Problem AProVE 06 tower

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputAProVE 06 tower

stdout:

MAYBE

Problem:
 plus(0(),x) -> x
 plus(s(x),y) -> s(plus(p(s(x)),y))
 times(0(),y) -> 0()
 times(s(x),y) -> plus(y,times(p(s(x)),y))
 exp(x,0()) -> s(0())
 exp(x,s(y)) -> times(x,exp(x,y))
 p(s(0())) -> 0()
 p(s(s(x))) -> s(p(s(x)))
 tower(x,y) -> towerIter(x,y,s(0()))
 towerIter(0(),y,z) -> z
 towerIter(s(x),y,z) -> towerIter(p(s(x)),y,exp(y,z))

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputAProVE 06 tower

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputAProVE 06 tower

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  plus(0(), x) -> x
     , plus(s(x), y) -> s(plus(p(s(x)), y))
     , times(0(), y) -> 0()
     , times(s(x), y) -> plus(y, times(p(s(x)), y))
     , exp(x, 0()) -> s(0())
     , exp(x, s(y)) -> times(x, exp(x, y))
     , p(s(0())) -> 0()
     , p(s(s(x))) -> s(p(s(x)))
     , tower(x, y) -> towerIter(x, y, s(0()))
     , towerIter(0(), y, z) -> z
     , towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(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: plus^#(0(), x) -> c_0()
              , 2: plus^#(s(x), y) -> c_1(plus^#(p(s(x)), y))
              , 3: times^#(0(), y) -> c_2()
              , 4: times^#(s(x), y) -> c_3(plus^#(y, times(p(s(x)), y)))
              , 5: exp^#(x, 0()) -> c_4()
              , 6: exp^#(x, s(y)) -> c_5(times^#(x, exp(x, y)))
              , 7: p^#(s(0())) -> c_6()
              , 8: p^#(s(s(x))) -> c_7(p^#(s(x)))
              , 9: tower^#(x, y) -> c_8(towerIter^#(x, y, s(0())))
              , 10: towerIter^#(0(), y, z) -> c_9()
              , 11: towerIter^#(s(x), y, z) ->
                    c_10(towerIter^#(p(s(x)), y, exp(y, z)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{9}                                                       [     inherited      ]
                |
                |->{10}                                                  [    YES(?,O(1))     ]
                |
                `->{11}                                                  [     inherited      ]
                    |
                    `->{10}                                              [         NA         ]
             
             ->{8}                                                       [   YES(?,O(n^1))    ]
                |
                `->{7}                                                   [   YES(?,O(n^1))    ]
             
             ->{6}                                                       [     inherited      ]
                |
                |->{3}                                                   [       MAYBE        ]
                |
                `->{4}                                                   [     inherited      ]
                    |
                    |->{1}                                               [         NA         ]
                    |
                    `->{2}                                               [     inherited      ]
                        |
                        `->{1}                                           [         NA         ]
             
             ->{5}                                                       [    YES(?,O(1))     ]
             
           
         
         Sub-problems:
         -------------
           * Path {5}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(plus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(times) = {},
                 Uargs(exp) = {}, Uargs(tower) = {}, Uargs(towerIter) = {},
                 Uargs(plus^#) = {}, Uargs(c_1) = {}, Uargs(times^#) = {},
                 Uargs(c_3) = {}, Uargs(exp^#) = {}, Uargs(c_5) = {},
                 Uargs(p^#) = {}, Uargs(c_7) = {}, Uargs(tower^#) = {},
                 Uargs(c_8) = {}, Uargs(towerIter^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(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]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                exp(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tower(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                towerIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                        [0 0]      [0 0]      [0 0]      [0]
                plus^#(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]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                exp^#(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]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                tower^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                towerIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                          [0 0]      [0 0]      [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {exp^#(x, 0()) -> c_4()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(exp^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                exp^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
                                [0 0]      [2 2]      [7]
                c_4() = [0]
                        [1]
           
           * Path {6}: inherited
             -------------------
             
             This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
           
           * Path {6}->{3}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  exp(x, 0()) -> s(0())
                , exp(x, s(y)) -> times(x, exp(x, y))
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(p(s(x)), y))
                , plus(0(), x) -> x
                , plus(s(x), y) -> s(plus(p(s(x)), y))
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  exp^#(x, s(y)) -> c_5(times^#(x, exp(x, y)))
                  , times^#(0(), y) -> c_2()
                  , exp(x, 0()) -> s(0())
                  , exp(x, s(y)) -> times(x, exp(x, y))
                  , times(0(), y) -> 0()
                  , times(s(x), y) -> plus(y, times(p(s(x)), y))
                  , plus(0(), x) -> x
                  , plus(s(x), y) -> s(plus(p(s(x)), y))
                  , p(s(0())) -> 0()
                  , p(s(s(x))) -> s(p(s(x)))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {6}->{4}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
           
