Problem SK90 4.53

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputSK90 4.53

stdout:

MAYBE

Problem:
 f(a()) -> b()
 f(c()) -> d()
 f(g(x,y)) -> g(f(x),f(y))
 f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
 g(x,x) -> h(e(),x)

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputSK90 4.53

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputSK90 4.53

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  f(a()) -> b()
     , f(c()) -> d()
     , f(g(x, y)) -> g(f(x), f(y))
     , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
     , g(x, x) -> h(e(), 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: f^#(a()) -> c_0()
              , 2: f^#(c()) -> c_1()
              , 3: f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
              , 4: f^#(h(x, y)) -> c_3(g^#(h(y, f(x)), h(x, f(y))))
              , 5: g^#(x, x) -> c_4()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{4}                                                       [     inherited      ]
                |
                `->{5}                                                   [         NA         ]
             
             ->{3}                                                       [     inherited      ]
                |
                `->{5}                                                   [       MAYBE        ]
             
             ->{2}                                                       [    YES(?,O(1))     ]
             
             ->{1}                                                       [    YES(?,O(1))     ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: 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_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                d() = [0]
                      [0]
                      [0]
                g(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]
                h(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]
                e() = [0]
                      [0]
                      [0]
                f^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                g^#(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_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: {f^#(a()) -> c_0()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(f^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a() = [2]
                      [2]
                      [2]
                f^#(x1) = [0 2 0] x1 + [7]
                          [2 2 0]      [3]
                          [2 2 2]      [3]
                c_0() = [0]
                        [1]
                        [1]
           
           * Path {2}: 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_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                d() = [0]
                      [0]
                      [0]
                g(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]
                h(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]
                e() = [0]
                      [0]
                      [0]
                f^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                g^#(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_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: {f^#(c()) -> c_1()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(f^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [2]
                      [2]
                      [2]
                f^#(x1) = [0 2 0] x1 + [7]
                          [2 2 0]      [3]
                          [2 2 2]      [3]
                c_1() = [0]
                        [1]
                        [1]
           
           * Path {3}: inherited
             -------------------
             
             This path is subsumed by the proof of path {3}->{5}.
           
           * Path {3}->{5}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  f(a()) -> b()
                , f(c()) -> d()
                , f(g(x, y)) -> g(f(x), f(y))
                , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
                , g(x, x) -> h(e(), x)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
                  , g^#(x, x) -> c_4()
                  , f(a()) -> b()
                  , f(c()) -> d()
                  , f(g(x, y)) -> g(f(x), f(y))
                  , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
                  , g(x, x) -> h(e(), x)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {4}: inherited
             -------------------
             
             This path is subsumed by the proof of path {4}->{5}.
           
           * Path {4}->{5}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  f(a()) -> b()
                , f(c()) -> d()
                , f(g(x, y)) -> g(f(x), f(y))
                , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
                , g(x, x) -> h(e(), x)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(a()) -> c_0()
              , 2: f^#(c()) -> c_1()
              , 3: f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
              , 4: f^#(h(x, y)) -> c_3(g^#(h(y, f(x)), h(x, f(y))))
              , 5: g^#(x, x) -> c_4()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{4}                                                       [     inherited      ]
                |
                `->{5}                                                   [         NA         ]
             
             ->{3}                                                       [     inherited      ]
                |
                `->{5}                                                   [       MAYBE        ]
             
             ->{2}                                                       [    YES(?,O(1))     ]
             
             ->{1}                                                       [    YES(?,O(1))     ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: 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_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                d() = [0]
                      [0]
                g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                h(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                e() = [0]
                      [0]
                f^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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: {f^#(a()) -> c_0()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(f^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a() = [2]
                      [2]
                f^#(x1) = [2 0] x1 + [7]
                          [2 2]      [7]
                c_0() = [0]
                        [1]
           
           * Path {2}: 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_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                d() = [0]
                      [0]
                g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                h(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                e() = [0]
                      [0]
                f^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [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: {f^#(c()) -> c_1()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(f^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [2]
                      [2]
                f^#(x1) = [2 0] x1 + [7]
                          [2 2]      [7]
                c_1() = [0]
                        [1]
           
           * Path {3}: inherited
             -------------------
             
             This path is subsumed by the proof of path {3}->{5}.
           
