Problem Mixed TRS gcd

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputMixed TRS gcd

stdout:

MAYBE

Problem:
 min(x,0()) -> 0()
 min(0(),y) -> 0()
 min(s(x),s(y)) -> s(min(x,y))
 max(x,0()) -> x
 max(0(),y) -> y
 max(s(x),s(y)) -> s(max(x,y))
 -(x,0()) -> x
 -(s(x),s(y)) -> -(x,y)
 gcd(s(x),0()) -> s(x)
 gcd(0(),s(x)) -> s(x)
 gcd(s(x),s(y)) -> gcd(-(max(x,y),min(x,y)),s(min(x,y)))

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputMixed TRS gcd

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputMixed TRS gcd

stdout:

MAYBE

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

Proof Output:    
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: min^#(x, 0()) -> c_0()
              , 2: min^#(0(), y) -> c_1()
              , 3: min^#(s(x), s(y)) -> c_2(min^#(x, y))
              , 4: max^#(x, 0()) -> c_3()
              , 5: max^#(0(), y) -> c_4()
              , 6: max^#(s(x), s(y)) -> c_5(max^#(x, y))
              , 7: -^#(x, 0()) -> c_6()
              , 8: -^#(s(x), s(y)) -> c_7(-^#(x, y))
              , 9: gcd^#(s(x), 0()) -> c_8()
              , 10: gcd^#(0(), s(x)) -> c_9()
              , 11: gcd^#(s(x), s(y)) ->
                    c_10(gcd^#(-(max(x, y), min(x, y)), s(min(x, y))))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [       MAYBE        ]
                |
                `->{10}                                                  [         NA         ]
             
             ->{9}                                                       [    YES(?,O(1))     ]
             
             ->{8}                                                       [   YES(?,O(n^2))    ]
                |
                `->{7}                                                   [   YES(?,O(n^2))    ]
             
             ->{6}                                                       [   YES(?,O(n^2))    ]
                |
                |->{4}                                                   [   YES(?,O(n^2))    ]
                |
                `->{5}                                                   [   YES(?,O(n^2))    ]
             
             ->{3}                                                       [   YES(?,O(n^2))    ]
                |
                |->{1}                                                   [   YES(?,O(n^2))    ]
                |
                `->{2}                                                   [   YES(?,O(n^2))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1},
                 Uargs(max^#) = {}, Uargs(c_5) = {}, Uargs(-^#) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                min^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                min^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                                [0 2]      [0 0]      [0]
                c_2(x1) = [1 2] x1 + [5]
                          [0 0]      [3]
           
           * Path {3}->{1}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1},
                 Uargs(max^#) = {}, Uargs(c_5) = {}, Uargs(-^#) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {min^#(x, 0()) -> c_0()}
               Weak Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [0]
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [0]
                min^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
                                [0 0]      [4 1]      [0]
                c_0() = [1]
                        [0]
                c_2(x1) = [1 0] x1 + [6]
                          [0 0]      [7]
           
           * Path {3}->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1},
                 Uargs(max^#) = {}, Uargs(c_5) = {}, Uargs(-^#) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {min^#(0(), y) -> c_1()}
               Weak Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [0]
                min^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
                                [4 1]      [2 0]      [0]
                c_1() = [1]
                        [0]
                c_2(x1) = [1 0] x1 + [3]
                          [0 0]      [7]
           
           * Path {6}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_5) = {1}, Uargs(-^#) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                max^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                max^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                                [0 2]      [0 0]      [0]
                c_5(x1) = [1 2] x1 + [5]
                          [0 0]      [3]
           
           * Path {6}->{4}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_5) = {1}, Uargs(-^#) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {max^#(x, 0()) -> c_3()}
               Weak Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [0]
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [0]
                max^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
                                [0 0]      [4 1]      [0]
                c_3() = [1]
                        [0]
                c_5(x1) = [1 0] x1 + [6]
                          [0 0]      [7]
           
           * Path {6}->{5}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_5) = {1}, Uargs(-^#) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {max^#(0(), y) -> c_4()}
               Weak Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [0]
                max^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
                                [4 1]      [2 0]      [0]
                c_4() = [1]
                        [0]
                c_5(x1) = [1 0] x1 + [3]
                          [0 0]      [7]
           
