Problem AProVE 10 ex1

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputAProVE 10 ex1

stdout:

MAYBE

Problem:
 f(x,y) -> cond(lt(x,y),x,y)
 cond(tt(),x,y) -> f(s(x),s(y))
 lt(0(),y) -> tt()
 lt(s(x),s(y)) -> lt(x,y)

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputAProVE 10 ex1

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputAProVE 10 ex1

stdout:

MAYBE

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

Proof Output:    
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
              , 2: cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
              , 3: lt^#(0(), y) -> c_2()
              , 4: lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^2))    ]
             
             ->{1,2}                                                     [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1,2}: MAYBE
             -----------------
             
             The usable rules for this path are:
             
               {  lt(0(), y) -> tt()
                , lt(s(x), s(y)) -> lt(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:
                 {  f^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
                  , cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
                  , lt(0(), y) -> tt()
                  , lt(s(x), s(y)) -> lt(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {4}: 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(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                cond(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]
                lt(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cond^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                lt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [3 3 3]      [3 3 3]      [0]
                               [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]
             
             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: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 0] x1 + [2]
                        [0 0 2]      [2]
                        [0 0 0]      [0]
                lt^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
                               [2 2 0]      [0 2 0]      [0]
                               [4 0 0]      [0 2 0]      [0]
                c_3(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [7]
           
           * Path {4}->{3}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                cond(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]
                lt(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cond^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                lt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [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^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {lt^#(0(), y) -> c_2()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 4 2] x1 + [0]
                        [0 0 2]      [0]
                        [0 0 1]      [0]
                0() = [2]
                      [2]
                      [2]
                lt^#(x1, x2) = [2 2 2] x1 + [2 0 0] x2 + [0]
                               [2 2 2]      [0 0 4]      [0]
                               [2 2 2]      [0 0 0]      [0]
                c_2() = [1]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
              , 2: cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
              , 3: lt^#(0(), y) -> c_2()
              , 4: lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{4}                                                       [   YES(?,O(n^2))    ]
                |
                `->{3}                                                   [   YES(?,O(n^2))    ]
             
             ->{1,2}                                                     [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1,2}: MAYBE
             -----------------
             
             The usable rules for this path are:
             
               {  lt(0(), y) -> tt()
                , lt(s(x), s(y)) -> lt(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:
                 {  f^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
                  , cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
                  , lt(0(), y) -> tt()
                  , lt(s(x), s(y)) -> lt(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {4}: YES(?,O(n^2))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                lt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                               [3 3]      [3 3]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                lt^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                               [0 2]      [0 0]      [0]
                c_3(x1) = [1 2] x1 + [5]
                          [0 0]      [3]
           
           * Path {4}->{3}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {lt^#(0(), y) -> c_2()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [0]
                0() = [2]
                      [2]
                lt^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
                               [4 1]      [2 0]      [0]
                c_2() = [1]
                        [0]
                c_3(x1) = [1 0] x1 + [3]
                          [0 0]      [7]
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
              , 2: cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
              , 3: lt^#(0(), y) -> c_2()
              , 4: lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^1))    ]
             
             ->{1,2}                                                     [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1,2}: MAYBE
             -----------------
             
             The usable rules for this path are:
             
               {  lt(0(), y) -> tt()
                , lt(s(x), s(y)) -> lt(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:
                 {  f^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
                  , cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
                  , lt(0(), y) -> tt()
                  , lt(s(x), s(y)) -> lt(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {4}: 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(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                f^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1(x1) = [0] x1 + [0]
                lt^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                lt^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_3(x1) = [1] x1 + [7]
           
           * Path {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(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                f^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1(x1) = [0] x1 + [0]
                lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {lt^#(0(), y) -> c_2()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                lt^#(x1, x2) = [2] x1 + [0] x2 + [4]
                c_2() = [1]
                c_3(x1) = [1] x1 + [2]
    
    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 10 ex1

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputAProVE 10 ex1

stdout:

MAYBE

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

Proof Output:    
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
              , 2: cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
              , 3: lt^#(0(), y) -> c_2()
              , 4: lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^2))    ]
             