           * Path {6}->{4}->{1}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  exp(x, 0()) -> s(0())
                , exp(x, s(y)) -> times(x, exp(x, y))
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(p(s(x)), y))
                , plus(0(), x) -> x
                , plus(s(x), y) -> s(plus(p(s(x)), y))
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {6}->{4}->{2}: inherited
             -----------------------------
             
             This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
           
           * Path {6}->{4}->{2}->{1}: NA
             ---------------------------
             
             The usable rules for this path are:
             
               {  exp(x, 0()) -> s(0())
                , exp(x, s(y)) -> times(x, exp(x, y))
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(p(s(x)), y))
                , plus(0(), x) -> x
                , plus(s(x), y) -> s(plus(p(s(x)), y))
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {8}: YES(?,O(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(plus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(times) = {},
                 Uargs(exp) = {}, Uargs(tower) = {}, Uargs(towerIter) = {},
                 Uargs(plus^#) = {}, Uargs(c_1) = {}, Uargs(times^#) = {},
                 Uargs(c_3) = {}, Uargs(exp^#) = {}, Uargs(c_5) = {},
                 Uargs(p^#) = {}, Uargs(c_7) = {1}, Uargs(tower^#) = {},
                 Uargs(c_8) = {}, Uargs(towerIter^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(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]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                exp(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tower(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                towerIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                        [0 0]      [0 0]      [0 0]      [0]
                plus^#(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]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                exp^#(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]
                p^#(x1) = [0 0] x1 + [0]
                          [3 3]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                tower^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                towerIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                          [0 0]      [0 0]      [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(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: {p^#(s(s(x))) -> c_7(p^#(s(x)))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [1]
                        [0 0]      [0]
                p^#(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
           
           * Path {8}->{7}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(plus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(times) = {},
                 Uargs(exp) = {}, Uargs(tower) = {}, Uargs(towerIter) = {},
                 Uargs(plus^#) = {}, Uargs(c_1) = {}, Uargs(times^#) = {},
                 Uargs(c_3) = {}, Uargs(exp^#) = {}, Uargs(c_5) = {},
                 Uargs(p^#) = {}, Uargs(c_7) = {1}, Uargs(tower^#) = {},
                 Uargs(c_8) = {}, Uargs(towerIter^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(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]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                exp(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tower(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                towerIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                        [0 0]      [0 0]      [0 0]      [0]
                plus^#(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]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                exp^#(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]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                tower^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                towerIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                          [0 0]      [0 0]      [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(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: {p^#(s(0())) -> c_6()}
               Weak Rules: {p^#(s(s(x))) -> c_7(p^#(s(x)))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [0]
                s(x1) = [0 2] x1 + [2]
                        [0 1]      [0]
                p^#(x1) = [2 0] x1 + [0]
                          [2 0]      [0]
                c_6() = [1]
                        [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 0]      [3]
           
           * Path {9}: inherited
             -------------------
             
             This path is subsumed by the proof of path {9}->{11}->{10}.
           
           * Path {9}->{10}: YES(?,O(1))
             ---------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(plus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(times) = {},
                 Uargs(exp) = {}, Uargs(tower) = {}, Uargs(towerIter) = {},
                 Uargs(plus^#) = {}, Uargs(c_1) = {}, Uargs(times^#) = {},
                 Uargs(c_3) = {}, Uargs(exp^#) = {}, Uargs(c_5) = {},
                 Uargs(p^#) = {}, Uargs(c_7) = {}, Uargs(tower^#) = {},
                 Uargs(c_8) = {1}, Uargs(towerIter^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(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]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                exp(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tower(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                towerIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                        [0 0]      [0 0]      [0 0]      [0]
                plus^#(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]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                exp^#(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]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                tower^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                towerIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                          [0 0]      [0 0]      [0 0]      [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {towerIter^#(0(), y, z) -> c_9()}
               Weak Rules: {tower^#(x, y) -> c_8(towerIter^#(x, y, s(0())))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(tower^#) = {}, Uargs(c_8) = {1},
                 Uargs(towerIter^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                tower^#(x1, x2) = [7 7] x1 + [7 7] x2 + [7]
                                  [7 7]      [7 7]      [7]
                c_8(x1) = [2 0] x1 + [7]
                          [2 0]      [7]
                towerIter^#(x1, x2, x3) = [2 2] x1 + [0 3] x2 + [2 0] x3 + [0]
                                          [0 2]      [0 7]      [0 0]      [0]
                c_9() = [1]
                        [0]
           
           * Path {9}->{11}: inherited
             -------------------------
             
             This path is subsumed by the proof of path {9}->{11}->{10}.
           