           * Path {8}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_5) = {}, Uargs(-^#) = {},
                 Uargs(c_7) = {1}, Uargs(gcd^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                -^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                              [3 3]      [3 3]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {-^#(s(x), s(y)) -> c_7(-^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(-^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                -^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                              [0 2]      [0 0]      [0]
                c_7(x1) = [1 2] x1 + [5]
                          [0 0]      [3]
           
           * Path {8}->{7}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_5) = {}, Uargs(-^#) = {},
                 Uargs(c_7) = {1}, Uargs(gcd^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {-^#(x, 0()) -> c_6()}
               Weak Rules: {-^#(s(x), s(y)) -> c_7(-^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(-^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [0]
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [0]
                -^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
                              [0 0]      [4 1]      [0]
                c_6() = [1]
                        [0]
                c_7(x1) = [1 0] x1 + [6]
                          [0 0]      [7]
           
           * Path {9}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_5) = {}, Uargs(-^#) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {gcd^#(s(x), 0()) -> c_8()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gcd^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [0]
                s(x1) = [0 0] x1 + [2]
                        [0 0]      [2]
                gcd^#(x1, x2) = [2 2] x1 + [2 0] x2 + [3]
                                [0 0]      [0 0]      [3]
                c_8() = [0]
                        [1]
           
           * Path {11}: MAYBE
             ----------------
             
             The usable rules for this path are:
             
               {  min(x, 0()) -> 0()
                , min(0(), y) -> 0()
                , min(s(x), s(y)) -> s(min(x, y))
                , max(x, 0()) -> x
                , max(0(), y) -> y
                , max(s(x), s(y)) -> s(max(x, y))
                , -(x, 0()) -> x
                , -(s(x), s(y)) -> -(x, y)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  gcd^#(s(x), s(y)) ->
                    c_10(gcd^#(-(max(x, y), min(x, y)), s(min(x, y))))
                  , min(x, 0()) -> 0()
                  , min(0(), y) -> 0()
                  , min(s(x), s(y)) -> s(min(x, y))
                  , max(x, 0()) -> x
                  , max(0(), y) -> y
                  , max(s(x), s(y)) -> s(max(x, y))
                  , -(x, 0()) -> x
                  , -(s(x), s(y)) -> -(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {11}->{10}: NA
             -------------------
             
             The usable rules for this path are:
             
               {  min(x, 0()) -> 0()
                , min(0(), y) -> 0()
                , min(s(x), s(y)) -> s(min(x, y))
                , max(x, 0()) -> x
                , max(0(), y) -> y
                , max(s(x), s(y)) -> s(max(x, y))
                , -(x, 0()) -> x
                , -(s(x), s(y)) -> -(x, y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(min) = {}, Uargs(s) = {1}, Uargs(max) = {},
                 Uargs(-) = {1, 2}, Uargs(gcd) = {}, Uargs(min^#) = {},
                 Uargs(c_2) = {}, Uargs(max^#) = {}, Uargs(c_5) = {},
                 Uargs(-^#) = {}, Uargs(c_7) = {}, Uargs(gcd^#) = {1, 2},
                 Uargs(c_10) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0 0] x1 + [0 1] x2 + [1]
                              [0 2]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [2]
                max(x1, x2) = [2 1] x1 + [1 0] x2 + [1]
                              [0 1]      [0 2]      [2]
                -(x1, x2) = [2 2] x1 + [2 0] x2 + [3]
                            [0 1]      [0 0]      [3]
                gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3() = [0]
                        [0]
                c_4() = [0]
                        [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_6() = [0]
                        [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gcd^#(x1, x2) = [3 0] x1 + [3 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_8() = [0]
                        [0]
                c_9() = [0]
                        [0]
                c_10(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: min^#(x, 0()) -> c_0()
              , 2: min^#(0(), y) -> c_1()
              , 3: min^#(s(x), s(y)) -> c_2(min^#(x, y))
              , 4: max^#(x, 0()) -> c_3()
              , 5: max^#(0(), y) -> c_4()
              , 6: max^#(s(x), s(y)) -> c_5(max^#(x, y))
              , 7: -^#(x, 0()) -> c_6()
              , 8: -^#(s(x), s(y)) -> c_7(-^#(x, y))
              , 9: gcd^#(s(x), 0()) -> c_8()
              , 10: gcd^#(0(), s(x)) -> c_9()
              , 11: gcd^#(s(x), s(y)) ->
                    c_10(gcd^#(-(max(x, y), min(x, y)), s(min(x, y))))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [       MAYBE        ]
                |
                `->{10}                                                  [         NA         ]
             