           * Path {3}->{5}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  f(a()) -> b()
                , f(c()) -> d()
                , f(g(x, y)) -> g(f(x), f(y))
                , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
                , g(x, x) -> h(e(), x)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
                  , g^#(x, x) -> c_4()
                  , f(a()) -> b()
                  , f(c()) -> d()
                  , f(g(x, y)) -> g(f(x), f(y))
                  , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
                  , g(x, x) -> h(e(), x)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {4}: inherited
             -------------------
             
             This path is subsumed by the proof of path {4}->{5}.
           
           * Path {4}->{5}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  f(a()) -> b()
                , f(c()) -> d()
                , f(g(x, y)) -> g(f(x), f(y))
                , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
                , g(x, x) -> h(e(), x)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(a()) -> c_0()
              , 2: f^#(c()) -> c_1()
              , 3: f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
              , 4: f^#(h(x, y)) -> c_3(g^#(h(y, f(x)), h(x, f(y))))
              , 5: g^#(x, x) -> c_4()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{4}                                                       [     inherited      ]
                |
                `->{5}                                                   [         NA         ]
             
             ->{3}                                                       [     inherited      ]
                |
                `->{5}                                                   [       MAYBE        ]
             
             ->{2}                                                       [    YES(?,O(1))     ]
             
             ->{1}                                                       [    YES(?,O(1))     ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: 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_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0] x1 + [0]
                a() = [0]
                b() = [0]
                c() = [0]
                d() = [0]
                g(x1, x2) = [0] x1 + [0] x2 + [0]
                h(x1, x2) = [0] x1 + [0] x2 + [0]
                e() = [0]
                f^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                g^#(x1, x2) = [0] x1 + [0] x2 + [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: {f^#(a()) -> c_0()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(f^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a() = [7]
                f^#(x1) = [1] x1 + [7]
                c_0() = [1]
           
           * Path {2}: 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_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0] x1 + [0]
                a() = [0]
                b() = [0]
                c() = [0]
                d() = [0]
                g(x1, x2) = [0] x1 + [0] x2 + [0]
                h(x1, x2) = [0] x1 + [0] x2 + [0]
                e() = [0]
                f^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                g^#(x1, x2) = [0] x1 + [0] x2 + [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: {f^#(c()) -> c_1()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(f^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [7]
                f^#(x1) = [1] x1 + [7]
                c_1() = [1]
           
           * Path {3}: inherited
             -------------------
             
             This path is subsumed by the proof of path {3}->{5}.
           
           * Path {3}->{5}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  f(a()) -> b()
                , f(c()) -> d()
                , f(g(x, y)) -> g(f(x), f(y))
                , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
                , g(x, x) -> h(e(), x)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
                  , g^#(x, x) -> c_4()
                  , f(a()) -> b()
                  , f(c()) -> d()
                  , f(g(x, y)) -> g(f(x), f(y))
                  , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
                  , g(x, x) -> h(e(), x)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {4}: inherited
             -------------------
             
             This path is subsumed by the proof of path {4}->{5}.
           
           * Path {4}->{5}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  f(a()) -> b()
                , f(c()) -> d()
                , f(g(x, y)) -> g(f(x), f(y))
                , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
                , g(x, x) -> h(e(), x)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    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 4.53

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputSK90 4.53

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  f(a()) -> b()
     , f(c()) -> d()
     , f(g(x, y)) -> g(f(x), f(y))
     , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
     , g(x, x) -> h(e(), 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: f^#(a()) -> c_0()
              , 2: f^#(c()) -> c_1()
              , 3: f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
              , 4: f^#(h(x, y)) -> c_3(g^#(h(y, f(x)), h(x, f(y))))
              , 5: g^#(x, x) -> c_4(x)}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{4}                                                       [     inherited      ]
                |
                `->{5}                                                   [         NA         ]
             
             ->{3}                                                       [     inherited      ]
                |
                `->{5}                                                   [       MAYBE        ]
             
             ->{2}                                                       [    YES(?,O(1))     ]
             
