                `->{5}                                                   [       MAYBE        ]
             
             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^1))    ]
             
             ->{2}                                                       [   YES(?,O(n^1))    ]
                |
                `->{1}                                                   [   YES(?,O(n^1))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {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(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {1}, Uargs(double^#) = {}, Uargs(c_3) = {},
                 Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                double(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                minus^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1(x1) = [1] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(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: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                minus^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_1(x1) = [1] x1 + [7]
           
           * Path {2}->{1}: 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(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {1}, Uargs(double^#) = {}, Uargs(c_3) = {},
                 Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                double(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                minus^#(x1, x2) = [3] x1 + [0] x2 + [0]
                c_0(x1) = [1] x1 + [0]
                c_1(x1) = [1] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(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: {minus^#(x, 0()) -> c_0(x)}
               Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                s(x1) = [1] x1 + [2]
                minus^#(x1, x2) = [2] x1 + [2] x2 + [2]
                c_0(x1) = [0] x1 + [1]
                c_1(x1) = [1] x1 + [5]
           
           * 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(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1},
                 Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                double(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                double^#(x1) = [3] x1 + [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(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: {double^#(s(x)) -> c_3(double^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [4]
                double^#(x1) = [2] x1 + [0]
                c_3(x1) = [1] x1 + [7]
           
           * Path {4}->{3}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {}, Uargs(s) = {}, Uargs(double) = {},
                 Uargs(plus) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1},
                 Uargs(plus^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                double(x1) = [0] x1 + [0]
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                double^#(x1) = [0] x1 + [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                c_8(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: {double^#(0()) -> c_2()}
               Weak Rules: {double^#(s(x)) -> c_3(double^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [0]
                double^#(x1) = [2] x1 + [0]
                c_2() = [1]
                c_3(x1) = [1] x1 + [0]
           
           * Path {6,9,8,7}: inherited
             -------------------------
             
             This path is subsumed by the proof of path {6,9,8,7}->{5}.
           
           * Path {6,9,8,7}->{5}: MAYBE
             --------------------------
             
             The usable rules for this path are:
             
               {  minus(x, 0()) -> x
                , minus(s(x), s(y)) -> minus(x, y)
                , double(0()) -> 0()
                , double(s(x)) -> s(s(double(x)))
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(x, y))
                , plus(s(x), y) -> plus(x, s(y))
                , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
                , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}
             
             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:
                 {  plus^#(s(x), y) -> c_5(plus^#(x, y))
                  , plus^#(s(plus(x, y)), z) -> c_8(plus^#(plus(x, y), z))
                  , plus^#(s(x), y) -> c_7(plus^#(minus(x, y), double(y)))
                  , plus^#(s(x), y) -> c_6(plus^#(x, s(y)))
                  , plus^#(0(), y) -> c_4(y)
                  , minus(x, 0()) -> x
                  , minus(s(x), s(y)) -> minus(x, y)
                  , double(0()) -> 0()
                  , double(s(x)) -> s(s(double(x)))
                  , plus(0(), y) -> y
                  , plus(s(x), y) -> s(plus(x, y))
                  , plus(s(x), y) -> plus(x, s(y))
                  , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
                  , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}
             
             Proof Output:    
               The input cannot be shown compatible
    
    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 pair1rc

Execution TimeUnknown
Answer
MAYBE
InputAG01 3.19

stdout:

MAYBE

We consider the following Problem:

  Strict Trs:
    {  minus(x, 0()) -> x
     , minus(s(x), s(y)) -> minus(x, y)
     , double(0()) -> 0()
     , double(s(x)) -> s(s(double(x)))
     , plus(0(), y) -> y
     , plus(s(x), y) -> s(plus(x, y))
     , plus(s(x), y) -> plus(x, s(y))
     , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
     , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}
  StartTerms: basic terms
  Strategy: none

Certificate: MAYBE

Application of 'pair1 (timeout of 60.0 seconds)':
-------------------------------------------------
  The processor is not applicable, reason is:
   Input problem is not restricted to innermost rewriting
  
  We abort the transformation and continue with the subprocessor on the problem
  
  Strict Trs:
    {  minus(x, 0()) -> x
     , minus(s(x), s(y)) -> minus(x, y)
     , double(0()) -> 0()
     , double(s(x)) -> s(s(double(x)))
     , plus(0(), y) -> y
     , plus(s(x), y) -> s(plus(x, y))
     , plus(s(x), y) -> plus(x, s(y))
     , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
     , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}
  StartTerms: basic terms
  Strategy: none
  
  1) None of the processors succeeded.
     