           * Path {9}->{11}->{10}: NA
             ------------------------
             
             The usable rules for this path are:
             
               {  exp(x, 0()) -> s(0())
                , exp(x, s(y)) -> times(x, exp(x, y))
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(p(s(x)), y))
                , plus(0(), x) -> x
                , plus(s(x), y) -> s(plus(p(s(x)), y))}
             
             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: plus^#(0(), x) -> c_0()
              , 2: plus^#(s(x), y) -> c_1(plus^#(p(s(x)), y))
              , 3: times^#(0(), y) -> c_2()
              , 4: times^#(s(x), y) -> c_3(plus^#(y, times(p(s(x)), y)))
              , 5: exp^#(x, 0()) -> c_4()
              , 6: exp^#(x, s(y)) -> c_5(times^#(x, exp(x, y)))
              , 7: p^#(s(0())) -> c_6()
              , 8: p^#(s(s(x))) -> c_7(p^#(s(x)))
              , 9: tower^#(x, y) -> c_8(towerIter^#(x, y, s(0())))
              , 10: towerIter^#(0(), y, z) -> c_9()
              , 11: towerIter^#(s(x), y, z) ->
                    c_10(towerIter^#(p(s(x)), y, exp(y, z)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{9}                                                       [     inherited      ]
                |
                |->{10}                                                  [    YES(?,O(1))     ]
                |
                `->{11}                                                  [     inherited      ]
                    |
                    `->{10}                                              [         NA         ]
             
             ->{8}                                                       [   YES(?,O(n^1))    ]
                |
                `->{7}                                                   [    YES(?,O(1))     ]
             
             ->{6}                                                       [     inherited      ]
                |
                |->{3}                                                   [       MAYBE        ]
                |
                `->{4}                                                   [     inherited      ]
                    |
                    |->{1}                                               [         NA         ]
                    |
                    `->{2}                                               [     inherited      ]
                        |
                        `->{1}                                           [         NA         ]
             
             ->{5}                                                       [    YES(?,O(1))     ]
             
           
         
         Sub-problems:
         -------------
           * Path {5}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(plus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(times) = {},
                 Uargs(exp) = {}, Uargs(tower) = {}, Uargs(towerIter) = {},
                 Uargs(plus^#) = {}, Uargs(c_1) = {}, Uargs(times^#) = {},
                 Uargs(c_3) = {}, Uargs(exp^#) = {}, Uargs(c_5) = {},
                 Uargs(p^#) = {}, Uargs(c_7) = {}, Uargs(tower^#) = {},
                 Uargs(c_8) = {}, Uargs(towerIter^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                p(x1) = [0] x1 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                exp(x1, x2) = [0] x1 + [0] x2 + [0]
                tower(x1, x2) = [0] x1 + [0] x2 + [0]
                towerIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                exp^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                p^#(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                tower^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8(x1) = [0] x1 + [0]
                towerIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {exp^#(x, 0()) -> c_4()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(exp^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [7]
                exp^#(x1, x2) = [0] x1 + [1] x2 + [7]
                c_4() = [1]
           
           * Path {6}: inherited
             -------------------
             
             This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
           
           * Path {6}->{3}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  exp(x, 0()) -> s(0())
                , exp(x, s(y)) -> times(x, exp(x, y))
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(p(s(x)), y))
                , plus(0(), x) -> x
                , plus(s(x), y) -> s(plus(p(s(x)), y))
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  exp^#(x, s(y)) -> c_5(times^#(x, exp(x, y)))
                  , times^#(0(), y) -> c_2()
                  , exp(x, 0()) -> s(0())
                  , exp(x, s(y)) -> times(x, exp(x, y))
                  , times(0(), y) -> 0()
                  , times(s(x), y) -> plus(y, times(p(s(x)), y))
                  , plus(0(), x) -> x
                  , plus(s(x), y) -> s(plus(p(s(x)), y))
                  , p(s(0())) -> 0()
                  , p(s(s(x))) -> s(p(s(x)))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {6}->{4}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
           
           * Path {6}->{4}->{1}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  exp(x, 0()) -> s(0())
                , exp(x, s(y)) -> times(x, exp(x, y))
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(p(s(x)), y))
                , plus(0(), x) -> x
                , plus(s(x), y) -> s(plus(p(s(x)), y))
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {6}->{4}->{2}: inherited
             -----------------------------
             
