Problem GTSSK07 cade05t

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputGTSSK07 cade05t

stdout:

MAYBE

Problem:
 minus(x,y) -> cond(equal(min(x,y),y),x,y)
 cond(true(),x,y) -> s(minus(x,s(y)))
 min(0(),v) -> 0()
 min(u,0()) -> 0()
 min(s(u),s(v)) -> s(min(u,v))
 equal(0(),0()) -> true()
 equal(s(x),0()) -> false()
 equal(0(),s(y)) -> false()
 equal(s(x),s(y)) -> equal(x,y)

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputGTSSK07 cade05t

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputGTSSK07 cade05t

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  minus(x, y) -> cond(equal(min(x, y), y), x, y)
     , cond(true(), x, y) -> s(minus(x, s(y)))
     , min(0(), v) -> 0()
     , min(u, 0()) -> 0()
     , min(s(u), s(v)) -> s(min(u, v))
     , equal(0(), 0()) -> true()
     , equal(s(x), 0()) -> false()
     , equal(0(), s(y)) -> false()
     , equal(s(x), s(y)) -> equal(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: minus^#(x, y) -> c_0(cond^#(equal(min(x, y), y), x, y))
              , 2: cond^#(true(), x, y) -> c_1(minus^#(x, s(y)))
              , 3: min^#(0(), v) -> c_2()
              , 4: min^#(u, 0()) -> c_3()
              , 5: min^#(s(u), s(v)) -> c_4(min^#(u, v))
              , 6: equal^#(0(), 0()) -> c_5()
              , 7: equal^#(s(x), 0()) -> c_6()
              , 8: equal^#(0(), s(y)) -> c_7()
              , 9: equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{9}                                                       [   YES(?,O(n^1))    ]
                |
                |->{6}                                                   [   YES(?,O(n^2))    ]
                |
                |->{7}                                                   [   YES(?,O(n^3))    ]
                |
                `->{8}                                                   [   YES(?,O(n^2))    ]
             
             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                |->{3}                                                   [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [   YES(?,O(n^2))    ]
             
             ->{1,2}                                                     [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {1,2}: NA
             --------------
             
             The usable rules for this path are:
             
               {  min(0(), v) -> 0()
                , min(u, 0()) -> 0()
                , min(s(u), s(v)) -> s(min(u, v))
                , equal(0(), 0()) -> true()
                , equal(s(x), 0()) -> false()
                , equal(0(), s(y)) -> false()
                , equal(s(x), s(y)) -> equal(x, y)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {1}, Uargs(equal^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                equal(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                min(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                min^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                equal^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 3] x1 + [0]
                        [0 0 1]      [4]
                        [0 0 0]      [2]
                min^#(x1, x2) = [1 1 0] x1 + [1 0 2] x2 + [0]
                                [0 0 2]      [0 0 0]      [2]
                                [1 1 0]      [4 1 0]      [0]
                c_4(x1) = [1 2 0] x1 + [3]
                          [0 0 0]      [2]
                          [0 2 0]      [3]
           
           * Path {5}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {1}, Uargs(equal^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                equal(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                min(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                min^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                equal^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {min^#(0(), v) -> c_2()}
               Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 4 2] x1 + [0]
                        [0 0 2]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [2]
                      [0]
                min^#(x1, x2) = [2 2 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [2 0 0]      [0 4 0]      [0]
                c_2() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
           
           * Path {5}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {1}, Uargs(equal^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                equal(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                min(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                min^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                equal^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {min^#(u, 0()) -> c_3()}
               Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 2] x1 + [2]
                        [0 1 2]      [0]
                        [0 0 0]      [3]
                0() = [2]
                      [2]
                      [2]
                min^#(x1, x2) = [4 0 0] x1 + [3 3 0] x2 + [0]
                                [2 0 2]      [2 0 0]      [0]
                                [0 4 0]      [2 2 3]      [0]
                c_3() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [2]
                          [0 0 0]      [2]
                          [0 0 0]      [7]
           