     Details of failed attempt(s):
     -----------------------------
       1) 'dp' failed due to the following reason:
            We have computed the following dependency pairs
            
            Strict Dependency Pairs:
              {  minus^#(x, 0()) -> c_1(x)
               , minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
               , double^#(0()) -> c_3()
               , double^#(s(x)) -> c_4(double^#(x))
               , plus^#(0(), y) -> c_5(y)
               , plus^#(s(x), y) -> c_6(plus^#(x, y))
               , plus^#(s(x), y) -> c_7(plus^#(x, s(y)))
               , plus^#(s(x), y) -> c_8(plus^#(minus(x, y), double(y)))
               , plus^#(s(plus(x, y)), z) -> c_9(plus^#(plus(x, y), z))}
            
            We consider the following Problem:
            
              Strict DPs:
                {  minus^#(x, 0()) -> c_1(x)
                 , minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
                 , double^#(0()) -> c_3()
                 , double^#(s(x)) -> c_4(double^#(x))
                 , plus^#(0(), y) -> c_5(y)
                 , plus^#(s(x), y) -> c_6(plus^#(x, y))
                 , plus^#(s(x), y) -> c_7(plus^#(x, s(y)))
                 , plus^#(s(x), y) -> c_8(plus^#(minus(x, y), double(y)))
                 , plus^#(s(plus(x, y)), z) -> c_9(plus^#(plus(x, y), z))}
              Strict Trs:
                {  minus(x, 0()) -> x
                 , minus(s(x), s(y)) -> minus(x, y)
                 , double(0()) -> 0()
                 , double(s(x)) -> s(s(double(x)))
                 , plus(0(), y) -> y
                 , plus(s(x), y) -> s(plus(x, y))
                 , plus(s(x), y) -> plus(x, s(y))
                 , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
                 , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}
              StartTerms: basic terms
              Strategy: none
            
            Certificate: MAYBE
            
            Application of 'Fastest':
            -------------------------
              None of the processors succeeded.
              
              Details of failed attempt(s):
              -----------------------------
                'Sequentially' failed due to the following reason:
                  None of the processors succeeded.
                  
                  Details of failed attempt(s):
                  -----------------------------
                    1) 'empty' failed due to the following reason:
                         Empty strict component of the problem is NOT empty.
                    
                    2) 'Fastest' failed due to the following reason:
                         None of the processors succeeded.
                         
                         Details of failed attempt(s):
                         -----------------------------
                           1) 'matrix-interpretation of dimension 3 (timeout of 100.0 seconds)' failed due to the following reason:
                                The input cannot be shown compatible
                           
                           2) 'matrix-interpretation of dimension 2 (timeout of 100.0 seconds)' failed due to the following reason:
                                The input cannot be shown compatible
                           
                    
       
       2) 'Fastest' failed due to the following reason:
            None of the processors succeeded.
            
            Details of failed attempt(s):
            -----------------------------
              1) 'Sequentially' failed due to the following reason:
                   None of the processors succeeded.
                   
                   Details of failed attempt(s):
                   -----------------------------
                     1) 'empty' failed due to the following reason:
                          Empty strict component of the problem is NOT empty.
                     
                     2) 'Fastest' failed due to the following reason:
                          None of the processors succeeded.
                          
                          Details of failed attempt(s):
                          -----------------------------
                            1) 'matrix-interpretation of dimension 4 (timeout of 100.0 seconds)' failed due to the following reason:
                                 The input cannot be shown compatible
                            
                            2) 'matrix-interpretation of dimension 3 (timeout of 100.0 seconds)' failed due to the following reason:
                                 The input cannot be shown compatible
                            
                            3) 'matrix-interpretation of dimension 2 (timeout of 100.0 seconds)' failed due to the following reason:
                                 The input cannot be shown compatible
                            
                     
              
              2) 'Fastest' failed due to the following reason:
                   None of the processors succeeded.
                   