             This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
           
           * Path {6}->{4}->{2}->{1}: NA
             ---------------------------
             
             The usable rules for this path are:
             
               {  exp(x, 0()) -> s(0())
                , exp(x, s(y)) -> times(x, exp(x, y))
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(p(s(x)), y))
                , plus(0(), x) -> x
                , plus(s(x), y) -> s(plus(p(s(x)), y))
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {8}: YES(?,O(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(plus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(times) = {},
                 Uargs(exp) = {}, Uargs(tower) = {}, Uargs(towerIter) = {},
                 Uargs(plus^#) = {}, Uargs(c_1) = {}, Uargs(times^#) = {},
                 Uargs(c_3) = {}, Uargs(exp^#) = {}, Uargs(c_5) = {},
                 Uargs(p^#) = {}, Uargs(c_7) = {1}, Uargs(tower^#) = {},
                 Uargs(c_8) = {}, Uargs(towerIter^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                p(x1) = [0] x1 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                exp(x1, x2) = [0] x1 + [0] x2 + [0]
                tower(x1, x2) = [0] x1 + [0] x2 + [0]
                towerIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                exp^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                p^#(x1) = [3] x1 + [0]
                c_6() = [0]
                c_7(x1) = [1] x1 + [0]
                tower^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8(x1) = [0] x1 + [0]
                towerIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_9() = [0]
                c_10(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: {p^#(s(s(x))) -> c_7(p^#(s(x)))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [4]
                p^#(x1) = [1] x1 + [0]
                c_7(x1) = [1] x1 + [3]
           
           * Path {8}->{7}: YES(?,O(1))
             --------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(plus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(times) = {},
                 Uargs(exp) = {}, Uargs(tower) = {}, Uargs(towerIter) = {},
                 Uargs(plus^#) = {}, Uargs(c_1) = {}, Uargs(times^#) = {},
                 Uargs(c_3) = {}, Uargs(exp^#) = {}, Uargs(c_5) = {},
                 Uargs(p^#) = {}, Uargs(c_7) = {1}, Uargs(tower^#) = {},
                 Uargs(c_8) = {}, Uargs(towerIter^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                p(x1) = [0] x1 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                exp(x1, x2) = [0] x1 + [0] x2 + [0]
                tower(x1, x2) = [0] x1 + [0] x2 + [0]
                towerIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                exp^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                p^#(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(x1) = [1] x1 + [0]
                tower^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8(x1) = [0] x1 + [0]
                towerIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {p^#(s(0())) -> c_6()}
               Weak Rules: {p^#(s(s(x))) -> c_7(p^#(s(x)))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                s(x1) = [0] x1 + [3]
                p^#(x1) = [2] x1 + [2]
                c_6() = [1]
                c_7(x1) = [1] x1 + [0]
           
           * Path {9}: inherited
             -------------------
             
             This path is subsumed by the proof of path {9}->{11}->{10}.
           
           * Path {9}->{10}: YES(?,O(1))
             ---------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(plus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(times) = {},
                 Uargs(exp) = {}, Uargs(tower) = {}, Uargs(towerIter) = {},
                 Uargs(plus^#) = {}, Uargs(c_1) = {}, Uargs(times^#) = {},
                 Uargs(c_3) = {}, Uargs(exp^#) = {}, Uargs(c_5) = {},
                 Uargs(p^#) = {}, Uargs(c_7) = {}, Uargs(tower^#) = {},
                 Uargs(c_8) = {1}, Uargs(towerIter^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                p(x1) = [0] x1 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                exp(x1, x2) = [0] x1 + [0] x2 + [0]
                tower(x1, x2) = [0] x1 + [0] x2 + [0]
                towerIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                exp^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                p^#(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                tower^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8(x1) = [1] x1 + [0]
                towerIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {towerIter^#(0(), y, z) -> c_9()}
               Weak Rules: {tower^#(x, y) -> c_8(towerIter^#(x, y, s(0())))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(tower^#) = {}, Uargs(c_8) = {1},
                 Uargs(towerIter^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [0] x1 + [4]
                tower^#(x1, x2) = [7] x1 + [7] x2 + [7]
                c_8(x1) = [1] x1 + [3]
                towerIter^#(x1, x2, x3) = [2] x1 + [7] x2 + [0] x3 + [4]
                c_9() = [1]
           
           * Path {9}->{11}: inherited
             -------------------------
             
             This path is subsumed by the proof of path {9}->{11}->{10}.
           