           * 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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                equal(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                min(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                min^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                equal^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [3 3 3]      [3 3 3]      [0]
                                  [3 3 3]      [3 3 3]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 0] x1 + [2]
                        [0 0 2]      [2]
                        [0 0 0]      [0]
                equal^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
                                  [2 2 0]      [0 2 0]      [0]
                                  [4 0 0]      [0 2 0]      [0]
                c_8(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [7]
           
           * Path {9}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                equal(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                min(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                min^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                equal^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {equal^#(0(), 0()) -> c_5()}
               Weak Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 3 2] x1 + [0]
                        [0 1 1]      [0]
                        [0 0 0]      [0]
                0() = [2]
                      [0]
                      [0]
                equal^#(x1, x2) = [2 0 0] x1 + [2 0 0] x2 + [0]
                                  [0 4 0]      [0 0 0]      [2]
                                  [4 0 0]      [0 0 0]      [4]
                c_5() = [1]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [1]
                          [0 2 0]      [0]
           
           * Path {9}->{7}: YES(?,O(n^3))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                equal(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                min(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                min^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                equal^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^3))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {equal^#(s(x), 0()) -> c_6()}
               Weak Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 3 2] x1 + [2]
                        [0 1 2]      [0]
                        [0 0 1]      [2]
                0() = [0]
                      [0]
                      [0]
                equal^#(x1, x2) = [2 2 0] x1 + [0 1 0] x2 + [0]
                                  [0 0 0]      [0 0 2]      [0]
                                  [0 1 0]      [0 2 4]      [0]
                c_6() = [1]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [0]
                          [0 0 0]      [6]
           
           * Path {9}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                equal(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                min(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                min^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                equal^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {equal^#(0(), s(y)) -> c_7()}
               Weak Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 2] x1 + [0]
                        [0 1 0]      [2]
                        [0 0 0]      [0]
                0() = [0]
                      [2]
                      [0]
                equal^#(x1, x2) = [0 2 0] x1 + [1 0 2] x2 + [0]
                                  [7 1 0]      [4 0 0]      [0]
                                  [4 2 0]      [4 0 0]      [0]
                c_7() = [1]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [3]
                          [2 0 0]      [2]
                          [0 0 0]      [2]
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: minus^#(x, y) -> c_0(cond^#(equal(min(x, y), y), x, y))
              , 2: cond^#(true(), x, y) -> c_1(minus^#(x, s(y)))
              , 3: min^#(0(), v) -> c_2()
              , 4: min^#(u, 0()) -> c_3()
              , 5: min^#(s(u), s(v)) -> c_4(min^#(u, v))
              , 6: equal^#(0(), 0()) -> c_5()
              , 7: equal^#(s(x), 0()) -> c_6()
              , 8: equal^#(0(), s(y)) -> c_7()
              , 9: equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{9}                                                       [   YES(?,O(n^2))    ]
                |
                |->{6}                                                   [   YES(?,O(n^1))    ]
                |
                |->{7}                                                   [   YES(?,O(n^1))    ]
                |
                `->{8}                                                   [   YES(?,O(n^2))    ]
             
             ->{5}                                                       [   YES(?,O(n^2))    ]
                |
                |->{3}                                                   [   YES(?,O(n^2))    ]
                |
                `->{4}                                                   [   YES(?,O(n^2))    ]
             
             ->{1,2}                                                     [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1,2}: MAYBE
             -----------------
             
             The usable rules for this path are:
             
               {  min(0(), v) -> 0()
                , min(u, 0()) -> 0()
                , min(s(u), s(v)) -> s(min(u, v))
                , equal(0(), 0()) -> true()
                , equal(s(x), 0()) -> false()
                , equal(0(), s(y)) -> false()
                , equal(s(x), s(y)) -> equal(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:
                 {  minus^#(x, y) -> c_0(cond^#(equal(min(x, y), y), x, y))
                  , cond^#(true(), x, y) -> c_1(minus^#(x, s(y)))
                  , min(0(), v) -> 0()
                  , min(u, 0()) -> 0()
                  , min(s(u), s(v)) -> s(min(u, v))
                  , equal(0(), 0()) -> true()
                  , equal(s(x), 0()) -> false()
                  , equal(0(), s(y)) -> false()
                  , equal(s(x), s(y)) -> equal(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {1}, Uargs(equal^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                equal(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]
                true() = [0]
                         [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                min^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7() = [0]
                        [0]
                c_8(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(u), s(v)) -> c_4(min^#(u, v))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 6] x1 + [2]
                        [0 1]      [0]
                min^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
                                [0 0]      [0 0]      [4]
                c_4(x1) = [1 0] x1 + [1]
                          [0 0]      [3]
           