                   Details of failed attempt(s):
                   -----------------------------
                     1) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
                          match-boundness of the problem could not be verified.
                     
                     2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
                          match-boundness of the problem could not be verified.
                     
              
       
  

Arrrr..

Tool pair2rc

Execution TimeUnknown
Answer
MAYBE
InputAG01 3.19

stdout:

MAYBE

We consider the following Problem:

  Strict Trs:
    {  minus(x, 0()) -> x
     , minus(s(x), s(y)) -> minus(x, y)
     , double(0()) -> 0()
     , double(s(x)) -> s(s(double(x)))
     , plus(0(), y) -> y
     , plus(s(x), y) -> s(plus(x, y))
     , plus(s(x), y) -> plus(x, s(y))
     , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
     , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}
  StartTerms: basic terms
  Strategy: none

Certificate: MAYBE

Application of 'pair2 (timeout of 60.0 seconds)':
-------------------------------------------------
  The processor is not applicable, reason is:
   Input problem is not restricted to innermost rewriting
  
  We abort the transformation and continue with the subprocessor on the problem
  
  Strict Trs:
    {  minus(x, 0()) -> x
     , minus(s(x), s(y)) -> minus(x, y)
     , double(0()) -> 0()
     , double(s(x)) -> s(s(double(x)))
     , plus(0(), y) -> y
     , plus(s(x), y) -> s(plus(x, y))
     , plus(s(x), y) -> plus(x, s(y))
     , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
     , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}
  StartTerms: basic terms
  Strategy: none
  
  1) None of the processors succeeded.
     
     Details of failed attempt(s):
     -----------------------------
       1) 'dp' failed due to the following reason:
            We have computed the following dependency pairs
            
            Strict Dependency Pairs:
              {  minus^#(x, 0()) -> c_1(x)
               , minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
               , double^#(0()) -> c_3()
               , double^#(s(x)) -> c_4(double^#(x))
               , plus^#(0(), y) -> c_5(y)
               , plus^#(s(x), y) -> c_6(plus^#(x, y))
               , plus^#(s(x), y) -> c_7(plus^#(x, s(y)))
               , plus^#(s(x), y) -> c_8(plus^#(minus(x, y), double(y)))
               , plus^#(s(plus(x, y)), z) -> c_9(plus^#(plus(x, y), z))}
            
            We consider the following Problem:
            
              Strict DPs:
                {  minus^#(x, 0()) -> c_1(x)
                 , minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
                 , double^#(0()) -> c_3()
                 , double^#(s(x)) -> c_4(double^#(x))
                 , plus^#(0(), y) -> c_5(y)
                 , plus^#(s(x), y) -> c_6(plus^#(x, y))
                 , plus^#(s(x), y) -> c_7(plus^#(x, s(y)))
                 , plus^#(s(x), y) -> c_8(plus^#(minus(x, y), double(y)))
                 , plus^#(s(plus(x, y)), z) -> c_9(plus^#(plus(x, y), z))}
              Strict Trs:
                {  minus(x, 0()) -> x
                 , minus(s(x), s(y)) -> minus(x, y)
                 , double(0()) -> 0()
                 , double(s(x)) -> s(s(double(x)))
                 , plus(0(), y) -> y
                 , plus(s(x), y) -> s(plus(x, y))
                 , plus(s(x), y) -> plus(x, s(y))
                 , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
                 , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}
              StartTerms: basic terms
              Strategy: none
            
            Certificate: MAYBE
            
            Application of 'Fastest':
            -------------------------
              None of the processors succeeded.
              
              Details of failed attempt(s):
              -----------------------------
                'Sequentially' failed due to the following reason:
                  None of the processors succeeded.
                  
                  Details of failed attempt(s):
                  -----------------------------
                    1) 'empty' failed due to the following reason:
                         Empty strict component of the problem is NOT empty.
                    
                    2) 'Fastest' failed due to the following reason:
                         None of the processors succeeded.
                         