           * Path {9}->{11}->{10}: NA
             ------------------------
             
             The usable rules for this path are:
             
               {  exp(x, 0()) -> s(0())
                , exp(x, s(y)) -> times(x, exp(x, y))
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(p(s(x)), y))
                , plus(0(), x) -> x
                , plus(s(x), y) -> s(plus(p(s(x)), y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    3) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    

Tool RC1

Execution TimeUnknown
Answer
MAYBE
InputAProVE 06 tower

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputAProVE 06 tower

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  plus(0(), x) -> x
     , plus(s(x), y) -> s(plus(p(s(x)), y))
     , times(0(), y) -> 0()
     , times(s(x), y) -> plus(y, times(p(s(x)), y))
     , exp(x, 0()) -> s(0())
     , exp(x, s(y)) -> times(x, exp(x, y))
     , p(s(0())) -> 0()
     , p(s(s(x))) -> s(p(s(x)))
     , tower(x, y) -> towerIter(x, y, s(0()))
     , towerIter(0(), y, z) -> z
     , towerIter(s(x), y, z) -> towerIter(p(s(x)), y, exp(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: plus^#(0(), x) -> c_0(x)
              , 2: plus^#(s(x), y) -> c_1(plus^#(p(s(x)), y))
              , 3: times^#(0(), y) -> c_2()
              , 4: times^#(s(x), y) -> c_3(plus^#(y, times(p(s(x)), y)))
              , 5: exp^#(x, 0()) -> c_4()
              , 6: exp^#(x, s(y)) -> c_5(times^#(x, exp(x, y)))
              , 7: p^#(s(0())) -> c_6()
              , 8: p^#(s(s(x))) -> c_7(p^#(s(x)))
              , 9: tower^#(x, y) -> c_8(towerIter^#(x, y, s(0())))
              , 10: towerIter^#(0(), y, z) -> c_9(z)
              , 11: towerIter^#(s(x), y, z) ->
                    c_10(towerIter^#(p(s(x)), y, exp(y, z)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{9}                                                       [     inherited      ]
                |
                |->{10}                                                  [    YES(?,O(1))     ]
                |
                `->{11}                                                  [     inherited      ]
                    |
                    `->{10}                                              [         NA         ]
             
             ->{8}                                                       [   YES(?,O(n^1))    ]
                |
                `->{7}                                                   [   YES(?,O(n^1))    ]
             
             ->{6}                                                       [     inherited      ]
                |
                |->{3}                                                   [       MAYBE        ]
                |
                `->{4}                                                   [     inherited      ]
                    |
                    |->{1}                                               [         NA         ]
                    |
                    `->{2}                                               [     inherited      ]
                        |
                        `->{1}                                           [         NA         ]
             
             ->{5}                                                       [    YES(?,O(1))     ]
             
           
         
         Sub-problems:
         -------------
           * Path {5}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(plus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(times) = {},
                 Uargs(exp) = {}, Uargs(tower) = {}, Uargs(towerIter) = {},
                 Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
                 Uargs(times^#) = {}, Uargs(c_3) = {}, Uargs(exp^#) = {},
                 Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
                 Uargs(tower^#) = {}, Uargs(c_8) = {}, Uargs(towerIter^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(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]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                exp(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tower(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                towerIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                        [0 0]      [0 0]      [0 0]      [0]
                plus^#(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]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                exp^#(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]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                tower^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                towerIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                          [0 0]      [0 0]      [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {exp^#(x, 0()) -> c_4()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(exp^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                exp^#(x1, x2) = [0 0] x1 + [2 0] x2 + [7]
                                [0 0]      [2 2]      [7]
                c_4() = [0]
                        [1]
           
           * Path {6}: inherited
             -------------------
             
             This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
           
           * Path {6}->{3}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  exp(x, 0()) -> s(0())
                , exp(x, s(y)) -> times(x, exp(x, y))
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(p(s(x)), y))
                , plus(0(), x) -> x
                , plus(s(x), y) -> s(plus(p(s(x)), y))
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  exp^#(x, s(y)) -> c_5(times^#(x, exp(x, y)))
                  , times^#(0(), y) -> c_2()
                  , exp(x, 0()) -> s(0())
                  , exp(x, s(y)) -> times(x, exp(x, y))
                  , times(0(), y) -> 0()
                  , times(s(x), y) -> plus(y, times(p(s(x)), y))
                  , plus(0(), x) -> x
                  , plus(s(x), y) -> s(plus(p(s(x)), y))
                  , p(s(0())) -> 0()
                  , p(s(s(x))) -> s(p(s(x)))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {6}->{4}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
           
