Problem Beerendonk 07 7

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputBeerendonk 07 7

stdout:

MAYBE

Problem:
 cond(true(),x) -> cond(odd(x),p(p(p(x))))
 odd(0()) -> false()
 odd(s(0())) -> true()
 odd(s(s(x))) -> odd(x)
 p(0()) -> 0()
 p(s(x)) -> x

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputBeerendonk 07 7

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputBeerendonk 07 7

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  cond(true(), x) -> cond(odd(x), p(p(p(x))))
     , odd(0()) -> false()
     , odd(s(0())) -> true()
     , odd(s(s(x))) -> odd(x)
     , p(0()) -> 0()
     , p(s(x)) -> x}

Proof Output:    
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: cond^#(true(), x) -> c_0(cond^#(odd(x), p(p(p(x)))))
              , 2: odd^#(0()) -> c_1()
              , 3: odd^#(s(0())) -> c_2()
              , 4: odd^#(s(s(x))) -> c_3(odd^#(x))
              , 5: p^#(0()) -> c_4()
              , 6: p^#(s(x)) -> c_5()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6}                                                       [    YES(?,O(1))     ]
             
             ->{5}                                                       [    YES(?,O(1))     ]
             
             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                |->{2}                                                   [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^2))    ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules for this path are:
             
               {  odd(0()) -> false()
                , odd(s(0())) -> true()
                , odd(s(s(x))) -> odd(x)
                , p(0()) -> 0()
                , p(s(x)) -> x}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Induced complexity for the usable rules: 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:
                 {  cond^#(true(), x) -> c_0(cond^#(odd(x), p(p(p(x)))))
                  , odd(0()) -> false()
                  , odd(s(0())) -> true()
                  , odd(s(s(x))) -> odd(x)
                  , p(0()) -> 0()
                  , p(s(x)) -> x}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {4}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following TMI:
               The following argument positions are usable:
                 Uargs(cond) = {}, Uargs(true) = {}, Uargs(odd) = {}, Uargs(p) = {},
                 Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
                 Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(odd^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
                 Uargs(p^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                odd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                cond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_0(x1) = [3 0] x1 + [0]
                          [0 0]      [0]
                odd^#(x1) = [0 0] x1 + [0]
                            [3 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [2 0] x1 + [0]
                          [0 0]      [0]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
             Induced complexity for the usable rules: YES(?,O(1))
             
             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: {odd^#(s(s(x))) -> c_3(odd^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(odd^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [1]
                odd^#(x1) = [2 2] x1 + [2]
                            [6 0]      [0]
                c_3(x1) = [1 0] x1 + [5]
                          [2 0]      [7]
           
           * Path {4}->{2}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following TMI:
               The following argument positions are usable:
                 Uargs(cond) = {}, Uargs(true) = {}, Uargs(odd) = {}, Uargs(p) = {},
                 Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
                 Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(odd^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
                 Uargs(p^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                odd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                cond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_0(x1) = [3 0] x1 + [0]
                          [0 0]      [0]
                odd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [3 0] x1 + [0]
                          [0 0]      [0]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
             Induced complexity for the usable rules: YES(?,O(1))
             
             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: {odd^#(0()) -> c_1()}
               Weak Rules: {odd^#(s(s(x))) -> c_3(odd^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(0) = {}, Uargs(s) = {}, Uargs(odd^#) = {}, Uargs(c_1) = {},
                 Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [2]
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [0]
                odd^#(x1) = [2 2] x1 + [4]
                            [6 2]      [0]
                c_1() = [1]
                        [0]
                c_3(x1) = [1 0] x1 + [3]
                          [2 0]      [3]
           
           * Path {4}->{3}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following TMI:
               The following argument positions are usable:
                 Uargs(cond) = {}, Uargs(true) = {}, Uargs(odd) = {}, Uargs(p) = {},
                 Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
                 Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(odd^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
                 Uargs(p^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                odd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                cond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_0(x1) = [3 0] x1 + [0]
                          [0 0]      [0]
                odd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [3 0] x1 + [0]
                          [0 0]      [0]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
             Induced complexity for the usable rules: YES(?,O(1))
             