             ->{1,2}                                                     [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1,2}: MAYBE
             -----------------
             
             The usable rules for this path are:
             
               {  lt(0(), y) -> tt()
                , lt(s(x), s(y)) -> lt(x, y)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  f^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
                  , cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
                  , lt(0(), y) -> tt()
                  , lt(s(x), s(y)) -> lt(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {4}: 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(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                cond(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]
                lt(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cond^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                lt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [3 3 3]      [3 3 3]      [0]
                               [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]
             
             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: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 0] x1 + [2]
                        [0 0 2]      [2]
                        [0 0 0]      [0]
                lt^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
                               [2 2 0]      [0 2 0]      [0]
                               [4 0 0]      [0 2 0]      [0]
                c_3(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [7]
           
           * Path {4}->{3}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                            [0 0 0]      [0 0 0]      [0]
                            [0 0 0]      [0 0 0]      [0]
                cond(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]
                lt(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]
                tt() = [0]
                       [0]
                       [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                f^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cond^#(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_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                lt^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                               [0 0 0]      [0 0 0]      [0]
                               [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3(x1) = [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^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {lt^#(0(), y) -> c_2()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 4 2] x1 + [0]
                        [0 0 2]      [0]
                        [0 0 1]      [0]
                0() = [2]
                      [2]
                      [2]
                lt^#(x1, x2) = [2 2 2] x1 + [2 0 0] x2 + [0]
                               [2 2 2]      [0 0 4]      [0]
                               [2 2 2]      [0 0 0]      [0]
                c_2() = [1]
                        [0]
                        [0]
                c_3(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
              , 2: cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
              , 3: lt^#(0(), y) -> c_2()
              , 4: lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{4}                                                       [   YES(?,O(n^2))    ]
                |
                `->{3}                                                   [   YES(?,O(n^2))    ]
             
             ->{1,2}                                                     [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1,2}: MAYBE
             -----------------
             
             The usable rules for this path are:
             
               {  lt(0(), y) -> tt()
                , lt(s(x), s(y)) -> lt(x, y)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  f^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
                  , cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
                  , lt(0(), y) -> tt()
                  , lt(s(x), s(y)) -> lt(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {4}: YES(?,O(n^2))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                lt^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                               [3 3]      [3 3]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                lt^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                               [0 2]      [0 0]      [0]
                c_3(x1) = [1 2] x1 + [5]
                          [0 0]      [3]
           
           * Path {4}->{3}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                lt(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                f^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                lt^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {lt^#(0(), y) -> c_2()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [0]
                0() = [2]
                      [2]
                lt^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
                               [4 1]      [2 0]      [0]
                c_2() = [1]
                        [0]
                c_3(x1) = [1 0] x1 + [3]
                          [0 0]      [7]
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
              , 2: cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
              , 3: lt^#(0(), y) -> c_2()
              , 4: lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^1))    ]
             
             ->{1,2}                                                     [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1,2}: MAYBE
             -----------------
             
             The usable rules for this path are:
             
               {  lt(0(), y) -> tt()
                , lt(s(x), s(y)) -> lt(x, y)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  f^#(x, y) -> c_0(cond^#(lt(x, y), x, y))
                  , cond^#(tt(), x, y) -> c_1(f^#(s(x), s(y)))
                  , lt(0(), y) -> tt()
                  , lt(s(x), s(y)) -> lt(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {4}: 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(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                f^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1(x1) = [0] x1 + [0]
                lt^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                lt^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_3(x1) = [1] x1 + [7]
           
           * Path {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(f) = {}, Uargs(cond) = {}, Uargs(lt) = {}, Uargs(s) = {},
                 Uargs(f^#) = {}, Uargs(c_0) = {}, Uargs(cond^#) = {},
                 Uargs(c_1) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                f^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1(x1) = [0] x1 + [0]
                lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {lt^#(0(), y) -> c_2()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_3(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                lt^#(x1, x2) = [2] x1 + [0] x2 + [4]
                c_2() = [1]
                c_3(x1) = [1] x1 + [2]
    
    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.