Problem AProVE 04 IJCAR 12

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputAProVE 04 IJCAR 12

stdout:

MAYBE

Problem:
 plus(x,0()) -> x
 plus(0(),y) -> y
 plus(s(x),y) -> s(plus(x,y))
 times(0(),y) -> 0()
 times(s(0()),y) -> y
 times(s(x),y) -> plus(y,times(x,y))
 div(0(),y) -> 0()
 div(x,y) -> quot(x,y,y)
 quot(0(),s(y),z) -> 0()
 quot(s(x),s(y),z) -> quot(x,y,z)
 quot(x,0(),s(z)) -> s(div(x,s(z)))
 div(div(x,y),z) -> div(x,times(y,z))

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputAProVE 04 IJCAR 12

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputAProVE 04 IJCAR 12

stdout:

MAYBE

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

Proof Output:    
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: plus^#(x, 0()) -> c_0()
              , 2: plus^#(0(), y) -> c_1()
              , 3: plus^#(s(x), y) -> c_2(plus^#(x, y))
              , 4: times^#(0(), y) -> c_3()
              , 5: times^#(s(0()), y) -> c_4()
              , 6: times^#(s(x), y) -> c_5(plus^#(y, times(x, y)))
              , 7: div^#(0(), y) -> c_6()
              , 8: div^#(x, y) -> c_7(quot^#(x, y, y))
              , 9: quot^#(0(), s(y), z) -> c_8()
              , 10: quot^#(s(x), s(y), z) -> c_9(quot^#(x, y, z))
              , 11: quot^#(x, 0(), s(z)) -> c_10(div^#(x, s(z)))
              , 12: div^#(div(x, y), z) -> c_11(div^#(x, times(y, z)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{8,12,11,10}                                              [     inherited      ]
                |
                |->{7}                                                   [         NA         ]
                |
                `->{9}                                                   [         NA         ]
             
             ->{6}                                                       [     inherited      ]
                |
                |->{1}                                                   [       MAYBE        ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    |->{1}                                               [         NA         ]
                    |
                    `->{2}                                               [         NA         ]
             
             ->{5}                                                       [    YES(?,O(1))     ]
             
             ->{4}                                                       [    YES(?,O(1))     ]
             
           
         
         Sub-problems:
         -------------
           * Path {4}: 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(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(plus^#) = {},
                 Uargs(c_2) = {}, Uargs(times^#) = {}, Uargs(c_5) = {},
                 Uargs(div^#) = {}, Uargs(c_7) = {}, Uargs(quot^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(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]
                times(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]
                div(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]
                quot(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                plus^#(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() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                times^#(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() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                div^#(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_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                quot^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11(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: {times^#(0(), y) -> c_3()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(times^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                times^#(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_3() = [0]
                        [1]
                        [1]
           
           * Path {5}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(plus^#) = {},
                 Uargs(c_2) = {}, Uargs(times^#) = {}, Uargs(c_5) = {},
                 Uargs(div^#) = {}, Uargs(c_7) = {}, Uargs(quot^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(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]
                times(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]
                div(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]
                quot(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                plus^#(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() = [0]
                        [0]
                        [0]
                c_2(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                times^#(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() = [0]
                        [0]
                        [0]
                c_4() = [0]
                        [0]
                        [0]
                c_5(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                div^#(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_6() = [0]
                        [0]
                        [0]
                c_7(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                quot^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_8() = [0]
                        [0]
                        [0]
                c_9(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_10(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                c_11(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: {times^#(s(0()), y) -> c_4()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(times^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [2]
                      [0]
                s(x1) = [0 3 2] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                times^#(x1, x2) = [2 0 0] x1 + [0 0 0] x2 + [3]
                                  [0 0 0]      [0 0 0]      [7]
                                  [0 0 0]      [0 0 0]      [7]
                c_4() = [0]
                        [1]
                        [1]
           
           * Path {6}: inherited
             -------------------
             
             This path is subsumed by the proof of path {6}->{3}->{1}.
           
           * Path {6}->{1}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(0()), y) -> y
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(x, 0()) -> x
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(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:
                 {  times^#(s(x), y) -> c_5(plus^#(y, times(x, y)))
                  , plus^#(x, 0()) -> c_0()
                  , times(0(), y) -> 0()
                  , times(s(0()), y) -> y
                  , times(s(x), y) -> plus(y, times(x, y))
                  , plus(x, 0()) -> x
                  , plus(0(), y) -> y
                  , plus(s(x), y) -> s(plus(x, y))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {6}->{2}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(0()), y) -> y
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(x, 0()) -> x
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(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.
           
           * Path {6}->{3}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {6}->{3}->{1}.
           
           * Path {6}->{3}->{1}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(0()), y) -> y
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(x, 0()) -> x
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(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.
           
           * Path {6}->{3}->{2}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(0()), y) -> y
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(x, 0()) -> x
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(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.
           
