Problem Strategy outermost added 08 4.14

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 4.14

stdout:

MAYBE

Problem:
f(g(x),s(0())) -> f(g(x),g(x))
g(s(x)) -> s(g(x))
g(0()) -> 0()

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 4.14

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	Strategy outermost added 08 4.14

stdout:

YES(?,O(n^1))

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  f(g(x), s(0())) -> f(g(x), g(x))
     , g(s(x)) -> s(g(x))
     , g(0()) -> 0()}

Proof Output:
  'wdg' proved the best result:

  Details:
  --------
    'wdg' succeeded with the following output:
     'wdg'
     -----
     Answer:           YES(?,O(n^1))
     Input Problem:    innermost runtime-complexity with respect to
       Rules:
         {  f(g(x), s(0())) -> f(g(x), g(x))
          , g(s(x)) -> s(g(x))
          , g(0()) -> 0()}

     Proof Output:
       Transformation Details:
       -----------------------
         We have computed the following set of weak (innermost) dependency pairs:

           {  1: f^#(g(x), s(0())) -> c_0(f^#(g(x), g(x)))
            , 2: g^#(s(x)) -> c_1(g^#(x))
            , 3: g^#(0()) -> c_2()}

         Following Dependency Graph (modulo SCCs) was computed. (Answers to
         subproofs are indicated to the right.)

           ->{2}                                                       [   YES(?,O(n^1))    ]
              |
              `->{3}                                                   [   YES(?,O(n^1))    ]

           ->{1}                                                       [   YES(?,O(n^1))    ]



       Sub-problems:
       -------------
         * Path {1}: YES(?,O(n^1))
           -----------------------

           The usable rules for this path are:

             {  g(s(x)) -> s(g(x))
              , g(0()) -> 0()}

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(f) = {}, Uargs(g) = {}, Uargs(s) = {1}, Uargs(f^#) = {},
               Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_1) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              f(x1, x2) = [0] x1 + [0] x2 + [0]
              g(x1) = [2] x1 + [1]
              s(x1) = [1] x1 + [2]
              0() = [3]
              f^#(x1, x2) = [2] x1 + [0] x2 + [0]
              c_0(x1) = [0] x1 + [0]
              g^#(x1) = [0] x1 + [0]
              c_1(x1) = [0] x1 + [0]
              c_2() = [0]
           Complexity induced by the adequate RMI: YES(?,O(n^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: {f^#(g(x), s(0())) -> c_0(f^#(g(x), g(x)))}
             Weak Rules:
               {  g(s(x)) -> s(g(x))
                , g(0()) -> 0()}

           Proof Output:
             The following argument positions are usable:
               Uargs(g) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              g(x1) = [2] x1 + [1]
              s(x1) = [0] x1 + [4]
              0() = [0]
              f^#(x1, x2) = [2] x1 + [0] x2 + [6]
              c_0(x1) = [0] x1 + [7]

         * Path {2}: YES(?,O(n^1))
           -----------------------

           The usable rules of this path are empty.

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(f) = {}, Uargs(g) = {}, Uargs(s) = {}, Uargs(f^#) = {},
               Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_1) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              f(x1, x2) = [0] x1 + [0] x2 + [0]
              g(x1) = [0] x1 + [0]
              s(x1) = [1] x1 + [0]
              0() = [0]
              f^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_0(x1) = [0] x1 + [0]
              g^#(x1) = [3] x1 + [0]
              c_1(x1) = [1] x1 + [0]
              c_2() = [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: {g^#(s(x)) -> c_1(g^#(x))}
             Weak Rules: {}

           Proof Output:
             The following argument positions are usable:
               Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_1) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              s(x1) = [1] x1 + [4]
              g^#(x1) = [2] x1 + [0]
              c_1(x1) = [1] x1 + [7]

         * Path {2}->{3}: YES(?,O(n^1))
           ----------------------------

           The usable rules of this path are empty.

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(f) = {}, Uargs(g) = {}, Uargs(s) = {}, Uargs(f^#) = {},
               Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_1) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              f(x1, x2) = [0] x1 + [0] x2 + [0]
              g(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              0() = [0]
              f^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_0(x1) = [0] x1 + [0]
              g^#(x1) = [0] x1 + [0]
              c_1(x1) = [1] x1 + [0]
              c_2() = [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: {g^#(0()) -> c_2()}
             Weak Rules: {g^#(s(x)) -> c_1(g^#(x))}

           Proof Output:
             The following argument positions are usable:
               Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_1) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              s(x1) = [1] x1 + [0]
              0() = [2]
              g^#(x1) = [2] x1 + [0]
              c_1(x1) = [1] x1 + [0]
              c_2() = [1]

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Strategy outermost added 08 4.14

stdout:

MAYBE
Warning when parsing problem:

Unsupported strategy 'OUTERMOST'

Tool RC2

Execution Time	Unknown
Answer	YES(?,O(n^1))
Input	Strategy outermost added 08 4.14

stdout:

YES(?,O(n^1))