             ->{9}                                                       [    YES(?,O(1))     ]
             
             ->{8}                                                       [   YES(?,O(n^1))    ]
                |
                `->{7}                                                   [   YES(?,O(n^1))    ]
             
             ->{6}                                                       [   YES(?,O(n^1))    ]
                |
                |->{4}                                                   [   YES(?,O(n^1))    ]
                |
                `->{5}                                                   [   YES(?,O(n^1))    ]
             
             ->{3}                                                       [   YES(?,O(n^1))    ]
                |
                |->{1}                                                   [   YES(?,O(n^1))    ]
                |
                `->{2}                                                   [   YES(?,O(n^1))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1},
                 Uargs(max^#) = {}, Uargs(c_5) = {}, Uargs(-^#) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                max(x1, x2) = [0] x1 + [0] x2 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                gcd(x1, x2) = [0] x1 + [0] x2 + [0]
                min^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                max^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                min^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_2(x1) = [1] x1 + [7]
           
           * Path {3}->{1}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1},
                 Uargs(max^#) = {}, Uargs(c_5) = {}, Uargs(-^#) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                max(x1, x2) = [0] x1 + [0] x2 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                gcd(x1, x2) = [0] x1 + [0] x2 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                max^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {min^#(x, 0()) -> c_0()}
               Weak Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                min^#(x1, x2) = [2] x1 + [2] x2 + [4]
                c_0() = [1]
                c_2(x1) = [1] x1 + [7]
           
           * Path {3}->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1},
                 Uargs(max^#) = {}, Uargs(c_5) = {}, Uargs(-^#) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                max(x1, x2) = [0] x1 + [0] x2 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                gcd(x1, x2) = [0] x1 + [0] x2 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                max^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {min^#(0(), y) -> c_1()}
               Weak Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                min^#(x1, x2) = [2] x1 + [0] x2 + [4]
                c_1() = [1]
                c_2(x1) = [1] x1 + [2]
           
           * 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_5) = {1}, Uargs(-^#) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                max(x1, x2) = [0] x1 + [0] x2 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                gcd(x1, x2) = [0] x1 + [0] x2 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                max^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [1] x1 + [0]
                -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                max^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_5(x1) = [1] x1 + [7]
           
           * Path {6}->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_5) = {1}, Uargs(-^#) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                max(x1, x2) = [0] x1 + [0] x2 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                gcd(x1, x2) = [0] x1 + [0] x2 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                max^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [1] x1 + [0]
                -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {max^#(x, 0()) -> c_3()}
               Weak Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                max^#(x1, x2) = [2] x1 + [2] x2 + [4]
                c_3() = [1]
                c_5(x1) = [1] x1 + [7]
           
           * Path {6}->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_5) = {1}, Uargs(-^#) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                max(x1, x2) = [0] x1 + [0] x2 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                gcd(x1, x2) = [0] x1 + [0] x2 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                max^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [1] x1 + [0]
                -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {max^#(0(), y) -> c_4()}
               Weak Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                max^#(x1, x2) = [2] x1 + [0] x2 + [4]
                c_4() = [1]
                c_5(x1) = [1] x1 + [2]
           
           * Path {8}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_5) = {}, Uargs(-^#) = {},
                 Uargs(c_7) = {1}, Uargs(gcd^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                max(x1, x2) = [0] x1 + [0] x2 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                gcd(x1, x2) = [0] x1 + [0] x2 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                max^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                -^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_6() = [0]
                c_7(x1) = [1] x1 + [0]
                gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {-^#(s(x), s(y)) -> c_7(-^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(-^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                -^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_7(x1) = [1] x1 + [7]
           