             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: {odd^#(s(0())) -> c_2()}
               Weak Rules: {odd^#(s(s(x))) -> c_3(odd^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(0) = {}, Uargs(s) = {}, Uargs(odd^#) = {}, Uargs(c_2) = {},
                 Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [1]
                      [0]
                s(x1) = [1 1] x1 + [2]
                        [0 1]      [1]
                odd^#(x1) = [2 1] x1 + [1]
                            [0 0]      [7]
                c_2() = [1]
                        [1]
                c_3(x1) = [1 1] x1 + [3]
                          [0 0]      [3]
           
           * Path {5}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following TMI:
               The following argument positions are usable:
                 Uargs(cond) = {}, Uargs(true) = {}, Uargs(odd) = {}, Uargs(p) = {},
                 Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
                 Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(odd^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(p^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                odd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                cond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_0(x1) = [3 0] x1 + [0]
                          [0 0]      [0]
                odd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [3 0] x1 + [0]
                          [0 0]      [0]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
             Induced complexity for the usable rules: YES(?,O(1))
             
             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: {p^#(0()) -> c_4()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(0) = {}, Uargs(p^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                p^#(x1) = [2 0] x1 + [7]
                          [2 2]      [7]
                c_4() = [0]
                        [1]
           
           * Path {6}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following TMI:
               The following argument positions are usable:
                 Uargs(cond) = {}, Uargs(true) = {}, Uargs(odd) = {}, Uargs(p) = {},
                 Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
                 Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(odd^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(p^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cond(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                odd(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                p(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                cond^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_0(x1) = [3 0] x1 + [0]
                          [0 0]      [0]
                odd^#(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                c_1() = [0]
                        [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [3 0] x1 + [0]
                          [0 0]      [0]
                p^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
             Induced complexity for the usable rules: YES(?,O(1))
             
             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: {p^#(s(x)) -> c_5()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0 0] x1 + [2]
                        [0 0]      [2]
                p^#(x1) = [2 0] x1 + [7]
                          [2 2]      [7]
                c_5() = [0]
                        [1]
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: cond^#(true(), x) -> c_0(cond^#(odd(x), p(p(p(x)))))
              , 2: odd^#(0()) -> c_1()
              , 3: odd^#(s(0())) -> c_2()
              , 4: odd^#(s(s(x))) -> c_3(odd^#(x))
              , 5: p^#(0()) -> c_4()
              , 6: p^#(s(x)) -> c_5()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6}                                                       [    YES(?,O(1))     ]
             
             ->{5}                                                       [    YES(?,O(1))     ]
             
             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                |->{2}                                                   [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [    YES(?,O(1))     ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules for this path are:
             
               {  odd(0()) -> false()
                , odd(s(0())) -> true()
                , odd(s(s(x))) -> odd(x)
                , p(0()) -> 0()
                , p(s(x)) -> x}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Induced complexity for the usable rules: 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:
                 {  cond^#(true(), x) -> c_0(cond^#(odd(x), p(p(p(x)))))
                  , odd(0()) -> false()
                  , odd(s(0())) -> true()
                  , odd(s(s(x))) -> odd(x)
                  , p(0()) -> 0()
                  , p(s(x)) -> x}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {4}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following TMI:
               The following argument positions are usable:
                 Uargs(cond) = {}, Uargs(true) = {}, Uargs(odd) = {}, Uargs(p) = {},
                 Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
                 Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(odd^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
                 Uargs(p^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cond(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                odd(x1) = [0] x1 + [0]
                p(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                s(x1) = [1] x1 + [0]
                cond^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [3] x1 + [0]
                odd^#(x1) = [3] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                p^#(x1) = [0] x1 + [0]
                c_4() = [0]
                c_5() = [0]
             Induced complexity for the usable rules: YES(?,O(n^1))
             
             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: {odd^#(s(s(x))) -> c_3(odd^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(odd^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                odd^#(x1) = [2] x1 + [0]
                c_3(x1) = [1] x1 + [7]
           