             ->{1}                                                       [    YES(?,O(1))     ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: 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_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                d() = [0]
                      [0]
                      [0]
                g(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]
                h(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]
                e() = [0]
                      [0]
                      [0]
                f^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                g^#(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_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             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: {f^#(a()) -> c_0()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(f^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a() = [2]
                      [2]
                      [2]
                f^#(x1) = [0 2 0] x1 + [7]
                          [2 2 0]      [3]
                          [2 2 2]      [3]
                c_0() = [0]
                        [1]
                        [1]
           
           * Path {2}: 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_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                a() = [0]
                      [0]
                      [0]
                b() = [0]
                      [0]
                      [0]
                c() = [0]
                      [0]
                      [0]
                d() = [0]
                      [0]
                      [0]
                g(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]
                h(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]
                e() = [0]
                      [0]
                      [0]
                f^#(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                g^#(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_3(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             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: {f^#(c()) -> c_1()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(f^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [2]
                      [2]
                      [2]
                f^#(x1) = [0 2 0] x1 + [7]
                          [2 2 0]      [3]
                          [2 2 2]      [3]
                c_1() = [0]
                        [1]
                        [1]
           
           * Path {3}: inherited
             -------------------
             
             This path is subsumed by the proof of path {3}->{5}.
           
           * Path {3}->{5}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  f(a()) -> b()
                , f(c()) -> d()
                , f(g(x, y)) -> g(f(x), f(y))
                , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
                , g(x, x) -> h(e(), x)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
                  , g^#(x, x) -> c_4(x)
                  , f(a()) -> b()
                  , f(c()) -> d()
                  , f(g(x, y)) -> g(f(x), f(y))
                  , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
                  , g(x, x) -> h(e(), x)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {4}: inherited
             -------------------
             
             This path is subsumed by the proof of path {4}->{5}.
           
           * Path {4}->{5}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  f(a()) -> b()
                , f(c()) -> d()
                , f(g(x, y)) -> g(f(x), f(y))
                , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
                , g(x, x) -> h(e(), x)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(a()) -> c_0()
              , 2: f^#(c()) -> c_1()
              , 3: f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
              , 4: f^#(h(x, y)) -> c_3(g^#(h(y, f(x)), h(x, f(y))))
              , 5: g^#(x, x) -> c_4(x)}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{4}                                                       [     inherited      ]
                |
                `->{5}                                                   [         NA         ]
             
             ->{3}                                                       [     inherited      ]
                |
                `->{5}                                                   [       MAYBE        ]
             
             ->{2}                                                       [    YES(?,O(1))     ]
             
             ->{1}                                                       [    YES(?,O(1))     ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: 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_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                d() = [0]
                      [0]
                g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                h(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                e() = [0]
                      [0]
                f^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {f^#(a()) -> c_0()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(f^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a() = [2]
                      [2]
                f^#(x1) = [2 0] x1 + [7]
                          [2 2]      [7]
                c_0() = [0]
                        [1]
           
           * Path {2}: 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_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                a() = [0]
                      [0]
                b() = [0]
                      [0]
                c() = [0]
                      [0]
                d() = [0]
                      [0]
                g(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                h(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                e() = [0]
                      [0]
                f^#(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                g^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {f^#(c()) -> c_1()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(f^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [2]
                      [2]
                f^#(x1) = [2 0] x1 + [7]
                          [2 2]      [7]
                c_1() = [0]
                        [1]
           
           * Path {3}: inherited
             -------------------
             
             This path is subsumed by the proof of path {3}->{5}.
           
           * Path {3}->{5}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  f(a()) -> b()
                , f(c()) -> d()
                , f(g(x, y)) -> g(f(x), f(y))
                , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
                , g(x, x) -> h(e(), x)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
                  , g^#(x, x) -> c_4(x)
                  , f(a()) -> b()
                  , f(c()) -> d()
                  , f(g(x, y)) -> g(f(x), f(y))
                  , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
                  , g(x, x) -> h(e(), x)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {4}: inherited
             -------------------
             
             This path is subsumed by the proof of path {4}->{5}.
           
