Problem AProVE 10 halfdouble

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputAProVE 10 halfdouble

stdout:

MAYBE

Problem:
 f(tt(),x) -> f(eq(x,half(double(x))),s(x))
 eq(s(x),s(y)) -> eq(x,y)
 eq(0(),0()) -> tt()
 double(s(x)) -> s(s(double(x)))
 double(0()) -> 0()
 half(s(s(x))) -> s(half(x))
 half(0()) -> 0()

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputAProVE 10 halfdouble

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputAProVE 10 halfdouble

stdout:

MAYBE

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

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^#(eq(x, half(double(x))), s(x)))
              , 2: eq^#(s(x), s(y)) -> c_1(eq^#(x, y))
              , 3: eq^#(0(), 0()) -> c_2()
              , 4: double^#(s(x)) -> c_3(double^#(x))
              , 5: double^#(0()) -> c_4()
              , 6: half^#(s(s(x))) -> c_5(half^#(x))
              , 7: half^#(0()) -> c_6()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6}                                                       [   YES(?,O(n^1))    ]
                |
                `->{7}                                                   [   YES(?,O(n^1))    ]
             
             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                `->{5}                                                   [   YES(?,O(n^1))    ]
             
             ->{2}                                                       [   YES(?,O(n^2))    ]
                |
                `->{3}                                                   [   YES(?,O(n^1))    ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules for this path are:
             
               {  eq(s(x), s(y)) -> eq(x, y)
                , eq(0(), 0()) -> tt()
                , double(s(x)) -> s(s(double(x)))
                , double(0()) -> 0()
                , half(s(s(x))) -> s(half(x))
                , half(0()) -> 0()}
             
             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^#(eq(x, half(double(x))), s(x)))
                  , eq(s(x), s(y)) -> eq(x, y)
                  , eq(0(), 0()) -> tt()
                  , double(s(x)) -> s(s(double(x)))
                  , double(0()) -> 0()
                  , half(s(s(x))) -> s(half(x))
                  , half(0()) -> 0()}
             
             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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
                 Uargs(c_5) = {}
               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]
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                double(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]
                eq^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                               [3 3]      [3 3]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_2() = [0]
                        [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6() = [0]
                        [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: {eq^#(s(x), s(y)) -> c_1(eq^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                eq^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                               [0 2]      [0 0]      [0]
                c_1(x1) = [1 2] x1 + [5]
                          [0 0]      [3]
           
           * 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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
                 Uargs(c_5) = {}
               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]
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                double(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]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_2() = [0]
                        [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6() = [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: {eq^#(0(), 0()) -> c_2()}
               Weak Rules: {eq^#(s(x), s(y)) -> c_1(eq^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [2]
                        [0 1]      [0]
                0() = [2]
                      [2]
                eq^#(x1, x2) = [2 2] x1 + [2 0] x2 + [0]
                               [2 0]      [0 0]      [4]
                c_1(x1) = [1 0] x1 + [7]
                          [0 0]      [7]
                c_2() = [1]
                        [0]
           
           * 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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(half^#) = {},
                 Uargs(c_5) = {}
               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]
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                double(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]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                double^#(x1) = [3 3] x1 + [0]
                               [3 3]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_4() = [0]
                        [0]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6() = [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: {double^#(s(x)) -> c_3(double^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [1]
                double^#(x1) = [0 1] x1 + [1]
                               [0 0]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
           
           * Path {4}->{5}: 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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(half^#) = {},
                 Uargs(c_5) = {}
               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]
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                double(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]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_4() = [0]
                        [0]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6() = [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: {double^#(0()) -> c_4()}
               Weak Rules: {double^#(s(x)) -> c_3(double^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [3]
                0() = [2]
                      [2]
                double^#(x1) = [1 2] x1 + [2]
                               [6 1]      [0]
                c_3(x1) = [1 0] x1 + [5]
                          [2 0]      [3]
                c_4() = [1]
                        [0]
           
           * Path {6}: 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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
                 Uargs(c_5) = {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]
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                s(x1) = [1 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]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                half^#(x1) = [0 0] x1 + [0]
                             [3 3]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_6() = [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: {half^#(s(s(x))) -> c_5(half^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [1]
                half^#(x1) = [2 2] x1 + [2]
                             [6 0]      [0]
                c_5(x1) = [1 0] x1 + [5]
                          [2 0]      [7]
           