           * Path {6}->{4}->{1}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  exp(x, 0()) -> s(0())
                , exp(x, s(y)) -> times(x, exp(x, y))
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(p(s(x)), y))
                , plus(0(), x) -> x
                , plus(s(x), y) -> s(plus(p(s(x)), y))
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {6}->{4}->{2}: inherited
             -----------------------------
             
             This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
           
           * Path {6}->{4}->{2}->{1}: NA
             ---------------------------
             
             The usable rules for this path are:
             
               {  exp(x, 0()) -> s(0())
                , exp(x, s(y)) -> times(x, exp(x, y))
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(p(s(x)), y))
                , plus(0(), x) -> x
                , plus(s(x), y) -> s(plus(p(s(x)), y))
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {8}: YES(?,O(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(plus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(times) = {},
                 Uargs(exp) = {}, Uargs(tower) = {}, Uargs(towerIter) = {},
                 Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
                 Uargs(times^#) = {}, Uargs(c_3) = {}, Uargs(exp^#) = {},
                 Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1},
                 Uargs(tower^#) = {}, Uargs(c_8) = {}, Uargs(towerIter^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(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]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                exp(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tower(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                towerIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                        [0 0]      [0 0]      [0 0]      [0]
                plus^#(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]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                exp^#(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]
                p^#(x1) = [0 0] x1 + [0]
                          [3 3]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                tower^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                towerIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                          [0 0]      [0 0]      [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(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: {p^#(s(s(x))) -> c_7(p^#(s(x)))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [1]
                        [0 0]      [0]
                p^#(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
           
           * Path {8}->{7}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(plus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(times) = {},
                 Uargs(exp) = {}, Uargs(tower) = {}, Uargs(towerIter) = {},
                 Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
                 Uargs(times^#) = {}, Uargs(c_3) = {}, Uargs(exp^#) = {},
                 Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1},
                 Uargs(tower^#) = {}, Uargs(c_8) = {}, Uargs(towerIter^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(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]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                exp(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tower(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                towerIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                        [0 0]      [0 0]      [0 0]      [0]
                plus^#(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]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                exp^#(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]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                tower^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                towerIter^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                          [0 0]      [0 0]      [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(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: {p^#(s(0())) -> c_6()}
               Weak Rules: {p^#(s(s(x))) -> c_7(p^#(s(x)))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [0]
                s(x1) = [0 2] x1 + [2]
                        [0 1]      [0]
                p^#(x1) = [2 0] x1 + [0]
                          [2 0]      [0]
                c_6() = [1]
                        [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 0]      [3]
           
           * Path {9}: inherited
             -------------------
             
             This path is subsumed by the proof of path {9}->{11}->{10}.
           
           * Path {9}->{10}: YES(?,O(1))
             ---------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(plus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(times) = {},
                 Uargs(exp) = {}, Uargs(tower) = {}, Uargs(towerIter) = {},
                 Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
                 Uargs(times^#) = {}, Uargs(c_3) = {}, Uargs(exp^#) = {},
                 Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
                 Uargs(tower^#) = {}, Uargs(c_8) = {1}, Uargs(towerIter^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(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]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                exp(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                tower(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                towerIter(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                        [0 0]      [0 0]      [0 0]      [0]
                plus^#(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]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                exp^#(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]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                tower^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                towerIter^#(x1, x2, x3) = [0 0] x1 + [3 3] x2 + [0 0] x3 + [0]
                                          [0 0]      [0 0]      [0 0]      [0]
                c_9(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {towerIter^#(0(), y, z) -> c_9(z)}
               Weak Rules: {tower^#(x, y) -> c_8(towerIter^#(x, y, s(0())))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(tower^#) = {}, Uargs(c_8) = {1},
                 Uargs(towerIter^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [3]
                      [0]
                s(x1) = [0 0] x1 + [2]
                        [0 0]      [0]
                tower^#(x1, x2) = [7 7] x1 + [7 7] x2 + [7]
                                  [7 7]      [7 7]      [7]
                c_8(x1) = [1 0] x1 + [1]
                          [0 0]      [7]
                towerIter^#(x1, x2, x3) = [2 0] x1 + [0 0] x2 + [2 0] x3 + [2]
                                          [0 0]      [0 0]      [0 0]      [0]
                c_9(x1) = [0 0] x1 + [1]
                          [0 0]      [0]
           
           * Path {9}->{11}: inherited
             -------------------------
             
             This path is subsumed by the proof of path {9}->{11}->{10}.
           