           * Path {8}->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_5) = {}, Uargs(-^#) = {},
                 Uargs(c_7) = {1}, Uargs(gcd^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                max(x1, x2) = [0] x1 + [0] x2 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                gcd(x1, x2) = [0] x1 + [0] x2 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                max^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [1] x1 + [0]
                gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {-^#(x, 0()) -> c_6()}
               Weak Rules: {-^#(s(x), s(y)) -> c_7(-^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(-^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                -^#(x1, x2) = [2] x1 + [2] x2 + [4]
                c_6() = [1]
                c_7(x1) = [1] x1 + [7]
           
           * Path {9}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_5) = {}, Uargs(-^#) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                max(x1, x2) = [0] x1 + [0] x2 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                gcd(x1, x2) = [0] x1 + [0] x2 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                max^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {gcd^#(s(x), 0()) -> c_8()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gcd^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [0] x1 + [2]
                gcd^#(x1, x2) = [2] x1 + [2] x2 + [7]
                c_8() = [0]
           
           * Path {11}: MAYBE
             ----------------
             
             The usable rules for this path are:
             
               {  min(x, 0()) -> 0()
                , min(0(), y) -> 0()
                , min(s(x), s(y)) -> s(min(x, y))
                , max(x, 0()) -> x
                , max(0(), y) -> y
                , max(s(x), s(y)) -> s(max(x, y))
                , -(x, 0()) -> x
                , -(s(x), s(y)) -> -(x, y)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  gcd^#(s(x), s(y)) ->
                    c_10(gcd^#(-(max(x, y), min(x, y)), s(min(x, y))))
                  , min(x, 0()) -> 0()
                  , min(0(), y) -> 0()
                  , min(s(x), s(y)) -> s(min(x, y))
                  , max(x, 0()) -> x
                  , max(0(), y) -> y
                  , max(s(x), s(y)) -> s(max(x, y))
                  , -(x, 0()) -> x
                  , -(s(x), s(y)) -> -(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {11}->{10}: NA
             -------------------
             
             The usable rules for this path are:
             
               {  min(x, 0()) -> 0()
                , min(0(), y) -> 0()
                , min(s(x), s(y)) -> s(min(x, y))
                , max(x, 0()) -> x
                , max(0(), y) -> y
                , max(s(x), s(y)) -> s(max(x, y))
                , -(x, 0()) -> x
                , -(s(x), s(y)) -> -(x, y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(min) = {}, Uargs(s) = {1}, Uargs(max) = {},
                 Uargs(-) = {1, 2}, Uargs(gcd) = {}, Uargs(min^#) = {},
                 Uargs(c_2) = {}, Uargs(max^#) = {}, Uargs(c_5) = {},
                 Uargs(-^#) = {}, Uargs(c_7) = {}, Uargs(gcd^#) = {1, 2},
                 Uargs(c_10) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0] x1 + [2] x2 + [2]
                0() = [0]
                s(x1) = [1] x1 + [2]
                max(x1, x2) = [1] x1 + [1] x2 + [2]
                -(x1, x2) = [3] x1 + [3] x2 + [3]
                gcd(x1, x2) = [0] x1 + [0] x2 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                max^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3() = [0]
                c_4() = [0]
                c_5(x1) = [0] x1 + [0]
                -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6() = [0]
                c_7(x1) = [0] x1 + [0]
                gcd^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_8() = [0]
                c_9() = [0]
                c_10(x1) = [1] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
    
    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
InputMixed TRS gcd

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputMixed TRS gcd

stdout:

MAYBE

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

Proof Output:    
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: min^#(x, 0()) -> c_0()
              , 2: min^#(0(), y) -> c_1()
              , 3: min^#(s(x), s(y)) -> c_2(min^#(x, y))
              , 4: max^#(x, 0()) -> c_3(x)
              , 5: max^#(0(), y) -> c_4(y)
              , 6: max^#(s(x), s(y)) -> c_5(max^#(x, y))
              , 7: -^#(x, 0()) -> c_6(x)
              , 8: -^#(s(x), s(y)) -> c_7(-^#(x, y))
              , 9: gcd^#(s(x), 0()) -> c_8(x)
              , 10: gcd^#(0(), s(x)) -> c_9(x)
              , 11: gcd^#(s(x), s(y)) ->
                    c_10(gcd^#(-(max(x, y), min(x, y)), s(min(x, y))))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [       MAYBE        ]
                |
                `->{10}                                                  [         NA         ]
             
             ->{9}                                                       [   YES(?,O(n^2))    ]
             
             ->{8}                                                       [   YES(?,O(n^2))    ]
                |
                `->{7}                                                   [   YES(?,O(n^2))    ]
             
             ->{6}                                                       [   YES(?,O(n^2))    ]
                |
                |->{4}                                                   [   YES(?,O(n^2))    ]
                |
                `->{5}                                                   [   YES(?,O(n^1))    ]
             
             ->{3}                                                       [   YES(?,O(n^2))    ]
                |
                |->{1}                                                   [   YES(?,O(n^2))    ]
                |
                `->{2}                                                   [   YES(?,O(n^2))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1},
                 Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(-^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
                 Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                min^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                min^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                                [0 2]      [0 0]      [0]
                c_2(x1) = [1 2] x1 + [5]
                          [0 0]      [3]
           
           * Path {3}->{1}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1},
                 Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(-^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
                 Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {min^#(x, 0()) -> c_0()}
               Weak Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [0]
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [0]
                min^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
                                [0 0]      [4 1]      [0]
                c_0() = [1]
                        [0]
                c_2(x1) = [1 0] x1 + [6]
                          [0 0]      [7]
           
           * Path {3}->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1},
                 Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(-^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
                 Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {min^#(0(), y) -> c_1()}
               Weak Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [0]
                min^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
                                [4 1]      [2 0]      [0]
                c_1() = [1]
                        [0]
                c_2(x1) = [1 0] x1 + [3]
                          [0 0]      [7]
           
           * Path {6}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {1}, Uargs(-^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_8) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                max^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                max^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                                [0 2]      [0 0]      [0]
                c_5(x1) = [1 2] x1 + [5]
                          [0 0]      [3]
           
           * Path {6}->{4}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {1}, Uargs(-^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_8) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                max^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {max^#(x, 0()) -> c_3(x)}
               Weak Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [2]
                s(x1) = [1 2] x1 + [0]
                        [0 1]      [0]
                max^#(x1, x2) = [2 2] x1 + [0 2] x2 + [0]
                                [4 1]      [3 2]      [0]
                c_3(x1) = [0 0] x1 + [1]
                          [0 1]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [2 0]      [0]
           
           * Path {6}->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {1}, Uargs(-^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_8) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                max^#(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {max^#(0(), y) -> c_4(y)}
               Weak Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [2]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
                max^#(x1, x2) = [0 2] x1 + [0 0] x2 + [4]
                                [0 2]      [0 1]      [0]
                c_4(x1) = [0 0] x1 + [1]
                          [0 1]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
           
           * Path {8}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(-^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {1}, Uargs(gcd^#) = {}, Uargs(c_8) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                -^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                              [3 3]      [3 3]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {-^#(s(x), s(y)) -> c_7(-^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(-^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                -^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                              [0 2]      [0 0]      [0]
                c_7(x1) = [1 2] x1 + [5]
                          [0 0]      [3]
           
           * Path {8}->{7}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(-^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {1}, Uargs(gcd^#) = {}, Uargs(c_8) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                -^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_6(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                gcd^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {-^#(x, 0()) -> c_6(x)}
               Weak Rules: {-^#(s(x), s(y)) -> c_7(-^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(-^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [2]
                s(x1) = [1 2] x1 + [0]
                        [0 1]      [0]
                -^#(x1, x2) = [2 2] x1 + [0 2] x2 + [0]
                              [4 1]      [3 2]      [0]
                c_6(x1) = [0 0] x1 + [1]
                          [0 1]      [0]
                c_7(x1) = [1 0] x1 + [0]
                          [2 0]      [0]
           