           * Path {8,12,11,10}: inherited
             ----------------------------
             
             This path is subsumed by the proof of path {8,12,11,10}->{7}.
           
           * Path {8,12,11,10}->{7}: NA
             --------------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(0()), y) -> y
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(x, 0()) -> x
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(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.
           
           * Path {8,12,11,10}->{9}: NA
             --------------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(0()), y) -> y
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(x, 0()) -> x
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(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: plus^#(x, 0()) -> c_0()
              , 2: plus^#(0(), y) -> c_1()
              , 3: plus^#(s(x), y) -> c_2(plus^#(x, y))
              , 4: times^#(0(), y) -> c_3()
              , 5: times^#(s(0()), y) -> c_4()
              , 6: times^#(s(x), y) -> c_5(plus^#(y, times(x, y)))
              , 7: div^#(0(), y) -> c_6()
              , 8: div^#(x, y) -> c_7(quot^#(x, y, y))
              , 9: quot^#(0(), s(y), z) -> c_8()
              , 10: quot^#(s(x), s(y), z) -> c_9(quot^#(x, y, z))
              , 11: quot^#(x, 0(), s(z)) -> c_10(div^#(x, s(z)))
              , 12: div^#(div(x, y), z) -> c_11(div^#(x, times(y, z)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{8,12,11,10}                                              [     inherited      ]
                |
                |->{7}                                                   [         NA         ]
                |
                `->{9}                                                   [         NA         ]
             
             ->{6}                                                       [     inherited      ]
                |
                |->{1}                                                   [       MAYBE        ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    |->{1}                                               [         NA         ]
                    |
                    `->{2}                                               [         NA         ]
             
             ->{5}                                                       [    YES(?,O(1))     ]
             
             ->{4}                                                       [    YES(?,O(1))     ]
             
           
         
         Sub-problems:
         -------------
           * Path {4}: 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(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(plus^#) = {},
                 Uargs(c_2) = {}, Uargs(times^#) = {}, Uargs(c_5) = {},
                 Uargs(div^#) = {}, Uargs(c_7) = {}, Uargs(quot^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(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]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11(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: {times^#(0(), y) -> c_3()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(times^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                times^#(x1, x2) = [2 0] x1 + [0 0] x2 + [7]
                                  [2 2]      [0 0]      [7]
                c_3() = [0]
                        [1]
           
           * Path {5}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(plus^#) = {},
                 Uargs(c_2) = {}, Uargs(times^#) = {}, Uargs(c_5) = {},
                 Uargs(div^#) = {}, Uargs(c_7) = {}, Uargs(quot^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(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]
                times(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                plus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                times^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                c_11(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: {times^#(s(0()), y) -> c_4()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(times^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [2]
                s(x1) = [0 1] x1 + [2]
                        [0 0]      [2]
                times^#(x1, x2) = [2 2] x1 + [0 0] x2 + [3]
                                  [2 2]      [0 0]      [3]
                c_4() = [0]
                        [1]
           
           * Path {6}: inherited
             -------------------
             
             This path is subsumed by the proof of path {6}->{3}->{1}.
           
           * Path {6}->{1}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(0()), y) -> y
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(x, 0()) -> x
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(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:
                 {  times^#(s(x), y) -> c_5(plus^#(y, times(x, y)))
                  , plus^#(x, 0()) -> c_0()
                  , times(0(), y) -> 0()
                  , times(s(0()), y) -> y
                  , times(s(x), y) -> plus(y, times(x, y))
                  , plus(x, 0()) -> x
                  , plus(0(), y) -> y
                  , plus(s(x), y) -> s(plus(x, y))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {6}->{2}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(0()), y) -> y
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(x, 0()) -> x
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(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.
           
           * Path {6}->{3}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {6}->{3}->{1}.
           
           * Path {6}->{3}->{1}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(0()), y) -> y
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(x, 0()) -> x
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(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.
           
           * Path {6}->{3}->{2}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(0()), y) -> y
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(x, 0()) -> x
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(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.
           
           * Path {8,12,11,10}: inherited
             ----------------------------
             
             This path is subsumed by the proof of path {8,12,11,10}->{7}.
           
           * Path {8,12,11,10}->{7}: NA
             --------------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(0()), y) -> y
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(x, 0()) -> x
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(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.
           