           * Path {9}->{11}->{10}: NA
             ------------------------
             
             The usable rules for this path are:
             
               {  exp(x, 0()) -> s(0())
                , exp(x, s(y)) -> times(x, exp(x, y))
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(p(s(x)), y))
                , plus(0(), x) -> x
                , plus(s(x), y) -> s(plus(p(s(x)), y))}
             
             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: plus^#(0(), x) -> c_0(x)
              , 2: plus^#(s(x), y) -> c_1(plus^#(p(s(x)), y))
              , 3: times^#(0(), y) -> c_2()
              , 4: times^#(s(x), y) -> c_3(plus^#(y, times(p(s(x)), y)))
              , 5: exp^#(x, 0()) -> c_4()
              , 6: exp^#(x, s(y)) -> c_5(times^#(x, exp(x, y)))
              , 7: p^#(s(0())) -> c_6()
              , 8: p^#(s(s(x))) -> c_7(p^#(s(x)))
              , 9: tower^#(x, y) -> c_8(towerIter^#(x, y, s(0())))
              , 10: towerIter^#(0(), y, z) -> c_9(z)
              , 11: towerIter^#(s(x), y, z) ->
                    c_10(towerIter^#(p(s(x)), y, exp(y, z)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{9}                                                       [     inherited      ]
                |
                |->{10}                                                  [    YES(?,O(1))     ]
                |
                `->{11}                                                  [     inherited      ]
                    |
                    `->{10}                                              [         NA         ]
             
             ->{8}                                                       [   YES(?,O(n^1))    ]
                |
                `->{7}                                                   [    YES(?,O(1))     ]
             
             ->{6}                                                       [     inherited      ]
                |
                |->{3}                                                   [       MAYBE        ]
                |
                `->{4}                                                   [     inherited      ]
                    |
                    |->{1}                                               [         NA         ]
                    |
                    `->{2}                                               [     inherited      ]
                        |
                        `->{1}                                           [         NA         ]
             
             ->{5}                                                       [    YES(?,O(1))     ]
             
           
         
         Sub-problems:
         -------------
           * Path {5}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(plus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(times) = {},
                 Uargs(exp) = {}, Uargs(tower) = {}, Uargs(towerIter) = {},
                 Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
                 Uargs(times^#) = {}, Uargs(c_3) = {}, Uargs(exp^#) = {},
                 Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
                 Uargs(tower^#) = {}, Uargs(c_8) = {}, Uargs(towerIter^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                p(x1) = [0] x1 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                exp(x1, x2) = [0] x1 + [0] x2 + [0]
                tower(x1, x2) = [0] x1 + [0] x2 + [0]
                towerIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                exp^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                p^#(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                tower^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8(x1) = [0] x1 + [0]
                towerIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {exp^#(x, 0()) -> c_4()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(exp^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [7]
                exp^#(x1, x2) = [0] x1 + [1] x2 + [7]
                c_4() = [1]
           
           * Path {6}: inherited
             -------------------
             
             This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
           
           * Path {6}->{3}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  exp(x, 0()) -> s(0())
                , exp(x, s(y)) -> times(x, exp(x, y))
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(p(s(x)), y))
                , plus(0(), x) -> x
                , plus(s(x), y) -> s(plus(p(s(x)), y))
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  exp^#(x, s(y)) -> c_5(times^#(x, exp(x, y)))
                  , times^#(0(), y) -> c_2()
                  , exp(x, 0()) -> s(0())
                  , exp(x, s(y)) -> times(x, exp(x, y))
                  , times(0(), y) -> 0()
                  , times(s(x), y) -> plus(y, times(p(s(x)), y))
                  , plus(0(), x) -> x
                  , plus(s(x), y) -> s(plus(p(s(x)), y))
                  , p(s(0())) -> 0()
                  , p(s(s(x))) -> s(p(s(x)))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {6}->{4}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
           
           * Path {6}->{4}->{1}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  exp(x, 0()) -> s(0())
                , exp(x, s(y)) -> times(x, exp(x, y))
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(p(s(x)), y))
                , plus(0(), x) -> x
                , plus(s(x), y) -> s(plus(p(s(x)), y))
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {6}->{4}->{2}: inherited
             -----------------------------
             
             This path is subsumed by the proof of path {6}->{4}->{2}->{1}.
           