                         Details of failed attempt(s):
                         -----------------------------
                           1) 'matrix-interpretation of dimension 3 (timeout of 100.0 seconds)' failed due to the following reason:
                                The input cannot be shown compatible
                           
                           2) 'matrix-interpretation of dimension 2 (timeout of 100.0 seconds)' failed due to the following reason:
                                The input cannot be shown compatible
                           
                    
       
       2) 'Fastest' failed due to the following reason:
            None of the processors succeeded.
            
            Details of failed attempt(s):
            -----------------------------
              1) 'Sequentially' failed due to the following reason:
                   None of the processors succeeded.
                   
                   Details of failed attempt(s):
                   -----------------------------
                     1) 'empty' failed due to the following reason:
                          Empty strict component of the problem is NOT empty.
                     
                     2) 'Fastest' failed due to the following reason:
                          None of the processors succeeded.
                          
                          Details of failed attempt(s):
                          -----------------------------
                            1) 'matrix-interpretation of dimension 4 (timeout of 100.0 seconds)' failed due to the following reason:
                                 The input cannot be shown compatible
                            
                            2) 'matrix-interpretation of dimension 3 (timeout of 100.0 seconds)' failed due to the following reason:
                                 The input cannot be shown compatible
                            
                            3) 'matrix-interpretation of dimension 2 (timeout of 100.0 seconds)' failed due to the following reason:
                                 The input cannot be shown compatible
                            
                     
              
              2) 'Fastest' failed due to the following reason:
                   None of the processors succeeded.
                   
                   Details of failed attempt(s):
                   -----------------------------
                     1) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
                          match-boundness of the problem could not be verified.
                     
                     2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
                          match-boundness of the problem could not be verified.
                     
              
       
  

Arrrr..

Tool pair3irc

Execution TimeUnknown
Answer
TIMEOUT
InputAG01 3.19

stdout:

TIMEOUT

We consider the following Problem:

  Strict Trs:
    {  minus(x, 0()) -> x
     , minus(s(x), s(y)) -> minus(x, y)
     , double(0()) -> 0()
     , double(s(x)) -> s(s(double(x)))
     , plus(0(), y) -> y
     , plus(s(x), y) -> s(plus(x, y))
     , plus(s(x), y) -> plus(x, s(y))
     , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
     , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}
  StartTerms: basic terms
  Strategy: innermost

Certificate: TIMEOUT

Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
  Computation stopped due to timeout after 60.0 seconds

Arrrr..

Tool pair3rc

Execution TimeUnknown
Answer
TIMEOUT
InputAG01 3.19

stdout:

TIMEOUT

We consider the following Problem:

  Strict Trs:
    {  minus(x, 0()) -> x
     , minus(s(x), s(y)) -> minus(x, y)
     , double(0()) -> 0()
     , double(s(x)) -> s(s(double(x)))
     , plus(0(), y) -> y
     , plus(s(x), y) -> s(plus(x, y))
     , plus(s(x), y) -> plus(x, s(y))
     , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
     , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}
  StartTerms: basic terms
  Strategy: none

Certificate: TIMEOUT

Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
  Computation stopped due to timeout after 60.0 seconds

Arrrr..

Tool rc

Execution TimeUnknown
Answer
MAYBE
InputAG01 3.19

stdout:

MAYBE

We consider the following Problem:

  Strict Trs:
    {  minus(x, 0()) -> x
     , minus(s(x), s(y)) -> minus(x, y)
     , double(0()) -> 0()
     , double(s(x)) -> s(s(double(x)))
     , plus(0(), y) -> y
     , plus(s(x), y) -> s(plus(x, y))
     , plus(s(x), y) -> plus(x, s(y))
     , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
     , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}
  StartTerms: basic terms
  Strategy: none

Certificate: MAYBE

Application of 'rc (timeout of 60.0 seconds)':
----------------------------------------------
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'Fastest' failed due to the following reason:
         None of the processors succeeded.
         
         Details of failed attempt(s):
         -----------------------------
           1) 'Sequentially' failed due to the following reason:
                None of the processors succeeded.
                
                Details of failed attempt(s):
                -----------------------------
                  1) 'empty' failed due to the following reason:
                       Empty strict component of the problem is NOT empty.
                  