           * Path {4}->{5}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  f(a()) -> b()
                , f(c()) -> d()
                , f(g(x, y)) -> g(f(x), f(y))
                , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
                , g(x, x) -> h(e(), x)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(a()) -> c_0()
              , 2: f^#(c()) -> c_1()
              , 3: f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
              , 4: f^#(h(x, y)) -> c_3(g^#(h(y, f(x)), h(x, f(y))))
              , 5: g^#(x, x) -> c_4(x)}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{4}                                                       [     inherited      ]
                |
                `->{5}                                                   [         NA         ]
             
             ->{3}                                                       [     inherited      ]
                |
                `->{5}                                                   [       MAYBE        ]
             
             ->{2}                                                       [    YES(?,O(1))     ]
             
             ->{1}                                                       [    YES(?,O(1))     ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: 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_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0] x1 + [0]
                a() = [0]
                b() = [0]
                c() = [0]
                d() = [0]
                g(x1, x2) = [0] x1 + [0] x2 + [0]
                h(x1, x2) = [0] x1 + [0] x2 + [0]
                e() = [0]
                f^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                g^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {f^#(a()) -> c_0()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(f^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                a() = [7]
                f^#(x1) = [1] x1 + [7]
                c_0() = [1]
           
           * Path {2}: 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_2) = {}, Uargs(g^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1) = [0] x1 + [0]
                a() = [0]
                b() = [0]
                c() = [0]
                d() = [0]
                g(x1, x2) = [0] x1 + [0] x2 + [0]
                h(x1, x2) = [0] x1 + [0] x2 + [0]
                e() = [0]
                f^#(x1) = [0] x1 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                g^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {f^#(c()) -> c_1()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(f^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                c() = [7]
                f^#(x1) = [1] x1 + [7]
                c_1() = [1]
           
           * Path {3}: inherited
             -------------------
             
             This path is subsumed by the proof of path {3}->{5}.
           
           * Path {3}->{5}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  f(a()) -> b()
                , f(c()) -> d()
                , f(g(x, y)) -> g(f(x), f(y))
                , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
                , g(x, x) -> h(e(), x)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  f^#(g(x, y)) -> c_2(g^#(f(x), f(y)))
                  , g^#(x, x) -> c_4(x)
                  , f(a()) -> b()
                  , f(c()) -> d()
                  , f(g(x, y)) -> g(f(x), f(y))
                  , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
                  , g(x, x) -> h(e(), x)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {4}: inherited
             -------------------
             
             This path is subsumed by the proof of path {4}->{5}.
           
           * Path {4}->{5}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  f(a()) -> b()
                , f(c()) -> d()
                , f(g(x, y)) -> g(f(x), f(y))
                , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
                , g(x, x) -> h(e(), x)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    4) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    

Tool pair1rc

Execution TimeUnknown
Answer
TIMEOUT
InputSK90 4.53

stdout:

TIMEOUT

We consider the following Problem:

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

Certificate: TIMEOUT

Application of 'pair1 (timeout of 60.0 seconds)':
-------------------------------------------------
  Computation stopped due to timeout after 60.0 seconds

Arrrr..

Tool pair2rc

Execution TimeUnknown
Answer
TIMEOUT
InputSK90 4.53

stdout:

TIMEOUT

We consider the following Problem:

  Strict Trs:
    {  f(a()) -> b()
     , f(c()) -> d()
     , f(g(x, y)) -> g(f(x), f(y))
     , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
     , g(x, x) -> h(e(), x)}
  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 4.53

stdout:

TIMEOUT

We consider the following Problem:

  Strict Trs:
    {  f(a()) -> b()
     , f(c()) -> d()
     , f(g(x, y)) -> g(f(x), f(y))
     , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
     , g(x, x) -> h(e(), x)}
  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 rc

Execution TimeUnknown
Answer
TIMEOUT
InputSK90 4.53

stdout:

TIMEOUT

We consider the following Problem:

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

Certificate: TIMEOUT

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

Arrrr..

Tool tup3irc

Execution Time60.825024ms
Answer
TIMEOUT
InputSK90 4.53

stdout:

TIMEOUT

We consider the following Problem:

  Strict Trs:
    {  f(a()) -> b()
     , f(c()) -> d()
     , f(g(x, y)) -> g(f(x), f(y))
     , f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
     , g(x, x) -> h(e(), x)}
  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..