           * Path {8,12,11,10}->{9}: NA
             --------------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(0()), y) -> y
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(x, 0()) -> x
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(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: plus^#(x, 0()) -> c_0()
              , 2: plus^#(0(), y) -> c_1()
              , 3: plus^#(s(x), y) -> c_2(plus^#(x, y))
              , 4: times^#(0(), y) -> c_3()
              , 5: times^#(s(0()), y) -> c_4()
              , 6: times^#(s(x), y) -> c_5(plus^#(y, times(x, y)))
              , 7: div^#(0(), y) -> c_6()
              , 8: div^#(x, y) -> c_7(quot^#(x, y, y))
              , 9: quot^#(0(), s(y), z) -> c_8()
              , 10: quot^#(s(x), s(y), z) -> c_9(quot^#(x, y, z))
              , 11: quot^#(x, 0(), s(z)) -> c_10(div^#(x, s(z)))
              , 12: div^#(div(x, y), z) -> c_11(div^#(x, times(y, z)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{8,12,11,10}                                              [     inherited      ]
                |
                |->{7}                                                   [         NA         ]
                |
                `->{9}                                                   [         NA         ]
             
             ->{6}                                                       [     inherited      ]
                |
                |->{1}                                                   [       MAYBE        ]
                |
                |->{2}                                                   [         NA         ]
                |
                `->{3}                                                   [     inherited      ]
                    |
                    |->{1}                                               [         NA         ]
                    |
                    `->{2}                                               [         NA         ]
             
             ->{5}                                                       [    YES(?,O(1))     ]
             
             ->{4}                                                       [    YES(?,O(1))     ]
             
           
         
         Sub-problems:
         -------------
           * Path {4}: 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(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(plus^#) = {},
                 Uargs(c_2) = {}, Uargs(times^#) = {}, Uargs(c_5) = {},
                 Uargs(div^#) = {}, Uargs(c_7) = {}, Uargs(quot^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                div(x1, x2) = [0] x1 + [0] x2 + [0]
                quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                div^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_8() = [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
                c_11(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: {times^#(0(), y) -> c_3()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(times^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [7]
                times^#(x1, x2) = [1] x1 + [0] x2 + [7]
                c_3() = [1]
           
           * Path {5}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(plus) = {}, Uargs(s) = {}, Uargs(times) = {},
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(plus^#) = {},
                 Uargs(c_2) = {}, Uargs(times^#) = {}, Uargs(c_5) = {},
                 Uargs(div^#) = {}, Uargs(c_7) = {}, Uargs(quot^#) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}, Uargs(c_11) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                plus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                times(x1, x2) = [0] x1 + [0] x2 + [0]
                div(x1, x2) = [0] x1 + [0] x2 + [0]
                quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                plus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                times^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                div^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_8() = [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
                c_11(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: {times^#(s(0()), y) -> c_4()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(times^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                s(x1) = [0] x1 + [2]
                times^#(x1, x2) = [2] x1 + [0] x2 + [7]
                c_4() = [0]
           
           * Path {6}: inherited
             -------------------
             
             This path is subsumed by the proof of path {6}->{3}->{1}.
           
           * Path {6}->{1}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(0()), y) -> y
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(x, 0()) -> x
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(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:
                 {  times^#(s(x), y) -> c_5(plus^#(y, times(x, y)))
                  , plus^#(x, 0()) -> c_0()
                  , times(0(), y) -> 0()
                  , times(s(0()), y) -> y
                  , times(s(x), y) -> plus(y, times(x, y))
                  , plus(x, 0()) -> x
                  , plus(0(), y) -> y
                  , plus(s(x), y) -> s(plus(x, y))}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {6}->{2}: NA
             -----------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(0()), y) -> y
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(x, 0()) -> x
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(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.
           
           * Path {6}->{3}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {6}->{3}->{1}.
           
           * Path {6}->{3}->{1}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(0()), y) -> y
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(x, 0()) -> x
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(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.
           
           * Path {6}->{3}->{2}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(0()), y) -> y
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(x, 0()) -> x
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(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.
           
           * Path {8,12,11,10}: inherited
             ----------------------------
             
             This path is subsumed by the proof of path {8,12,11,10}->{7}.
           
           * Path {8,12,11,10}->{7}: NA
             --------------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(0()), y) -> y
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(x, 0()) -> x
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(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.
           
           * Path {8,12,11,10}->{9}: NA
             --------------------------
             
             The usable rules for this path are:
             
               {  times(0(), y) -> 0()
                , times(s(0()), y) -> y
                , times(s(x), y) -> plus(y, times(x, y))
                , plus(x, 0()) -> x
                , plus(0(), y) -> y
                , plus(s(x), y) -> s(plus(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
InputAProVE 04 IJCAR 12

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
TIMEOUT
InputAProVE 04 IJCAR 12

stdout:

TIMEOUT

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           TIMEOUT
Input Problem:    runtime-complexity with respect to
  Rules:
    {  plus(x, 0()) -> x
     , plus(0(), y) -> y
     , plus(s(x), y) -> s(plus(x, y))
     , times(0(), y) -> 0()
     , times(s(0()), y) -> y
     , times(s(x), y) -> plus(y, times(x, y))
     , div(0(), y) -> 0()
     , div(x, y) -> quot(x, y, y)
     , quot(0(), s(y), z) -> 0()
     , quot(s(x), s(y), z) -> quot(x, y, z)
     , quot(x, 0(), s(z)) -> s(div(x, s(z)))
     , div(div(x, y), z) -> div(x, times(y, z))}

Proof Output:    
  Computation stopped due to timeout after 60.0 seconds