                  2) 'Fastest' failed due to the following reason:
                       None of the processors succeeded.
                       
                       Details of failed attempt(s):
                       -----------------------------
                         1) 'matrix-interpretation of dimension 4 (timeout of 100.0 seconds)' failed due to the following reason:
                              The input cannot be shown compatible
                         
                         2) 'matrix-interpretation of dimension 3 (timeout of 100.0 seconds)' failed due to the following reason:
                              The input cannot be shown compatible
                         
                         3) 'matrix-interpretation of dimension 2 (timeout of 100.0 seconds)' failed due to the following reason:
                              The input cannot be shown compatible
                         
                  
           
           2) 'Fastest' failed due to the following reason:
                None of the processors succeeded.
                
                Details of failed attempt(s):
                -----------------------------
                  1) 'Bounds with minimal-enrichment and initial automaton 'match' (timeout of 100.0 seconds)' failed due to the following reason:
                       match-boundness of the problem could not be verified.
                  
                  2) 'Bounds with perSymbol-enrichment and initial automaton 'match' (timeout of 5.0 seconds)' failed due to the following reason:
                       match-boundness of the problem could not be verified.
                  
           
    
    2) 'dp' failed due to the following reason:
         We have computed the following dependency pairs
         
         Strict Dependency Pairs:
           {  minus^#(x, 0()) -> c_1(x)
            , minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
            , double^#(0()) -> c_3()
            , double^#(s(x)) -> c_4(double^#(x))
            , plus^#(0(), y) -> c_5(y)
            , plus^#(s(x), y) -> c_6(plus^#(x, y))
            , plus^#(s(x), y) -> c_7(plus^#(x, s(y)))
            , plus^#(s(x), y) -> c_8(plus^#(minus(x, y), double(y)))
            , plus^#(s(plus(x, y)), z) -> c_9(plus^#(plus(x, y), z))}
         
         We consider the following Problem:
         
           Strict DPs:
             {  minus^#(x, 0()) -> c_1(x)
              , minus^#(s(x), s(y)) -> c_2(minus^#(x, y))
              , double^#(0()) -> c_3()
              , double^#(s(x)) -> c_4(double^#(x))
              , plus^#(0(), y) -> c_5(y)
              , plus^#(s(x), y) -> c_6(plus^#(x, y))
              , plus^#(s(x), y) -> c_7(plus^#(x, s(y)))
              , plus^#(s(x), y) -> c_8(plus^#(minus(x, y), double(y)))
              , plus^#(s(plus(x, y)), z) -> c_9(plus^#(plus(x, y), z))}
           Strict Trs:
             {  minus(x, 0()) -> x
              , minus(s(x), s(y)) -> minus(x, y)
              , double(0()) -> 0()
              , double(s(x)) -> s(s(double(x)))
              , plus(0(), y) -> y
              , plus(s(x), y) -> s(plus(x, y))
              , plus(s(x), y) -> plus(x, s(y))
              , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
              , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}
           StartTerms: basic terms
           Strategy: none
         
         Certificate: MAYBE
         
         Application of 'usablerules':
         -----------------------------
           All rules are usable.
           
           No subproblems were generated.
    

Arrrr..

Tool tup3irc

Execution Time60.05861ms
Answer
TIMEOUT
InputAG01 3.19

stdout:

TIMEOUT

We consider the following Problem:

  Strict Trs:
    {  minus(x, 0()) -> x
     , minus(s(x), s(y)) -> minus(x, y)
     , double(0()) -> 0()
     , double(s(x)) -> s(s(double(x)))
     , plus(0(), y) -> y
     , plus(s(x), y) -> s(plus(x, y))
     , plus(s(x), y) -> plus(x, s(y))
     , plus(s(x), y) -> s(plus(minus(x, y), double(y)))
     , plus(s(plus(x, y)), z) -> s(plus(plus(x, y), z))}
  StartTerms: basic terms
  Strategy: innermost

Certificate: TIMEOUT

Application of 'tup3 (timeout of 60.0 seconds)':
------------------------------------------------
  Computation stopped due to timeout after 60.0 seconds

Arrrr..