Problem TCT 09 addmult

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputTCT 09 addmult

stdout:

MAYBE

Problem:
 add(0(),x) -> x
 add(s(x),y) -> s(add(x,y))
 mult(0(),x) -> 0()
 mult(s(x),y) -> add(y,mult(x,y))

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputTCT 09 addmult

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputTCT 09 addmult

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  add(0(), x) -> x
     , add(s(x), y) -> s(add(x, y))
     , mult(0(), x) -> 0()
     , mult(s(x), y) -> add(y, mult(x, y))}

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: add^#(0(), x) -> c_0()
              , 2: add^#(s(x), y) -> c_1(add^#(x, y))
              , 3: mult^#(0(), x) -> c_2()
              , 4: mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{4}                                                       [     inherited      ]
                |
                |->{1}                                                   [       MAYBE        ]
                |
                `->{2}                                                   [     inherited      ]
                    |
                    `->{1}                                               [         NA         ]
             
             ->{3}                                                       [    YES(?,O(1))     ]
             
           
         
         Sub-problems:
         -------------
           * Path {3}: 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(add) = {}, Uargs(s) = {}, Uargs(mult) = {},
                 Uargs(add^#) = {}, Uargs(c_1) = {}, Uargs(mult^#) = {},
                 Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                add(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]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                mult(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]
                add^#(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]
                mult^#(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_2() = [0]
                        [0]
                        [0]
                c_3(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:    innermost DP runtime-complexity with respect to
               Strict Rules: {mult^#(0(), x) -> c_2()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mult^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                mult^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
                                 [2 2 0]      [0 0 0]      [3]
                                 [2 2 2]      [0 0 0]      [3]
                c_2() = [0]
                        [1]
                        [1]
           
           * Path {4}: inherited
             -------------------
             
             This path is subsumed by the proof of path {4}->{2}->{1}.
           
           * Path {4}->{1}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  mult(0(), x) -> 0()
                , mult(s(x), y) -> add(y, mult(x, y))
                , add(0(), x) -> x
                , add(s(x), y) -> s(add(x, 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 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))
                  , add^#(0(), x) -> c_0()
                  , mult(0(), x) -> 0()
                  , mult(s(x), y) -> add(y, mult(x, y))
                  , add(0(), x) -> x
                  , add(s(x), y) -> s(add(x, y))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {4}->{2}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {4}->{2}->{1}.
           
           * Path {4}->{2}->{1}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  mult(0(), x) -> 0()
                , mult(s(x), y) -> add(y, mult(x, y))
                , add(0(), x) -> x
                , add(s(x), y) -> s(add(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: add^#(0(), x) -> c_0()
              , 2: add^#(s(x), y) -> c_1(add^#(x, y))
              , 3: mult^#(0(), x) -> c_2()
              , 4: mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{4}                                                       [     inherited      ]
                |
                |->{1}                                                   [       MAYBE        ]
                |
                `->{2}                                                   [     inherited      ]
                    |
                    `->{1}                                               [         NA         ]
             
             ->{3}                                                       [    YES(?,O(1))     ]
             
           
         
         Sub-problems:
         -------------
           * Path {3}: 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(add) = {}, Uargs(s) = {}, Uargs(mult) = {},
                 Uargs(add^#) = {}, Uargs(c_1) = {}, Uargs(mult^#) = {},
                 Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                mult(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                add^#(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]
                mult^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             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: {mult^#(0(), x) -> c_2()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mult^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                mult^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
                                 [2 2]      [0 0]      [7]
                c_2() = [0]
                        [1]
           
           * Path {4}: inherited
             -------------------
             
             This path is subsumed by the proof of path {4}->{2}->{1}.
           
           * Path {4}->{1}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  mult(0(), x) -> 0()
                , mult(s(x), y) -> add(y, mult(x, y))
                , add(0(), x) -> x
                , add(s(x), y) -> s(add(x, 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:
                 {  mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))
                  , add^#(0(), x) -> c_0()
                  , mult(0(), x) -> 0()
                  , mult(s(x), y) -> add(y, mult(x, y))
                  , add(0(), x) -> x
                  , add(s(x), y) -> s(add(x, y))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {4}->{2}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {4}->{2}->{1}.
           
           * Path {4}->{2}->{1}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  mult(0(), x) -> 0()
                , mult(s(x), y) -> add(y, mult(x, y))
                , add(0(), x) -> x
                , add(s(x), y) -> s(add(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: add^#(0(), x) -> c_0()
              , 2: add^#(s(x), y) -> c_1(add^#(x, y))
              , 3: mult^#(0(), x) -> c_2()
              , 4: mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{4}                                                       [     inherited      ]
                |
                |->{1}                                                   [       MAYBE        ]
                |
                `->{2}                                                   [     inherited      ]
                    |
                    `->{1}                                               [         NA         ]
             
             ->{3}                                                       [    YES(?,O(1))     ]
             
           
         
         Sub-problems:
         -------------
           * Path {3}: 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(add) = {}, Uargs(s) = {}, Uargs(mult) = {},
                 Uargs(add^#) = {}, Uargs(c_1) = {}, Uargs(mult^#) = {},
                 Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                mult(x1, x2) = [0] x1 + [0] x2 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                mult^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {mult^#(0(), x) -> c_2()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mult^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [7]
                mult^#(x1, x2) = [1] x1 + [0] x2 + [7]
                c_2() = [1]
           
           * Path {4}: inherited
             -------------------
             
             This path is subsumed by the proof of path {4}->{2}->{1}.
           
