Problem AProVE 04 IJCAR 1

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputAProVE 04 IJCAR 1

stdout:

MAYBE

Problem:
 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)))

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputAProVE 04 IJCAR 1

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputAProVE 04 IJCAR 1

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  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)))}

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: div^#(0(), y) -> c_0()
              , 2: div^#(x, y) -> c_1(quot^#(x, y, y))
              , 3: quot^#(0(), s(y), z) -> c_2()
              , 4: quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))
              , 5: quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{2,5,4}                                                   [       MAYBE        ]
                |
                |->{1}                                                   [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^1))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {2,5,4}: MAYBE
             -------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
                 Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                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]
                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]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                div^#(x1, x2) = [0 0 0] x1 + [0 0 3] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                quot^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [3 3 3]      [3 3 3]      [3 3 3]      [0]
                                     [3 3 3]      [3 3 3]      [3 3 3]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules:
                 {  div^#(x, y) -> c_1(quot^#(x, y, y))
                  , quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))
                  , quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {2,5,4}->{1}: YES(?,O(n^1))
             --------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
                 Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                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]
                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]
                s(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_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [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_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {div^#(0(), y) -> c_0()}
               Weak Rules:
                 {  div^#(x, y) -> c_1(quot^#(x, y, y))
                  , quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))
                  , quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(div^#) = {}, Uargs(c_1) = {1},
                 Uargs(quot^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [0]
                      [2]
                s(x1) = [0 0 1] x1 + [0]
                        [0 0 1]      [0]
                        [0 0 1]      [0]
                div^#(x1, x2) = [0 0 2] x1 + [0 0 0] x2 + [0]
                                [4 4 4]      [0 0 4]      [4]
                                [0 0 0]      [2 0 0]      [0]
                c_0() = [1]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [3]
                          [0 0 0]      [0]
                quot^#(x1, x2, x3) = [0 0 2] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [4 3 1]      [0 0 0]      [1 3 4]      [0]
                                     [4 0 0]      [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 0 2]      [0]
                          [0 0 0]      [0]
           
           * Path {2,5,4}->{3}: YES(?,O(n^1))
             --------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
                 Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                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]
                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]
                s(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_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [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_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {quot^#(0(), s(y), z) -> c_2()}
               Weak Rules:
                 {  div^#(x, y) -> c_1(quot^#(x, y, y))
                  , quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))
                  , quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(div^#) = {}, Uargs(c_1) = {1},
                 Uargs(quot^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [0]
                      [0]
                s(x1) = [1 0 4] x1 + [0]
                        [0 0 1]      [0]
                        [0 0 0]      [0]
                div^#(x1, x2) = [2 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 2 0]      [0]
                                [4 4 4]      [0 0 0]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                quot^#(x1, x2, x3) = [2 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [2 0 4]      [0 0 0]      [2 0 0]      [0]
                                     [0 4 0]      [0 0 0]      [2 0 0]      [0]
                c_2() = [1]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 2 0]      [0]
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: div^#(0(), y) -> c_0()
              , 2: div^#(x, y) -> c_1(quot^#(x, y, y))
              , 3: quot^#(0(), s(y), z) -> c_2()
              , 4: quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))
              , 5: quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{2,5,4}                                                   [       MAYBE        ]
                |
                |->{1}                                                   [         NA         ]
                |
                `->{3}                                                   [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {2,5,4}: MAYBE
             -------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
                 Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [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]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
                div^#(x1, x2) = [3 3] x1 + [0 1] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                quot^#(x1, x2, x3) = [3 3] x1 + [0 0] x2 + [0 1] x3 + [0]
                                     [3 3]      [3 3]      [3 3]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules:
                 {  div^#(x, y) -> c_1(quot^#(x, y, y))
                  , quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))
                  , quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {2,5,4}->{1}: NA
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
                 Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [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]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {2,5,4}->{3}: NA
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
                 Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [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]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             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: div^#(0(), y) -> c_0()
              , 2: div^#(x, y) -> c_1(quot^#(x, y, y))
              , 3: quot^#(0(), s(y), z) -> c_2()
              , 4: quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))
              , 5: quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{2,5,4}                                                   [       MAYBE        ]
                |
                |->{1}                                                   [         NA         ]
                |
                `->{3}                                                   [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {2,5,4}: MAYBE
             -------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
                 Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                div(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                s(x1) = [1] x1 + [0]
                div^#(x1, x2) = [3] x1 + [2] x2 + [0]
                c_0() = [0]
                c_1(x1) = [1] x1 + [0]
                quot^#(x1, x2, x3) = [3] x1 + [0] x2 + [2] x3 + [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                c_4(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules:
                 {  div^#(x, y) -> c_1(quot^#(x, y, y))
                  , quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))
                  , quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {2,5,4}->{1}: NA
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
                 Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                div(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                s(x1) = [0] x1 + [0]
                div^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [1] x1 + [0]
                quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                c_4(x1) = [1] x1 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {2,5,4}->{3}: NA
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
                 Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                div(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                s(x1) = [0] x1 + [0]
                div^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [1] x1 + [0]
                quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                c_4(x1) = [1] x1 + [0]
             
             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 1

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputAProVE 04 IJCAR 1

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  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)))}