           * Path {9}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(-^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
                 Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                0() = [0]
                      [0]
                s(x1) = [1 1] x1 + [0]
                        [0 1]      [0]
                max(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                -(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                            [0 0]      [0 0]      [0]
                gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gcd^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
                c_9(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_10(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {gcd^#(s(x), 0()) -> c_8(x)}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gcd^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [2]
                s(x1) = [1 2] x1 + [2]
                        [0 0]      [2]
                gcd^#(x1, x2) = [2 2] x1 + [0 2] x2 + [3]
                                [2 2]      [0 2]      [3]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [1]
           
           * Path {11}: MAYBE
             ----------------
             
             The usable rules for this path are:
             
               {  min(x, 0()) -> 0()
                , min(0(), y) -> 0()
                , min(s(x), s(y)) -> s(min(x, y))
                , max(x, 0()) -> x
                , max(0(), y) -> y
                , max(s(x), s(y)) -> s(max(x, y))
                , -(x, 0()) -> x
                , -(s(x), s(y)) -> -(x, y)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  gcd^#(s(x), s(y)) ->
                    c_10(gcd^#(-(max(x, y), min(x, y)), s(min(x, y))))
                  , min(x, 0()) -> 0()
                  , min(0(), y) -> 0()
                  , min(s(x), s(y)) -> s(min(x, y))
                  , max(x, 0()) -> x
                  , max(0(), y) -> y
                  , max(s(x), s(y)) -> s(max(x, y))
                  , -(x, 0()) -> x
                  , -(s(x), s(y)) -> -(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {11}->{10}: NA
             -------------------
             
             The usable rules for this path are:
             
               {  min(x, 0()) -> 0()
                , min(0(), y) -> 0()
                , min(s(x), s(y)) -> s(min(x, y))
                , max(x, 0()) -> x
                , max(0(), y) -> y
                , max(s(x), s(y)) -> s(max(x, y))
                , -(x, 0()) -> x
                , -(s(x), s(y)) -> -(x, y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(min) = {1, 2}, Uargs(s) = {1}, Uargs(max) = {1, 2},
                 Uargs(-) = {1, 2}, Uargs(gcd) = {}, Uargs(min^#) = {},
                 Uargs(c_2) = {}, Uargs(max^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(-^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {1, 2}, Uargs(c_8) = {},
                 Uargs(c_9) = {1}, Uargs(c_10) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [1 3] x1 + [2 1] x2 + [0]
                              [0 2]      [0 0]      [1]
                0() = [0]
                      [1]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [2]
                max(x1, x2) = [2 1] x1 + [1 1] x2 + [0]
                              [0 1]      [0 1]      [0]
                -(x1, x2) = [2 2] x1 + [1 0] x2 + [3]
                            [0 1]      [0 0]      [3]
                gcd(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                max^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                -^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                gcd^#(x1, x2) = [3 0] x1 + [3 3] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_9(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_10(x1) = [1 0] x1 + [0]
                           [0 1]      [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: min^#(x, 0()) -> c_0()
              , 2: min^#(0(), y) -> c_1()
              , 3: min^#(s(x), s(y)) -> c_2(min^#(x, y))
              , 4: max^#(x, 0()) -> c_3(x)
              , 5: max^#(0(), y) -> c_4(y)
              , 6: max^#(s(x), s(y)) -> c_5(max^#(x, y))
              , 7: -^#(x, 0()) -> c_6(x)
              , 8: -^#(s(x), s(y)) -> c_7(-^#(x, y))
              , 9: gcd^#(s(x), 0()) -> c_8(x)
              , 10: gcd^#(0(), s(x)) -> c_9(x)
              , 11: gcd^#(s(x), s(y)) ->
                    c_10(gcd^#(-(max(x, y), min(x, y)), s(min(x, y))))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{11}                                                      [       MAYBE        ]
                |
                `->{10}                                                  [         NA         ]
             
             ->{9}                                                       [   YES(?,O(n^1))    ]
             