           * Path {5}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {1}, Uargs(equal^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                equal(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]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7() = [0]
                        [0]
                c_8(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(), v) -> c_2()}
               Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [0]
                        [0 1]      [2]
                0() = [2]
                      [2]
                min^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
                                [3 3]      [0 1]      [0]
                c_2() = [1]
                        [0]
                c_4(x1) = [1 0] x1 + [3]
                          [0 0]      [7]
           
           * Path {5}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {1}, Uargs(equal^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                equal(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]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7() = [0]
                        [0]
                c_8(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^#(u, 0()) -> c_3()}
               Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 3] x1 + [2]
                        [0 1]      [2]
                0() = [2]
                      [0]
                min^#(x1, x2) = [2 0] x1 + [2 3] x2 + [0]
                                [0 2]      [3 2]      [0]
                c_3() = [1]
                        [0]
                c_4(x1) = [1 0] x1 + [7]
                          [0 0]      [7]
           
           * 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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                equal(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]
                true() = [0]
                         [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                equal^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                  [3 3]      [3 3]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                equal^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                                  [0 2]      [0 0]      [0]
                c_8(x1) = [1 2] x1 + [5]
                          [0 0]      [3]
           
           * Path {9}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                equal(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]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {equal^#(0(), 0()) -> c_5()}
               Weak Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [2]
                        [0 1]      [0]
                0() = [2]
                      [2]
                equal^#(x1, x2) = [2 2] x1 + [2 0] x2 + [0]
                                  [2 0]      [0 0]      [4]
                c_5() = [1]
                        [0]
                c_8(x1) = [1 0] x1 + [7]
                          [0 0]      [7]
           
           * Path {9}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                equal(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]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {equal^#(s(x), 0()) -> c_6()}
               Weak Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 4] x1 + [2]
                        [0 0]      [0]
                0() = [0]
                      [0]
                equal^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
                                  [0 0]      [0 2]      [0]
                c_6() = [1]
                        [0]
                c_8(x1) = [1 2] x1 + [3]
                          [0 0]      [0]
           
           * Path {9}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                equal(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]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {equal^#(0(), s(y)) -> c_7()}
               Weak Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 6] x1 + [2]
                        [0 1]      [2]
                0() = [2]
                      [2]
                equal^#(x1, x2) = [2 1] x1 + [0 1] x2 + [0]
                                  [1 2]      [2 0]      [0]
                c_7() = [1]
                        [0]
                c_8(x1) = [1 0] x1 + [7]
                          [0 0]      [7]
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: minus^#(x, y) -> c_0(cond^#(equal(min(x, y), y), x, y))
              , 2: cond^#(true(), x, y) -> c_1(minus^#(x, s(y)))
              , 3: min^#(0(), v) -> c_2()
              , 4: min^#(u, 0()) -> c_3()
              , 5: min^#(s(u), s(v)) -> c_4(min^#(u, v))
              , 6: equal^#(0(), 0()) -> c_5()
              , 7: equal^#(s(x), 0()) -> c_6()
              , 8: equal^#(0(), s(y)) -> c_7()
              , 9: equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{9}                                                       [   YES(?,O(n^1))    ]
                |
                |->{6}                                                   [   YES(?,O(n^1))    ]
                |
                |->{7}                                                   [   YES(?,O(n^1))    ]
                |
                `->{8}                                                   [   YES(?,O(n^1))    ]
             
             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                |->{3}                                                   [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [   YES(?,O(n^1))    ]
             
             ->{1,2}                                                     [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1,2}: MAYBE
             -----------------
             
             The usable rules for this path are:
             