           * Path {6}->{7}: 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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
                 Uargs(c_5) = {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]
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                double(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]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_6() = [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: {half^#(0()) -> c_6()}
               Weak Rules: {half^#(s(s(x))) -> c_5(half^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [0]
                0() = [0]
                      [2]
                half^#(x1) = [2 2] x1 + [4]
                             [6 2]      [0]
                c_5(x1) = [1 0] x1 + [3]
                          [2 0]      [3]
                c_6() = [1]
                        [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^#(eq(x, half(double(x))), s(x)))
              , 2: eq^#(s(x), s(y)) -> c_1(eq^#(x, y))
              , 3: eq^#(0(), 0()) -> c_2()
              , 4: double^#(s(x)) -> c_3(double^#(x))
              , 5: double^#(0()) -> c_4()
              , 6: half^#(s(s(x))) -> c_5(half^#(x))
              , 7: half^#(0()) -> c_6()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6}                                                       [   YES(?,O(n^1))    ]
                |
                `->{7}                                                   [   YES(?,O(n^1))    ]
             
             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                `->{5}                                                   [   YES(?,O(n^1))    ]
             
             ->{2}                                                       [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^1))    ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules for this path are:
             
               {  eq(s(x), s(y)) -> eq(x, y)
                , eq(0(), 0()) -> tt()
                , double(s(x)) -> s(s(double(x)))
                , double(0()) -> 0()
                , half(s(s(x))) -> s(half(x))
                , half(0()) -> 0()}
             
             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^#(eq(x, half(double(x))), s(x)))
                  , eq(s(x), s(y)) -> eq(x, y)
                  , eq(0(), 0()) -> tt()
                  , double(s(x)) -> s(s(double(x)))
                  , double(0()) -> 0()
                  , half(s(s(x))) -> s(half(x))
                  , half(0()) -> 0()}
             
             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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                half(x1) = [0] x1 + [0]
                double(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]
                eq^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_1(x1) = [1] x1 + [0]
                c_2() = [0]
                double^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4() = [0]
                half^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6() = [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: {eq^#(s(x), s(y)) -> c_1(eq^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                eq^#(x1, x2) = [2] x1 + [2] x2 + [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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                half(x1) = [0] x1 + [0]
                double(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]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [1] x1 + [0]
                c_2() = [0]
                double^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4() = [0]
                half^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6() = [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: {eq^#(0(), 0()) -> c_2()}
               Weak Rules: {eq^#(s(x), s(y)) -> c_1(eq^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [0]
                eq^#(x1, x2) = [2] x1 + [0] x2 + [4]
                c_1(x1) = [1] x1 + [3]
                c_2() = [1]
           
           * 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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(half^#) = {},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                half(x1) = [0] x1 + [0]
                double(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]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2() = [0]
                double^#(x1) = [3] x1 + [0]
                c_3(x1) = [1] x1 + [0]
                c_4() = [0]
                half^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6() = [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: {double^#(s(x)) -> c_3(double^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [4]
                double^#(x1) = [2] x1 + [0]
                c_3(x1) = [1] x1 + [7]
           
           * Path {4}->{5}: 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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(half^#) = {},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                half(x1) = [0] x1 + [0]
                double(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]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2() = [0]
                double^#(x1) = [0] x1 + [0]
                c_3(x1) = [1] x1 + [0]
                c_4() = [0]
                half^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6() = [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: {double^#(0()) -> c_4()}
               Weak Rules: {double^#(s(x)) -> c_3(double^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                0() = [2]
                double^#(x1) = [2] x1 + [0]
                c_3(x1) = [1] x1 + [0]
                c_4() = [1]
           
           * Path {6}: 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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
                 Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                half(x1) = [0] x1 + [0]
                double(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]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2() = [0]
                double^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4() = [0]
                half^#(x1) = [3] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                c_6() = [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: {half^#(s(s(x))) -> c_5(half^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                half^#(x1) = [2] x1 + [0]
                c_5(x1) = [1] x1 + [7]
           
           * Path {6}->{7}: 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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
                 Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                half(x1) = [0] x1 + [0]
                double(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]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2() = [0]
                double^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4() = [0]
                half^#(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                c_6() = [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: {half^#(0()) -> c_6()}
               Weak Rules: {half^#(s(s(x))) -> c_5(half^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                0() = [2]
                half^#(x1) = [2] x1 + [4]
                c_5(x1) = [1] x1 + [0]
                c_6() = [1]
    
    3) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    5) '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 halfdouble

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputAProVE 10 halfdouble

stdout:

MAYBE

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

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^#(eq(x, half(double(x))), s(x)))
              , 2: eq^#(s(x), s(y)) -> c_1(eq^#(x, y))
              , 3: eq^#(0(), 0()) -> c_2()
              , 4: double^#(s(x)) -> c_3(double^#(x))
              , 5: double^#(0()) -> c_4()
              , 6: half^#(s(s(x))) -> c_5(half^#(x))
              , 7: half^#(0()) -> c_6()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6}                                                       [   YES(?,O(n^1))    ]
                |
                `->{7}                                                   [   YES(?,O(n^1))    ]
             
             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                `->{5}                                                   [   YES(?,O(n^1))    ]
             
             ->{2}                                                       [   YES(?,O(n^2))    ]
                |
                `->{3}                                                   [   YES(?,O(n^1))    ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules for this path are:
             
               {  eq(s(x), s(y)) -> eq(x, y)
                , eq(0(), 0()) -> tt()
                , double(s(x)) -> s(s(double(x)))
                , double(0()) -> 0()
                , half(s(s(x))) -> s(half(x))
                , half(0()) -> 0()}
             
             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^#(eq(x, half(double(x))), s(x)))
                  , eq(s(x), s(y)) -> eq(x, y)
                  , eq(0(), 0()) -> tt()
                  , double(s(x)) -> s(s(double(x)))
                  , double(0()) -> 0()
                  , half(s(s(x))) -> s(half(x))
                  , half(0()) -> 0()}
             
             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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
                 Uargs(c_5) = {}
               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]
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                double(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]
                eq^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                               [3 3]      [3 3]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_2() = [0]
                        [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6() = [0]
                        [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: {eq^#(s(x), s(y)) -> c_1(eq^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                eq^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                               [0 2]      [0 0]      [0]
                c_1(x1) = [1 2] x1 + [5]
                          [0 0]      [3]
           
           * 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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
                 Uargs(c_5) = {}
               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]
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                double(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]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_2() = [0]
                        [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6() = [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: {eq^#(0(), 0()) -> c_2()}
               Weak Rules: {eq^#(s(x), s(y)) -> c_1(eq^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [2]
                        [0 1]      [0]
                0() = [2]
                      [2]
                eq^#(x1, x2) = [2 2] x1 + [2 0] x2 + [0]
                               [2 0]      [0 0]      [4]
                c_1(x1) = [1 0] x1 + [7]
                          [0 0]      [7]
                c_2() = [1]
                        [0]
           
           * 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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(half^#) = {},
                 Uargs(c_5) = {}
               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]
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                double(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]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                double^#(x1) = [3 3] x1 + [0]
                               [3 3]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_4() = [0]
                        [0]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6() = [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: {double^#(s(x)) -> c_3(double^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [1]
                double^#(x1) = [0 1] x1 + [1]
                               [0 0]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
           
           * Path {4}->{5}: 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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(half^#) = {},
                 Uargs(c_5) = {}
               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]
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                double(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]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_4() = [0]
                        [0]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6() = [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: {double^#(0()) -> c_4()}
               Weak Rules: {double^#(s(x)) -> c_3(double^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [3]
                0() = [2]
                      [2]
                double^#(x1) = [1 2] x1 + [2]
                               [6 1]      [0]
                c_3(x1) = [1 0] x1 + [5]
                          [2 0]      [3]
                c_4() = [1]
                        [0]
           
           * Path {6}: 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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
                 Uargs(c_5) = {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]
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                double(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                s(x1) = [1 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]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                half^#(x1) = [0 0] x1 + [0]
                             [3 3]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_6() = [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: {half^#(s(s(x))) -> c_5(half^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [1]
                half^#(x1) = [2 2] x1 + [2]
                             [6 0]      [0]
                c_5(x1) = [1 0] x1 + [5]
                          [2 0]      [7]
           
           * Path {6}->{7}: 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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
                 Uargs(c_5) = {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]
                eq(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                half(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                double(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]
                eq^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_2() = [0]
                        [0]
                double^#(x1) = [0 0] x1 + [0]
                               [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4() = [0]
                        [0]
                half^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_6() = [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: {half^#(0()) -> c_6()}
               Weak Rules: {half^#(s(s(x))) -> c_5(half^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 0]      [0]
                0() = [0]
                      [2]
                half^#(x1) = [2 2] x1 + [4]
                             [6 2]      [0]
                c_5(x1) = [1 0] x1 + [3]
                          [2 0]      [3]
                c_6() = [1]
                        [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^#(eq(x, half(double(x))), s(x)))
              , 2: eq^#(s(x), s(y)) -> c_1(eq^#(x, y))
              , 3: eq^#(0(), 0()) -> c_2()
              , 4: double^#(s(x)) -> c_3(double^#(x))
              , 5: double^#(0()) -> c_4()
              , 6: half^#(s(s(x))) -> c_5(half^#(x))
              , 7: half^#(0()) -> c_6()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6}                                                       [   YES(?,O(n^1))    ]
                |
                `->{7}                                                   [   YES(?,O(n^1))    ]
             