             ->{8}                                                       [   YES(?,O(n^1))    ]
                |
                `->{7}                                                   [   YES(?,O(n^1))    ]
             
             ->{6}                                                       [   YES(?,O(n^1))    ]
                |
                |->{4}                                                   [   YES(?,O(n^1))    ]
                |
                `->{5}                                                   [   YES(?,O(n^1))    ]
             
             ->{3}                                                       [   YES(?,O(n^1))    ]
                |
                |->{1}                                                   [   YES(?,O(n^1))    ]
                |
                `->{2}                                                   [   YES(?,O(n^1))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1},
                 Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(-^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
                 Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                max(x1, x2) = [0] x1 + [0] x2 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                gcd(x1, x2) = [0] x1 + [0] x2 + [0]
                min^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                max^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                min^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_2(x1) = [1] x1 + [7]
           
           * Path {3}->{1}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1},
                 Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(-^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
                 Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                max(x1, x2) = [0] x1 + [0] x2 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                gcd(x1, x2) = [0] x1 + [0] x2 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                max^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {min^#(x, 0()) -> c_0()}
               Weak Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                min^#(x1, x2) = [2] x1 + [2] x2 + [4]
                c_0() = [1]
                c_2(x1) = [1] x1 + [7]
           
           * Path {3}->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1},
                 Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(-^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
                 Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                max(x1, x2) = [0] x1 + [0] x2 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                gcd(x1, x2) = [0] x1 + [0] x2 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [1] x1 + [0]
                max^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {min^#(0(), y) -> c_1()}
               Weak Rules: {min^#(s(x), s(y)) -> c_2(min^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_2) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                min^#(x1, x2) = [2] x1 + [0] x2 + [4]
                c_1() = [1]
                c_2(x1) = [1] x1 + [2]
           
           * 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {1}, Uargs(-^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_8) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                max(x1, x2) = [0] x1 + [0] x2 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                gcd(x1, x2) = [0] x1 + [0] x2 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                max^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                max^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_5(x1) = [1] x1 + [7]
           
           * Path {6}->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {1}, Uargs(-^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_8) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                max(x1, x2) = [0] x1 + [0] x2 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                gcd(x1, x2) = [0] x1 + [0] x2 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                max^#(x1, x2) = [3] x1 + [0] x2 + [0]
                c_3(x1) = [1] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {max^#(x, 0()) -> c_3(x)}
               Weak Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                s(x1) = [1] x1 + [2]
                max^#(x1, x2) = [2] x1 + [2] x2 + [2]
                c_3(x1) = [0] x1 + [1]
                c_5(x1) = [1] x1 + [5]
           
           * Path {6}->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {1}, Uargs(-^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {}, Uargs(c_8) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                max(x1, x2) = [0] x1 + [0] x2 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                gcd(x1, x2) = [0] x1 + [0] x2 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                max^#(x1, x2) = [0] x1 + [3] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [1] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {max^#(0(), y) -> c_4(y)}
               Weak Rules: {max^#(s(x), s(y)) -> c_5(max^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(max^#) = {}, Uargs(c_4) = {}, Uargs(c_5) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                s(x1) = [1] x1 + [2]
                max^#(x1, x2) = [2] x1 + [2] x2 + [1]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]
           
           * Path {8}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(-^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {1}, Uargs(gcd^#) = {}, Uargs(c_8) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                max(x1, x2) = [0] x1 + [0] x2 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                gcd(x1, x2) = [0] x1 + [0] x2 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                max^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                -^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [1] x1 + [0]
                gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {-^#(s(x), s(y)) -> c_7(-^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(-^#) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                -^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_7(x1) = [1] x1 + [7]
           
           * Path {8}->{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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(-^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {1}, Uargs(gcd^#) = {}, Uargs(c_8) = {},
                 Uargs(c_9) = {}, Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                max(x1, x2) = [0] x1 + [0] x2 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                gcd(x1, x2) = [0] x1 + [0] x2 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                max^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                -^#(x1, x2) = [3] x1 + [0] x2 + [0]
                c_6(x1) = [1] x1 + [0]
                c_7(x1) = [1] x1 + [0]
                gcd^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {-^#(x, 0()) -> c_6(x)}
               Weak Rules: {-^#(s(x), s(y)) -> c_7(-^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(-^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                s(x1) = [1] x1 + [2]
                -^#(x1, x2) = [2] x1 + [2] x2 + [2]
                c_6(x1) = [0] x1 + [1]
                c_7(x1) = [1] x1 + [5]
           
