Problem AProVE 10 isNat

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputAProVE 10 isNat

stdout:

MAYBE

Problem:
 f(tt(),x) -> f(isNat(x),s(x))
 isNat(s(x)) -> isNat(x)
 isNat(0()) -> tt()

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputAProVE 10 isNat

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputAProVE 10 isNat

stdout:

MAYBE

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

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^#(tt(), x) -> c_0(f^#(isNat(x), s(x)))
              , 2: isNat^#(s(x)) -> c_1(isNat^#(x))
              , 3: isNat^#(0()) -> c_2()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{2}                                                       [   YES(?,O(n^2))    ]
                |
                `->{3}                                                   [   YES(?,O(n^2))    ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules for this path are:
             
               {  isNat(s(x)) -> isNat(x)
                , isNat(0()) -> tt()}
             
             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^#(tt(), x) -> c_0(f^#(isNat(x), s(x)))
                  , isNat(s(x)) -> isNat(x)
                  , isNat(0()) -> tt()}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {2}: 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(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {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]
                tt() = [0]
                       [0]
                       [0]
                isNat(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 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]
                isNat^#(x1) = [0 0 0] x1 + [0]
                              [3 3 3]      [0]
                              [3 3 3]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_2() = [0]
                        [0]
                        [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: {isNat^#(s(x)) -> c_1(isNat^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 2] x1 + [2]
                        [0 1 2]      [2]
                        [0 0 0]      [0]
                isNat^#(x1) = [0 1 0] x1 + [2]
                              [6 0 0]      [0]
                              [2 3 0]      [2]
                c_1(x1) = [1 0 0] x1 + [1]
                          [2 0 2]      [0]
                          [0 0 0]      [0]
           
           * Path {2}->{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(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {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]
                tt() = [0]
                       [0]
                       [0]
                isNat(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 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]
                isNat^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_2() = [0]
                        [0]
                        [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: {isNat^#(0()) -> c_2()}
               Weak Rules: {isNat^#(s(x)) -> c_1(isNat^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 0] x1 + [0]
                        [0 1 1]      [1]
                        [0 0 0]      [0]
                0() = [2]
                      [2]
                      [2]
                isNat^#(x1) = [2 2 2] x1 + [0]
                              [0 6 0]      [0]
                              [0 0 2]      [0]
                c_1(x1) = [1 0 0] x1 + [2]
                          [0 0 0]      [3]
                          [0 0 0]      [0]
                c_2() = [1]
                        [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^#(tt(), x) -> c_0(f^#(isNat(x), s(x)))
              , 2: isNat^#(s(x)) -> c_1(isNat^#(x))
              , 3: isNat^#(0()) -> c_2()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{2}                                                       [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^1))    ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules for this path are:
             
               {  isNat(s(x)) -> isNat(x)
                , isNat(0()) -> tt()}
             
             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^#(tt(), x) -> c_0(f^#(isNat(x), s(x)))
                  , isNat(s(x)) -> isNat(x)
                  , isNat(0()) -> tt()}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {2}: 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(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {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]
                tt() = [0]
                       [0]
                isNat(x1) = [0 0] x1 + [0]
                            [0 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]
                isNat^#(x1) = [3 3] x1 + [0]
                              [3 3]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_2() = [0]
                        [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {isNat^#(s(x)) -> c_1(isNat^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [1]
                isNat^#(x1) = [0 1] x1 + [1]
                              [0 0]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
           
           * Path {2}->{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(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {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]
                tt() = [0]
                       [0]
                isNat(x1) = [0 0] x1 + [0]
                            [0 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]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_2() = [0]
                        [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {isNat^#(0()) -> c_2()}
               Weak Rules: {isNat^#(s(x)) -> c_1(isNat^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [3]
                0() = [2]
                      [2]
                isNat^#(x1) = [1 2] x1 + [2]
                              [6 1]      [0]
                c_1(x1) = [1 0] x1 + [5]
                          [2 0]      [3]
                c_2() = [1]
                        [0]
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(tt(), x) -> c_0(f^#(isNat(x), s(x)))
              , 2: isNat^#(s(x)) -> c_1(isNat^#(x))
              , 3: isNat^#(0()) -> c_2()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{2}                                                       [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^1))    ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules for this path are:
             
               {  isNat(s(x)) -> isNat(x)
                , isNat(0()) -> tt()}
             
             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^#(tt(), x) -> c_0(f^#(isNat(x), s(x)))
                  , isNat(s(x)) -> isNat(x)
                  , isNat(0()) -> tt()}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {2}: 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(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                isNat(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                f^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                isNat^#(x1) = [3] x1 + [0]
                c_1(x1) = [1] x1 + [0]
                c_2() = [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: {isNat^#(s(x)) -> c_1(isNat^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [4]
                isNat^#(x1) = [2] x1 + [0]
                c_1(x1) = [1] x1 + [7]
           
           * Path {2}->{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(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                isNat(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                f^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                isNat^#(x1) = [0] x1 + [0]
                c_1(x1) = [1] x1 + [0]
                c_2() = [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: {isNat^#(0()) -> c_2()}
               Weak Rules: {isNat^#(s(x)) -> c_1(isNat^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                0() = [2]
                isNat^#(x1) = [2] x1 + [0]
                c_1(x1) = [1] x1 + [0]
                c_2() = [1]
    
    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 isNat

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputAProVE 10 isNat

stdout:

MAYBE

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

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^#(tt(), x) -> c_0(f^#(isNat(x), s(x)))
              , 2: isNat^#(s(x)) -> c_1(isNat^#(x))
              , 3: isNat^#(0()) -> c_2()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{2}                                                       [   YES(?,O(n^2))    ]
                |
                `->{3}                                                   [   YES(?,O(n^2))    ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules for this path are:
             
               {  isNat(s(x)) -> isNat(x)
                , isNat(0()) -> tt()}
             
             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^#(tt(), x) -> c_0(f^#(isNat(x), s(x)))
                  , isNat(s(x)) -> isNat(x)
                  , isNat(0()) -> tt()}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {2}: 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(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {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]
                tt() = [0]
                       [0]
                       [0]
                isNat(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 0 0]      [0]
                s(x1) = [1 3 0] x1 + [0]
                        [0 1 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]
                isNat^#(x1) = [0 0 0] x1 + [0]
                              [3 3 3]      [0]
                              [3 3 3]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_2() = [0]
                        [0]
                        [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: {isNat^#(s(x)) -> c_1(isNat^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 2] x1 + [2]
                        [0 1 2]      [2]
                        [0 0 0]      [0]
                isNat^#(x1) = [0 1 0] x1 + [2]
                              [6 0 0]      [0]
                              [2 3 0]      [2]
                c_1(x1) = [1 0 0] x1 + [1]
                          [2 0 2]      [0]
                          [0 0 0]      [0]
           
           * Path {2}->{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(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {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]
                tt() = [0]
                       [0]
                       [0]
                isNat(x1) = [0 0 0] x1 + [0]
                            [0 0 0]      [0]
                            [0 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]
                isNat^#(x1) = [0 0 0] x1 + [0]
                              [0 0 0]      [0]
                              [0 0 0]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_2() = [0]
                        [0]
                        [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: {isNat^#(0()) -> c_2()}
               Weak Rules: {isNat^#(s(x)) -> c_1(isNat^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 0] x1 + [0]
                        [0 1 1]      [1]
                        [0 0 0]      [0]
                0() = [2]
                      [2]
                      [2]
                isNat^#(x1) = [2 2 2] x1 + [0]
                              [0 6 0]      [0]
                              [0 0 2]      [0]
                c_1(x1) = [1 0 0] x1 + [2]
                          [0 0 0]      [3]
                          [0 0 0]      [0]
                c_2() = [1]
                        [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^#(tt(), x) -> c_0(f^#(isNat(x), s(x)))
              , 2: isNat^#(s(x)) -> c_1(isNat^#(x))
              , 3: isNat^#(0()) -> c_2()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{2}                                                       [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^1))    ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules for this path are:
             
               {  isNat(s(x)) -> isNat(x)
                , isNat(0()) -> tt()}
             
             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^#(tt(), x) -> c_0(f^#(isNat(x), s(x)))
                  , isNat(s(x)) -> isNat(x)
                  , isNat(0()) -> tt()}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {2}: 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(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {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]
                tt() = [0]
                       [0]
                isNat(x1) = [0 0] x1 + [0]
                            [0 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]
                isNat^#(x1) = [3 3] x1 + [0]
                              [3 3]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_2() = [0]
                        [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {isNat^#(s(x)) -> c_1(isNat^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [1]
                isNat^#(x1) = [0 1] x1 + [1]
                              [0 0]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
           
           * Path {2}->{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(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {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]
                tt() = [0]
                       [0]
                isNat(x1) = [0 0] x1 + [0]
                            [0 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]
                isNat^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_2() = [0]
                        [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {isNat^#(0()) -> c_2()}
               Weak Rules: {isNat^#(s(x)) -> c_1(isNat^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [3]
                0() = [2]
                      [2]
                isNat^#(x1) = [1 2] x1 + [2]
                              [6 1]      [0]
                c_1(x1) = [1 0] x1 + [5]
                          [2 0]      [3]
                c_2() = [1]
                        [0]
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: f^#(tt(), x) -> c_0(f^#(isNat(x), s(x)))
              , 2: isNat^#(s(x)) -> c_1(isNat^#(x))
              , 3: isNat^#(0()) -> c_2()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{2}                                                       [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^1))    ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules for this path are:
             
               {  isNat(s(x)) -> isNat(x)
                , isNat(0()) -> tt()}
             
             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^#(tt(), x) -> c_0(f^#(isNat(x), s(x)))
                  , isNat(s(x)) -> isNat(x)
                  , isNat(0()) -> tt()}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {2}: 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(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                isNat(x1) = [0] x1 + [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                f^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                isNat^#(x1) = [3] x1 + [0]
                c_1(x1) = [1] x1 + [0]
                c_2() = [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: {isNat^#(s(x)) -> c_1(isNat^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [4]
                isNat^#(x1) = [2] x1 + [0]
                c_1(x1) = [1] x1 + [7]
           
           * Path {2}->{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(isNat) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                isNat(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                f^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                isNat^#(x1) = [0] x1 + [0]
                c_1(x1) = [1] x1 + [0]
                c_2() = [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: {isNat^#(0()) -> c_2()}
               Weak Rules: {isNat^#(s(x)) -> c_1(isNat^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(isNat^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                0() = [2]
                isNat^#(x1) = [2] x1 + [0]
                c_1(x1) = [1] x1 + [0]
                c_2() = [1]
    
    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.