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           YES(?,O(n^1))
Input Problem:    runtime-complexity with respect to
  Rules:
    {  f(g(x), s(0())) -> f(g(x), g(x))
     , g(s(x)) -> s(g(x))
     , g(0()) -> 0()}

Proof Output:
  'wdg' proved the best result:

  Details:
  --------
    'wdg' succeeded with the following output:
     'wdg'
     -----
     Answer:           YES(?,O(n^1))
     Input Problem:    runtime-complexity with respect to
       Rules:
         {  f(g(x), s(0())) -> f(g(x), g(x))
          , g(s(x)) -> s(g(x))
          , g(0()) -> 0()}

     Proof Output:
       Transformation Details:
       -----------------------
         We have computed the following set of weak (innermost) dependency pairs:

           {  1: f^#(g(x), s(0())) -> c_0(f^#(g(x), g(x)))
            , 2: g^#(s(x)) -> c_1(g^#(x))
            , 3: g^#(0()) -> c_2()}

         Following Dependency Graph (modulo SCCs) was computed. (Answers to
         subproofs are indicated to the right.)

           ->{2}                                                       [   YES(?,O(n^1))    ]
              |
              `->{3}                                                   [   YES(?,O(n^1))    ]

           ->{1}                                                       [   YES(?,O(n^1))    ]



       Sub-problems:
       -------------
         * Path {1}: YES(?,O(n^1))
           -----------------------

           The usable rules for this path are:

             {  g(s(x)) -> s(g(x))
              , g(0()) -> 0()}

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(f) = {}, Uargs(g) = {}, Uargs(s) = {1}, Uargs(f^#) = {},
               Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_1) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              f(x1, x2) = [0] x1 + [0] x2 + [0]
              g(x1) = [2] x1 + [1]
              s(x1) = [1] x1 + [2]
              0() = [3]
              f^#(x1, x2) = [2] x1 + [0] x2 + [0]
              c_0(x1) = [0] x1 + [0]
              g^#(x1) = [0] x1 + [0]
              c_1(x1) = [0] x1 + [0]
              c_2() = [0]
           Complexity induced by the adequate RMI: YES(?,O(n^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: {f^#(g(x), s(0())) -> c_0(f^#(g(x), g(x)))}
             Weak Rules:
               {  g(s(x)) -> s(g(x))
                , g(0()) -> 0()}

           Proof Output:
             The following argument positions are usable:
               Uargs(g) = {}, Uargs(s) = {}, Uargs(f^#) = {}, Uargs(c_0) = {}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              g(x1) = [2] x1 + [1]
              s(x1) = [0] x1 + [4]
              0() = [0]
              f^#(x1, x2) = [2] x1 + [0] x2 + [6]
              c_0(x1) = [0] x1 + [7]

         * Path {2}: YES(?,O(n^1))
           -----------------------

           The usable rules of this path are empty.

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(f) = {}, Uargs(g) = {}, Uargs(s) = {}, Uargs(f^#) = {},
               Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_1) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              f(x1, x2) = [0] x1 + [0] x2 + [0]
              g(x1) = [0] x1 + [0]
              s(x1) = [1] x1 + [0]
              0() = [0]
              f^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_0(x1) = [0] x1 + [0]
              g^#(x1) = [3] x1 + [0]
              c_1(x1) = [1] x1 + [0]
              c_2() = [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: {g^#(s(x)) -> c_1(g^#(x))}
             Weak Rules: {}

           Proof Output:
             The following argument positions are usable:
               Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_1) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              s(x1) = [1] x1 + [4]
              g^#(x1) = [2] x1 + [0]
              c_1(x1) = [1] x1 + [7]

         * Path {2}->{3}: YES(?,O(n^1))
           ----------------------------

           The usable rules of this path are empty.

           The weightgap principle applies, using the following adequate RMI:
             The following argument positions are usable:
               Uargs(f) = {}, Uargs(g) = {}, Uargs(s) = {}, Uargs(f^#) = {},
               Uargs(c_0) = {}, Uargs(g^#) = {}, Uargs(c_1) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              f(x1, x2) = [0] x1 + [0] x2 + [0]
              g(x1) = [0] x1 + [0]
              s(x1) = [0] x1 + [0]
              0() = [0]
              f^#(x1, x2) = [0] x1 + [0] x2 + [0]
              c_0(x1) = [0] x1 + [0]
              g^#(x1) = [0] x1 + [0]
              c_1(x1) = [1] x1 + [0]
              c_2() = [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: {g^#(0()) -> c_2()}
             Weak Rules: {g^#(s(x)) -> c_1(g^#(x))}

           Proof Output:
             The following argument positions are usable:
               Uargs(s) = {}, Uargs(g^#) = {}, Uargs(c_1) = {1}
             We have the following constructor-restricted matrix interpretation:
             Interpretation Functions:
              s(x1) = [1] x1 + [0]
              0() = [2]
              g^#(x1) = [2] x1 + [0]
              c_1(x1) = [1] x1 + [0]
              c_2() = [1]