           * Path {9}: 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(min) = {}, Uargs(s) = {}, Uargs(max) = {}, Uargs(-) = {},
                 Uargs(gcd) = {}, Uargs(min^#) = {}, Uargs(c_2) = {},
                 Uargs(max^#) = {}, Uargs(c_3) = {}, Uargs(c_4) = {},
                 Uargs(c_5) = {}, Uargs(-^#) = {}, Uargs(c_6) = {}, Uargs(c_7) = {},
                 Uargs(gcd^#) = {}, Uargs(c_8) = {}, Uargs(c_9) = {},
                 Uargs(c_10) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                max(x1, x2) = [0] x1 + [0] x2 + [0]
                -(x1, x2) = [0] x1 + [0] x2 + [0]
                gcd(x1, x2) = [0] x1 + [0] x2 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                max^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                gcd^#(x1, x2) = [3] x1 + [0] x2 + [0]
                c_8(x1) = [1] x1 + [0]
                c_9(x1) = [0] x1 + [0]
                c_10(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {gcd^#(s(x), 0()) -> c_8(x)}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(gcd^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                gcd^#(x1, x2) = [2] x1 + [2] x2 + [7]
                c_8(x1) = [0] x1 + [0]
           
           * Path {11}: MAYBE
             ----------------
             
             The usable rules for this path are:
             
               {  min(x, 0()) -> 0()
                , min(0(), y) -> 0()
                , min(s(x), s(y)) -> s(min(x, y))
                , max(x, 0()) -> x
                , max(0(), y) -> y
                , max(s(x), s(y)) -> s(max(x, y))
                , -(x, 0()) -> x
                , -(s(x), s(y)) -> -(x, y)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  gcd^#(s(x), s(y)) ->
                    c_10(gcd^#(-(max(x, y), min(x, y)), s(min(x, y))))
                  , min(x, 0()) -> 0()
                  , min(0(), y) -> 0()
                  , min(s(x), s(y)) -> s(min(x, y))
                  , max(x, 0()) -> x
                  , max(0(), y) -> y
                  , max(s(x), s(y)) -> s(max(x, y))
                  , -(x, 0()) -> x
                  , -(s(x), s(y)) -> -(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {11}->{10}: NA
             -------------------
             
             The usable rules for this path are:
             
               {  min(x, 0()) -> 0()
                , min(0(), y) -> 0()
                , min(s(x), s(y)) -> s(min(x, y))
                , max(x, 0()) -> x
                , max(0(), y) -> y
                , max(s(x), s(y)) -> s(max(x, y))
                , -(x, 0()) -> x
                , -(s(x), s(y)) -> -(x, y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(min) = {1, 2}, Uargs(s) = {1}, Uargs(max) = {1, 2},
                 Uargs(-) = {1, 2}, Uargs(gcd) = {}, Uargs(min^#) = {},
                 Uargs(c_2) = {}, Uargs(max^#) = {}, Uargs(c_3) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(-^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}, Uargs(gcd^#) = {1, 2}, Uargs(c_8) = {},
                 Uargs(c_9) = {1}, Uargs(c_10) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                min(x1, x2) = [2] x1 + [2] x2 + [0]
                0() = [3]
                s(x1) = [1] x1 + [1]
                max(x1, x2) = [2] x1 + [2] x2 + [0]
                -(x1, x2) = [3] x1 + [3] x2 + [1]
                gcd(x1, x2) = [0] x1 + [0] x2 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1() = [0]
                c_2(x1) = [0] x1 + [0]
                max^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_3(x1) = [0] x1 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                -^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
                gcd^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_8(x1) = [0] x1 + [0]
                c_9(x1) = [1] x1 + [0]
                c_10(x1) = [1] x1 + [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
    
    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.