           * Path {4}->{1}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  mult(0(), x) -> 0()
                , mult(s(x), y) -> add(y, mult(x, y))
                , add(0(), x) -> x
                , add(s(x), y) -> s(add(x, 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:
                 {  mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))
                  , add^#(0(), x) -> c_0()
                  , mult(0(), x) -> 0()
                  , mult(s(x), y) -> add(y, mult(x, y))
                  , add(0(), x) -> x
                  , add(s(x), y) -> s(add(x, y))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {4}->{2}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {4}->{2}->{1}.
           
           * Path {4}->{2}->{1}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  mult(0(), x) -> 0()
                , mult(s(x), y) -> add(y, mult(x, y))
                , add(0(), x) -> x
                , add(s(x), y) -> s(add(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    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
InputTCT 09 addmult

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputTCT 09 addmult

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  add(0(), x) -> x
     , add(s(x), y) -> s(add(x, y))
     , mult(0(), x) -> 0()
     , mult(s(x), y) -> add(y, mult(x, y))}

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: add^#(0(), x) -> c_0(x)
              , 2: add^#(s(x), y) -> c_1(add^#(x, y))
              , 3: mult^#(0(), x) -> c_2()
              , 4: mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{4}                                                       [     inherited      ]
                |
                |->{1}                                                   [       MAYBE        ]
                |
                `->{2}                                                   [     inherited      ]
                    |
                    `->{1}                                               [         NA         ]
             
             ->{3}                                                       [    YES(?,O(1))     ]
             
           
         
         Sub-problems:
         -------------
           * Path {3}: 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(add) = {}, Uargs(s) = {}, Uargs(mult) = {},
                 Uargs(add^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
                 Uargs(mult^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                add(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]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                mult(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]
                add^#(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]
                mult^#(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_2() = [0]
                        [0]
                        [0]
                c_3(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: {mult^#(0(), x) -> c_2()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mult^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                mult^#(x1, x2) = [0 2 0] x1 + [0 0 0] x2 + [7]
                                 [2 2 0]      [0 0 0]      [3]
                                 [2 2 2]      [0 0 0]      [3]
                c_2() = [0]
                        [1]
                        [1]
           
           * Path {4}: inherited
             -------------------
             
             This path is subsumed by the proof of path {4}->{2}->{1}.
           
           * Path {4}->{1}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  mult(0(), x) -> 0()
                , mult(s(x), y) -> add(y, mult(x, y))
                , add(0(), x) -> x
                , add(s(x), y) -> s(add(x, 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 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))
                  , add^#(0(), x) -> c_0(x)
                  , mult(0(), x) -> 0()
                  , mult(s(x), y) -> add(y, mult(x, y))
                  , add(0(), x) -> x
                  , add(s(x), y) -> s(add(x, y))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {4}->{2}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {4}->{2}->{1}.
           
           * Path {4}->{2}->{1}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  mult(0(), x) -> 0()
                , mult(s(x), y) -> add(y, mult(x, y))
                , add(0(), x) -> x
                , add(s(x), y) -> s(add(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: add^#(0(), x) -> c_0(x)
              , 2: add^#(s(x), y) -> c_1(add^#(x, y))
              , 3: mult^#(0(), x) -> c_2()
              , 4: mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{4}                                                       [     inherited      ]
                |
                |->{1}                                                   [       MAYBE        ]
                |
                `->{2}                                                   [     inherited      ]
                    |
                    `->{1}                                               [         NA         ]
             
             ->{3}                                                       [    YES(?,O(1))     ]
             
           
         
         Sub-problems:
         -------------
           * Path {3}: 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(add) = {}, Uargs(s) = {}, Uargs(mult) = {},
                 Uargs(add^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
                 Uargs(mult^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                add(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                mult(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                add^#(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]
                mult^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             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: {mult^#(0(), x) -> c_2()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mult^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                mult^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
                                 [2 2]      [0 0]      [7]
                c_2() = [0]
                        [1]
           
           * Path {4}: inherited
             -------------------
             
             This path is subsumed by the proof of path {4}->{2}->{1}.
           
           * Path {4}->{1}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  mult(0(), x) -> 0()
                , mult(s(x), y) -> add(y, mult(x, y))
                , add(0(), x) -> x
                , add(s(x), y) -> s(add(x, 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:
                 {  mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))
                  , add^#(0(), x) -> c_0(x)
                  , mult(0(), x) -> 0()
                  , mult(s(x), y) -> add(y, mult(x, y))
                  , add(0(), x) -> x
                  , add(s(x), y) -> s(add(x, y))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {4}->{2}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {4}->{2}->{1}.
           
