Problem SK90 4.42

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputSK90 4.42

stdout:

MAYBE

Problem:
 f(a(),g(y)) -> g(g(y))
 f(g(x),a()) -> f(x,g(a()))
 f(g(x),g(y)) -> h(g(y),x,g(y))
 h(g(x),y,z) -> f(y,h(x,y,z))
 h(a(),y,z) -> z

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputSK90 4.42

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputSK90 4.42

stdout:

MAYBE

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

Tool RC1

Execution TimeUnknown
Answer
MAYBE
InputSK90 4.42

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputSK90 4.42

stdout:

MAYBE

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

Tool pair2rc

Execution TimeUnknown
Answer
TIMEOUT
InputSK90 4.42

stdout:

TIMEOUT

We consider the following Problem:

  Strict Trs:
    {  f(a(), g(y)) -> g(g(y))
     , f(g(x), a()) -> f(x, g(a()))
     , f(g(x), g(y)) -> h(g(y), x, g(y))
     , h(g(x), y, z) -> f(y, h(x, y, z))
     , h(a(), y, z) -> z}
  StartTerms: basic terms
  Strategy: none

Certificate: TIMEOUT

Application of 'pair2 (timeout of 60.0 seconds)':
-------------------------------------------------
  Computation stopped due to timeout after 60.0 seconds

Arrrr..

Tool pair3irc

Execution TimeUnknown
Answer
TIMEOUT
InputSK90 4.42

stdout:

TIMEOUT

We consider the following Problem:

  Strict Trs:
    {  f(a(), g(y)) -> g(g(y))
     , f(g(x), a()) -> f(x, g(a()))
     , f(g(x), g(y)) -> h(g(y), x, g(y))
     , h(g(x), y, z) -> f(y, h(x, y, z))
     , h(a(), y, z) -> z}
  StartTerms: basic terms
  Strategy: innermost

Certificate: TIMEOUT

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

Arrrr..

Tool pair3rc

Execution TimeUnknown
Answer
TIMEOUT
InputSK90 4.42

stdout:

TIMEOUT

We consider the following Problem:

  Strict Trs:
    {  f(a(), g(y)) -> g(g(y))
     , f(g(x), a()) -> f(x, g(a()))
     , f(g(x), g(y)) -> h(g(y), x, g(y))
     , h(g(x), y, z) -> f(y, h(x, y, z))
     , h(a(), y, z) -> z}
  StartTerms: basic terms
  Strategy: none

Certificate: TIMEOUT

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

Arrrr..

Tool rc

Execution TimeUnknown
Answer
MAYBE
InputSK90 4.42

stdout:

MAYBE

We consider the following Problem:

  Strict Trs:
    {  f(a(), g(y)) -> g(g(y))
     , f(g(x), a()) -> f(x, g(a()))
     , f(g(x), g(y)) -> h(g(y), x, g(y))
     , h(g(x), y, z) -> f(y, h(x, y, z))
     , h(a(), y, z) -> z}
  StartTerms: basic terms
  Strategy: none

Certificate: MAYBE

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

Arrrr..

Tool tup3irc

Execution Time61.71491ms
Answer
TIMEOUT
InputSK90 4.42

stdout:

TIMEOUT

We consider the following Problem:

  Strict Trs:
    {  f(a(), g(y)) -> g(g(y))
     , f(g(x), a()) -> f(x, g(a()))
     , f(g(x), g(y)) -> h(g(y), x, g(y))
     , h(g(x), y, z) -> f(y, h(x, y, z))
     , h(a(), y, z) -> z}
  StartTerms: basic terms
  Strategy: innermost

Certificate: TIMEOUT

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

Arrrr..