           * Path {6}->{4}->{2}->{1}: NA
             ---------------------------
             
             The usable rules for this path are:
             
               {  exp(x, 0()) -> s(0())
                , exp(x, s(y)) -> times(x, exp(x, y))
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(p(s(x)), y))
                , plus(0(), x) -> x
                , plus(s(x), y) -> s(plus(p(s(x)), y))
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {8}: YES(?,O(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(plus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(times) = {},
                 Uargs(exp) = {}, Uargs(tower) = {}, Uargs(towerIter) = {},
                 Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
                 Uargs(times^#) = {}, Uargs(c_3) = {}, Uargs(exp^#) = {},
                 Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1},
                 Uargs(tower^#) = {}, Uargs(c_8) = {}, Uargs(towerIter^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                p(x1) = [0] x1 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                exp(x1, x2) = [0] x1 + [0] x2 + [0]
                tower(x1, x2) = [0] x1 + [0] x2 + [0]
                towerIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                exp^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                p^#(x1) = [3] x1 + [0]
                c_6() = [0]
                c_7(x1) = [1] x1 + [0]
                tower^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8(x1) = [0] x1 + [0]
                towerIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(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: {p^#(s(s(x))) -> c_7(p^#(s(x)))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [4]
                p^#(x1) = [1] x1 + [0]
                c_7(x1) = [1] x1 + [3]
           
           * Path {8}->{7}: YES(?,O(1))
             --------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(plus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(times) = {},
                 Uargs(exp) = {}, Uargs(tower) = {}, Uargs(towerIter) = {},
                 Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
                 Uargs(times^#) = {}, Uargs(c_3) = {}, Uargs(exp^#) = {},
                 Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1},
                 Uargs(tower^#) = {}, Uargs(c_8) = {}, Uargs(towerIter^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                p(x1) = [0] x1 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                exp(x1, x2) = [0] x1 + [0] x2 + [0]
                tower(x1, x2) = [0] x1 + [0] x2 + [0]
                towerIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                exp^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                p^#(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(x1) = [1] x1 + [0]
                tower^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8(x1) = [0] x1 + [0]
                towerIter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {p^#(s(0())) -> c_6()}
               Weak Rules: {p^#(s(s(x))) -> c_7(p^#(s(x)))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                s(x1) = [0] x1 + [3]
                p^#(x1) = [2] x1 + [2]
                c_6() = [1]
                c_7(x1) = [1] x1 + [0]
           
           * Path {9}: inherited
             -------------------
             
             This path is subsumed by the proof of path {9}->{11}->{10}.
           
           * Path {9}->{10}: YES(?,O(1))
             ---------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(plus) = {}, Uargs(s) = {}, Uargs(p) = {}, Uargs(times) = {},
                 Uargs(exp) = {}, Uargs(tower) = {}, Uargs(towerIter) = {},
                 Uargs(plus^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
                 Uargs(times^#) = {}, Uargs(c_3) = {}, Uargs(exp^#) = {},
                 Uargs(c_5) = {}, Uargs(p^#) = {}, Uargs(c_7) = {},
                 Uargs(tower^#) = {}, Uargs(c_8) = {1}, Uargs(towerIter^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                p(x1) = [0] x1 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                exp(x1, x2) = [0] x1 + [0] x2 + [0]
                tower(x1, x2) = [0] x1 + [0] x2 + [0]
                towerIter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
                exp^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                p^#(x1) = [0] x1 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                tower^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8(x1) = [1] x1 + [0]
                towerIter^#(x1, x2, x3) = [0] x1 + [3] x2 + [0] x3 + [0]
                c_9(x1) = [1] x1 + [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {towerIter^#(0(), y, z) -> c_9(z)}
               Weak Rules: {tower^#(x, y) -> c_8(towerIter^#(x, y, s(0())))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(tower^#) = {}, Uargs(c_8) = {1},
                 Uargs(towerIter^#) = {}, Uargs(c_9) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                s(x1) = [0] x1 + [0]
                tower^#(x1, x2) = [7] x1 + [7] x2 + [7]
                c_8(x1) = [2] x1 + [3]
                towerIter^#(x1, x2, x3) = [0] x1 + [3] x2 + [2] x3 + [2]
                c_9(x1) = [0] x1 + [1]
           
           * Path {9}->{11}: inherited
             -------------------------
             
             This path is subsumed by the proof of path {9}->{11}->{10}.
           
           * Path {9}->{11}->{10}: NA
             ------------------------
             
             The usable rules for this path are:
             
               {  exp(x, 0()) -> s(0())
                , exp(x, s(y)) -> times(x, exp(x, y))
                , p(s(0())) -> 0()
                , p(s(s(x))) -> s(p(s(x)))
                , times(0(), y) -> 0()
                , times(s(x), y) -> plus(y, times(p(s(x)), y))
                , plus(0(), x) -> x
                , plus(s(x), y) -> s(plus(p(s(x)), y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    3) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.