           * Path {4}->{2}->{1}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  mult(0(), x) -> 0()
                , mult(s(x), y) -> add(y, mult(x, y))
                , add(0(), x) -> x
                , add(s(x), y) -> s(add(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: add^#(0(), x) -> c_0(x)
              , 2: add^#(s(x), y) -> c_1(add^#(x, y))
              , 3: mult^#(0(), x) -> c_2()
              , 4: mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{4}                                                       [     inherited      ]
                |
                |->{1}                                                   [       MAYBE        ]
                |
                `->{2}                                                   [     inherited      ]
                    |
                    `->{1}                                               [         NA         ]
             
             ->{3}                                                       [    YES(?,O(1))     ]
             
           
         
         Sub-problems:
         -------------
           * Path {3}: 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(add) = {}, Uargs(s) = {}, Uargs(mult) = {},
                 Uargs(add^#) = {}, Uargs(c_0) = {}, Uargs(c_1) = {},
                 Uargs(mult^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                add(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                mult(x1, x2) = [0] x1 + [0] x2 + [0]
                add^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                mult^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3(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: {mult^#(0(), x) -> c_2()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(mult^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [7]
                mult^#(x1, x2) = [1] x1 + [0] x2 + [7]
                c_2() = [1]
           
           * Path {4}: inherited
             -------------------
             
             This path is subsumed by the proof of path {4}->{2}->{1}.
           
           * Path {4}->{1}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  mult(0(), x) -> 0()
                , mult(s(x), y) -> add(y, mult(x, y))
                , add(0(), x) -> x
                , add(s(x), y) -> s(add(x, 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:
                 {  mult^#(s(x), y) -> c_3(add^#(y, mult(x, y)))
                  , add^#(0(), x) -> c_0(x)
                  , mult(0(), x) -> 0()
                  , mult(s(x), y) -> add(y, mult(x, y))
                  , add(0(), x) -> x
                  , add(s(x), y) -> s(add(x, y))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {4}->{2}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {4}->{2}->{1}.
           
           * Path {4}->{2}->{1}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  mult(0(), x) -> 0()
                , mult(s(x), y) -> add(y, mult(x, y))
                , add(0(), x) -> x
                , add(s(x), y) -> s(add(x, y))}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
    
    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
InputTCT 09 addmult

stdout:

TIMEOUT

We consider the following Problem:

  Strict Trs:
    {  add(0(), x) -> x
     , add(s(x), y) -> s(add(x, y))
     , mult(0(), x) -> 0()
     , mult(s(x), y) -> add(y, mult(x, y))}
  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
InputTCT 09 addmult

stdout:

TIMEOUT

We consider the following Problem:

  Strict Trs:
    {  add(0(), x) -> x
     , add(s(x), y) -> s(add(x, y))
     , mult(0(), x) -> 0()
     , mult(s(x), y) -> add(y, mult(x, y))}
  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
InputTCT 09 addmult

stdout:

TIMEOUT

We consider the following Problem:

  Strict Trs:
    {  add(0(), x) -> x
     , add(s(x), y) -> s(add(x, y))
     , mult(0(), x) -> 0()
     , mult(s(x), y) -> add(y, mult(x, y))}
  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
InputTCT 09 addmult

stdout:

TIMEOUT

We consider the following Problem:

  Strict Trs:
    {  add(0(), x) -> x
     , add(s(x), y) -> s(add(x, y))
     , mult(0(), x) -> 0()
     , mult(s(x), y) -> add(y, mult(x, y))}
  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
InputTCT 09 addmult

stdout:

MAYBE

We consider the following Problem:

  Strict Trs:
    {  add(0(), x) -> x
     , add(s(x), y) -> s(add(x, y))
     , mult(0(), x) -> 0()
     , mult(s(x), y) -> add(y, mult(x, y))}
  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:
           {  add^#(0(), x) -> c_1(x)
            , add^#(s(x), y) -> c_2(add^#(x, y))
            , mult^#(0(), x) -> c_3()
            , mult^#(s(x), y) -> c_4(add^#(y, mult(x, y)))}
         
         We consider the following Problem:
         
           Strict DPs:
             {  add^#(0(), x) -> c_1(x)
              , add^#(s(x), y) -> c_2(add^#(x, y))
              , mult^#(0(), x) -> c_3()
              , mult^#(s(x), y) -> c_4(add^#(y, mult(x, y)))}
           Strict Trs:
             {  add(0(), x) -> x
              , add(s(x), y) -> s(add(x, y))
              , mult(0(), x) -> 0()
              , mult(s(x), y) -> add(y, mult(x, y))}
           StartTerms: basic terms
           Strategy: none
         
         Certificate: MAYBE
         
         Application of 'usablerules':
         -----------------------------
           All rules are usable.
           
           No subproblems were generated.
    

Arrrr..

Tool tup3irc

Execution Time65.25244ms
Answer
TIMEOUT
InputTCT 09 addmult

stdout:

TIMEOUT

We consider the following Problem:

  Strict Trs:
    {  add(0(), x) -> x
     , add(s(x), y) -> s(add(x, y))
     , mult(0(), x) -> 0()
     , mult(s(x), y) -> add(y, mult(x, y))}
  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..