           * Path {4}->{2}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following TMI:
               The following argument positions are usable:
                 Uargs(cond) = {}, Uargs(true) = {}, Uargs(odd) = {}, Uargs(p) = {},
                 Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
                 Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(odd^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
                 Uargs(p^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cond(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                odd(x1) = [0] x1 + [0]
                p(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                s(x1) = [0] x1 + [0]
                cond^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [3] x1 + [0]
                odd^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3(x1) = [3] x1 + [0]
                p^#(x1) = [0] x1 + [0]
                c_4() = [0]
                c_5() = [0]
             Induced complexity for the usable rules: YES(?,O(1))
             
             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: {odd^#(0()) -> c_1()}
               Weak Rules: {odd^#(s(s(x))) -> c_3(odd^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(0) = {}, Uargs(s) = {}, Uargs(odd^#) = {}, Uargs(c_1) = {},
                 Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [0]
                odd^#(x1) = [2] x1 + [4]
                c_1() = [1]
                c_3(x1) = [1] x1 + [0]
           
           * Path {4}->{3}: YES(?,O(1))
             --------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following TMI:
               The following argument positions are usable:
                 Uargs(cond) = {}, Uargs(true) = {}, Uargs(odd) = {}, Uargs(p) = {},
                 Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
                 Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(odd^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {1},
                 Uargs(p^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cond(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                odd(x1) = [0] x1 + [0]
                p(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                s(x1) = [0] x1 + [0]
                cond^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [3] x1 + [0]
                odd^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3(x1) = [3] x1 + [0]
                p^#(x1) = [0] x1 + [0]
                c_4() = [0]
                c_5() = [0]
             Induced complexity for the usable rules: YES(?,O(1))
             
             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: {odd^#(s(0())) -> c_2()}
               Weak Rules: {odd^#(s(s(x))) -> c_3(odd^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(0) = {}, Uargs(s) = {}, Uargs(odd^#) = {}, Uargs(c_2) = {},
                 Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                s(x1) = [0] x1 + [0]
                odd^#(x1) = [0] x1 + [1]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
           
           * Path {5}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following TMI:
               The following argument positions are usable:
                 Uargs(cond) = {}, Uargs(true) = {}, Uargs(odd) = {}, Uargs(p) = {},
                 Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
                 Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(odd^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(p^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cond(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                odd(x1) = [0] x1 + [0]
                p(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                s(x1) = [0] x1 + [0]
                cond^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [3] x1 + [0]
                odd^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3(x1) = [3] x1 + [0]
                p^#(x1) = [0] x1 + [0]
                c_4() = [0]
                c_5() = [0]
             Induced complexity for the usable rules: YES(?,O(1))
             
             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^#(0()) -> c_4()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(0) = {}, Uargs(p^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [7]
                p^#(x1) = [1] x1 + [7]
                c_4() = [1]
           
           * Path {6}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following TMI:
               The following argument positions are usable:
                 Uargs(cond) = {}, Uargs(true) = {}, Uargs(odd) = {}, Uargs(p) = {},
                 Uargs(0) = {}, Uargs(false) = {}, Uargs(s) = {},
                 Uargs(cond^#) = {}, Uargs(c_0) = {}, Uargs(odd^#) = {},
                 Uargs(c_1) = {}, Uargs(c_2) = {}, Uargs(c_3) = {}, Uargs(p^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cond(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                odd(x1) = [0] x1 + [0]
                p(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                s(x1) = [0] x1 + [0]
                cond^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [3] x1 + [0]
                odd^#(x1) = [0] x1 + [0]
                c_1() = [0]
                c_2() = [0]
                c_3(x1) = [3] x1 + [0]
                p^#(x1) = [0] x1 + [0]
                c_4() = [0]
                c_5() = [0]
             Induced complexity for the usable rules: YES(?,O(1))
             
             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(x)) -> c_5()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(p^#) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [0] x1 + [7]
                p^#(x1) = [1] x1 + [7]
                c_5() = [1]
    
    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
InputBeerendonk 07 7

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputBeerendonk 07 7

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  cond(true(), x) -> cond(odd(x), p(p(p(x))))
     , odd(0()) -> false()
     , odd(s(0())) -> true()
     , odd(s(s(x))) -> odd(x)
     , p(0()) -> 0()
     , p(s(x)) -> x}

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