               {  min(0(), v) -> 0()
                , min(u, 0()) -> 0()
                , min(s(u), s(v)) -> s(min(u, v))
                , equal(0(), 0()) -> true()
                , equal(s(x), 0()) -> false()
                , equal(0(), s(y)) -> false()
                , equal(s(x), s(y)) -> equal(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:
                 {  minus^#(x, y) -> c_0(cond^#(equal(min(x, y), y), x, y))
                  , cond^#(true(), x, y) -> c_1(minus^#(x, s(y)))
                  , min(0(), v) -> 0()
                  , min(u, 0()) -> 0()
                  , min(s(u), s(v)) -> s(min(u, v))
                  , equal(0(), 0()) -> true()
                  , equal(s(x), 0()) -> false()
                  , equal(0(), s(y)) -> false()
                  , equal(s(x), s(y)) -> equal(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {1}, Uargs(equal^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                equal(x1, x2) = [0] x1 + [0] x2 + [0]
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                false() = [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1(x1) = [0] x1 + [0]
                min^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7() = [0]
                c_8(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(u), s(v)) -> c_4(min^#(u, v))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {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_4(x1) = [1] x1 + [7]
           
           * Path {5}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {1}, Uargs(equal^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                equal(x1, x2) = [0] x1 + [0] x2 + [0]
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1(x1) = [0] x1 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7() = [0]
                c_8(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(), v) -> c_2()}
               Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                min^#(x1, x2) = [3] x1 + [3] x2 + [2]
                c_2() = [1]
                c_4(x1) = [1] x1 + [5]
           
           * Path {5}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {1}, Uargs(equal^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                equal(x1, x2) = [0] x1 + [0] x2 + [0]
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1(x1) = [0] x1 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7() = [0]
                c_8(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^#(u, 0()) -> c_3()}
               Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                min^#(x1, x2) = [2] x1 + [2] x2 + [4]
                c_3() = [1]
                c_4(x1) = [1] x1 + [6]
           
           * 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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                equal(x1, x2) = [0] x1 + [0] x2 + [0]
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                false() = [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1(x1) = [0] x1 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                equal^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                equal^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_8(x1) = [1] x1 + [7]
           
           * Path {9}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                equal(x1, x2) = [0] x1 + [0] x2 + [0]
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1(x1) = [0] x1 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {equal^#(0(), 0()) -> c_5()}
               Weak Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [0]
                equal^#(x1, x2) = [2] x1 + [0] x2 + [4]
                c_5() = [1]
                c_8(x1) = [1] x1 + [3]
           
           * Path {9}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                equal(x1, x2) = [0] x1 + [0] x2 + [0]
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1(x1) = [0] x1 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {equal^#(s(x), 0()) -> c_6()}
               Weak Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                equal^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_6() = [1]
                c_8(x1) = [1] x1 + [7]
           
           * Path {9}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                equal(x1, x2) = [0] x1 + [0] x2 + [0]
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1(x1) = [0] x1 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {equal^#(0(), s(y)) -> c_7()}
               Weak Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                equal^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_7() = [1]
                c_8(x1) = [1] x1 + [7]
    
    4) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    

Tool RC1

Execution TimeUnknown
Answer
MAYBE
InputGTSSK07 cade05t

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputGTSSK07 cade05t

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  minus(x, y) -> cond(equal(min(x, y), y), x, y)
     , cond(true(), x, y) -> s(minus(x, s(y)))
     , min(0(), v) -> 0()
     , min(u, 0()) -> 0()
     , min(s(u), s(v)) -> s(min(u, v))
     , equal(0(), 0()) -> true()
     , equal(s(x), 0()) -> false()
     , equal(0(), s(y)) -> false()
     , equal(s(x), s(y)) -> equal(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: minus^#(x, y) -> c_0(cond^#(equal(min(x, y), y), x, y))
              , 2: cond^#(true(), x, y) -> c_1(minus^#(x, s(y)))
              , 3: min^#(0(), v) -> c_2()
              , 4: min^#(u, 0()) -> c_3()
              , 5: min^#(s(u), s(v)) -> c_4(min^#(u, v))
              , 6: equal^#(0(), 0()) -> c_5()
              , 7: equal^#(s(x), 0()) -> c_6()
              , 8: equal^#(0(), s(y)) -> c_7()
              , 9: equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{9}                                                       [   YES(?,O(n^1))    ]
                |
                |->{6}                                                   [   YES(?,O(n^2))    ]
                |
                |->{7}                                                   [   YES(?,O(n^3))    ]
                |
                `->{8}                                                   [   YES(?,O(n^2))    ]
             