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: div^#(0(), y) -> c_0()
              , 2: div^#(x, y) -> c_1(quot^#(x, y, y))
              , 3: quot^#(0(), s(y), z) -> c_2()
              , 4: quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))
              , 5: quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{2,5,4}                                                   [       MAYBE        ]
                |
                |->{1}                                                   [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^1))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {2,5,4}: MAYBE
             -------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
                 Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                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]
                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]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                div^#(x1, x2) = [0 0 0] x1 + [0 0 3] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                quot^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [3 3 3]      [3 3 3]      [3 3 3]      [0]
                                     [3 3 3]      [3 3 3]      [3 3 3]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules:
                 {  div^#(x, y) -> c_1(quot^#(x, y, y))
                  , quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))
                  , quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {2,5,4}->{1}: YES(?,O(n^1))
             --------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
                 Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                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]
                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]
                s(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_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [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_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {div^#(0(), y) -> c_0()}
               Weak Rules:
                 {  div^#(x, y) -> c_1(quot^#(x, y, y))
                  , quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))
                  , quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(div^#) = {}, Uargs(c_1) = {1},
                 Uargs(quot^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [0]
                      [2]
                s(x1) = [0 0 1] x1 + [0]
                        [0 0 1]      [0]
                        [0 0 1]      [0]
                div^#(x1, x2) = [0 0 2] x1 + [0 0 0] x2 + [0]
                                [4 4 4]      [0 0 4]      [4]
                                [0 0 0]      [2 0 0]      [0]
                c_0() = [1]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [3]
                          [0 0 0]      [0]
                quot^#(x1, x2, x3) = [0 0 2] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [4 3 1]      [0 0 0]      [1 3 4]      [0]
                                     [4 0 0]      [0 0 0]      [0 0 0]      [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 0 2]      [0]
                          [0 0 0]      [0]
           
           * Path {2,5,4}->{3}: YES(?,O(n^1))
             --------------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
                 Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                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]
                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]
                s(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_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [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_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {quot^#(0(), s(y), z) -> c_2()}
               Weak Rules:
                 {  div^#(x, y) -> c_1(quot^#(x, y, y))
                  , quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))
                  , quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(div^#) = {}, Uargs(c_1) = {1},
                 Uargs(quot^#) = {}, Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [0]
                      [0]
                s(x1) = [1 0 4] x1 + [0]
                        [0 0 1]      [0]
                        [0 0 0]      [0]
                div^#(x1, x2) = [2 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 2 0]      [0]
                                [4 4 4]      [0 0 0]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                quot^#(x1, x2, x3) = [2 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [2 0 4]      [0 0 0]      [2 0 0]      [0]
                                     [0 4 0]      [0 0 0]      [2 0 0]      [0]
                c_2() = [1]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 2 0]      [0]
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: div^#(0(), y) -> c_0()
              , 2: div^#(x, y) -> c_1(quot^#(x, y, y))
              , 3: quot^#(0(), s(y), z) -> c_2()
              , 4: quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))
              , 5: quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{2,5,4}                                                   [       MAYBE        ]
                |
                |->{1}                                                   [         NA         ]
                |
                `->{3}                                                   [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {2,5,4}: MAYBE
             -------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
                 Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [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]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
                div^#(x1, x2) = [3 3] x1 + [0 1] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                quot^#(x1, x2, x3) = [3 3] x1 + [0 0] x2 + [0 1] x3 + [0]
                                     [3 3]      [3 3]      [3 3]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules:
                 {  div^#(x, y) -> c_1(quot^#(x, y, y))
                  , quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))
                  , quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {2,5,4}->{1}: NA
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
                 Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [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]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {2,5,4}->{3}: NA
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
                 Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                div(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [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]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                div^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             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: div^#(0(), y) -> c_0()
              , 2: div^#(x, y) -> c_1(quot^#(x, y, y))
              , 3: quot^#(0(), s(y), z) -> c_2()
              , 4: quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))
              , 5: quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{2,5,4}                                                   [       MAYBE        ]
                |
                |->{1}                                                   [         NA         ]
                |
                `->{3}                                                   [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {2,5,4}: MAYBE
             -------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
                 Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                div(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                s(x1) = [1] x1 + [0]
                div^#(x1, x2) = [3] x1 + [2] x2 + [0]
                c_0() = [0]
                c_1(x1) = [1] x1 + [0]
                quot^#(x1, x2, x3) = [3] x1 + [0] x2 + [2] x3 + [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                c_4(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules:
                 {  div^#(x, y) -> c_1(quot^#(x, y, y))
                  , quot^#(x, 0(), s(z)) -> c_4(div^#(x, s(z)))
                  , quot^#(s(x), s(y), z) -> c_3(quot^#(x, y, z))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {2,5,4}->{1}: NA
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
                 Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                div(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                s(x1) = [0] x1 + [0]
                div^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [1] x1 + [0]
                quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                c_4(x1) = [1] x1 + [0]
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {2,5,4}->{3}: NA
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(div) = {}, Uargs(quot) = {}, Uargs(s) = {},
                 Uargs(div^#) = {}, Uargs(c_1) = {1}, Uargs(quot^#) = {},
                 Uargs(c_3) = {1}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                div(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                s(x1) = [0] x1 + [0]
                div^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [1] x1 + [0]
                quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
                c_4(x1) = [1] x1 + [0]
             
             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.