             ->{4}                                                       [   YES(?,O(n^1))    ]
                |
                `->{5}                                                   [   YES(?,O(n^1))    ]
             
             ->{2}                                                       [   YES(?,O(n^1))    ]
                |
                `->{3}                                                   [   YES(?,O(n^1))    ]
             
             ->{1}                                                       [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: MAYBE
             ---------------
             
             The usable rules for this path are:
             
               {  eq(s(x), s(y)) -> eq(x, y)
                , eq(0(), 0()) -> tt()
                , double(s(x)) -> s(s(double(x)))
                , double(0()) -> 0()
                , half(s(s(x))) -> s(half(x))
                , half(0()) -> 0()}
             
             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^#(eq(x, half(double(x))), s(x)))
                  , eq(s(x), s(y)) -> eq(x, y)
                  , eq(0(), 0()) -> tt()
                  , double(s(x)) -> s(s(double(x)))
                  , double(0()) -> 0()
                  , half(s(s(x))) -> s(half(x))
                  , half(0()) -> 0()}
             
             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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                half(x1) = [0] x1 + [0]
                double(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]
                eq^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_1(x1) = [1] x1 + [0]
                c_2() = [0]
                double^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4() = [0]
                half^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6() = [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: {eq^#(s(x), s(y)) -> c_1(eq^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                eq^#(x1, x2) = [2] x1 + [2] x2 + [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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                half(x1) = [0] x1 + [0]
                double(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]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [1] x1 + [0]
                c_2() = [0]
                double^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4() = [0]
                half^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6() = [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: {eq^#(0(), 0()) -> c_2()}
               Weak Rules: {eq^#(s(x), s(y)) -> c_1(eq^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [0]
                eq^#(x1, x2) = [2] x1 + [0] x2 + [4]
                c_1(x1) = [1] x1 + [3]
                c_2() = [1]
           
           * 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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(half^#) = {},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                half(x1) = [0] x1 + [0]
                double(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]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2() = [0]
                double^#(x1) = [3] x1 + [0]
                c_3(x1) = [1] x1 + [0]
                c_4() = [0]
                half^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6() = [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: {double^#(s(x)) -> c_3(double^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [4]
                double^#(x1) = [2] x1 + [0]
                c_3(x1) = [1] x1 + [7]
           
           * Path {4}->{5}: 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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {1}, Uargs(half^#) = {},
                 Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                half(x1) = [0] x1 + [0]
                double(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]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2() = [0]
                double^#(x1) = [0] x1 + [0]
                c_3(x1) = [1] x1 + [0]
                c_4() = [0]
                half^#(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6() = [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: {double^#(0()) -> c_4()}
               Weak Rules: {double^#(s(x)) -> c_3(double^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(double^#) = {}, Uargs(c_3) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                0() = [2]
                double^#(x1) = [2] x1 + [0]
                c_3(x1) = [1] x1 + [0]
                c_4() = [1]
           
           * Path {6}: 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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
                 Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                half(x1) = [0] x1 + [0]
                double(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]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2() = [0]
                double^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4() = [0]
                half^#(x1) = [3] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                c_6() = [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: {half^#(s(s(x))) -> c_5(half^#(x))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                half^#(x1) = [2] x1 + [0]
                c_5(x1) = [1] x1 + [7]
           
           * Path {6}->{7}: 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(eq) = {}, Uargs(half) = {},
                 Uargs(double) = {}, Uargs(s) = {}, Uargs(f^#) = {},
                 Uargs(c_0) = {}, Uargs(eq^#) = {}, Uargs(c_1) = {},
                 Uargs(double^#) = {}, Uargs(c_3) = {}, Uargs(half^#) = {},
                 Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                f(x1, x2) = [0] x1 + [0] x2 + [0]
                tt() = [0]
                eq(x1, x2) = [0] x1 + [0] x2 + [0]
                half(x1) = [0] x1 + [0]
                double(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]
                eq^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_1(x1) = [0] x1 + [0]
                c_2() = [0]
                double^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4() = [0]
                half^#(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                c_6() = [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: {half^#(0()) -> c_6()}
               Weak Rules: {half^#(s(s(x))) -> c_5(half^#(x))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(half^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [0]
                0() = [2]
                half^#(x1) = [2] x1 + [4]
                c_5(x1) = [1] x1 + [0]
                c_6() = [1]
    
    3) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.