             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                |->{3}                                                   [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [   YES(?,O(n^2))    ]
             
             ->{1,2}                                                     [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {1,2}: NA
             --------------
             
             The usable rules for this path are:
             
               {  min(0(), v) -> 0()
                , min(u, 0()) -> 0()
                , min(s(u), s(v)) -> s(min(u, v))
                , equal(0(), 0()) -> true()
                , equal(s(x), 0()) -> false()
                , equal(0(), s(y)) -> false()
                , equal(s(x), s(y)) -> equal(x, y)}
             
             The weight gap principle does not apply:
               The input cannot be shown compatible
             Complexity induced by the adequate RMI: MAYBE
             
             We have not generated a proof for the resulting sub-problem.
           
           * Path {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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {1}, Uargs(equal^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                equal(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                min(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                min^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [3 3 3]      [3 3 3]      [0]
                                [3 3 3]      [3 3 3]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                equal^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 3] x1 + [0]
                        [0 0 1]      [4]
                        [0 0 0]      [2]
                min^#(x1, x2) = [1 1 0] x1 + [1 0 2] x2 + [0]
                                [0 0 2]      [0 0 0]      [2]
                                [1 1 0]      [4 1 0]      [0]
                c_4(x1) = [1 2 0] x1 + [3]
                          [0 0 0]      [2]
                          [0 2 0]      [3]
           
           * Path {5}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {1}, Uargs(equal^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                equal(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                min(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                min^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                equal^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {min^#(0(), v) -> c_2()}
               Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 4 2] x1 + [0]
                        [0 0 2]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [2]
                      [0]
                min^#(x1, x2) = [2 2 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [2 0 0]      [0 4 0]      [0]
                c_2() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
           
           * Path {5}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {1}, Uargs(equal^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                equal(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                min(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                min^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                equal^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {min^#(u, 0()) -> c_3()}
               Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2 2] x1 + [2]
                        [0 1 2]      [0]
                        [0 0 0]      [3]
                0() = [2]
                      [2]
                      [2]
                min^#(x1, x2) = [4 0 0] x1 + [3 3 0] x2 + [0]
                                [2 0 2]      [2 0 0]      [0]
                                [0 4 0]      [2 2 3]      [0]
                c_3() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [2]
                          [0 0 0]      [2]
                          [0 0 0]      [7]
           
           * 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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                equal(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                min(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                min^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                equal^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [3 3 3]      [3 3 3]      [0]
                                  [3 3 3]      [3 3 3]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 0] x1 + [2]
                        [0 0 2]      [2]
                        [0 0 0]      [0]
                equal^#(x1, x2) = [1 0 0] x1 + [5 0 0] x2 + [0]
                                  [2 2 0]      [0 2 0]      [0]
                                  [4 0 0]      [0 2 0]      [0]
                c_8(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [7]
           
           * Path {9}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                equal(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                min(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                min^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                equal^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {equal^#(0(), 0()) -> c_5()}
               Weak Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 3 2] x1 + [0]
                        [0 1 1]      [0]
                        [0 0 0]      [0]
                0() = [2]
                      [0]
                      [0]
                equal^#(x1, x2) = [2 0 0] x1 + [2 0 0] x2 + [0]
                                  [0 4 0]      [0 0 0]      [2]
                                  [4 0 0]      [0 0 0]      [4]
                c_5() = [1]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [1]
                          [0 2 0]      [0]
           
           * Path {9}->{7}: YES(?,O(n^3))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                equal(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                min(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                min^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                equal^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^3))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {equal^#(s(x), 0()) -> c_6()}
               Weak Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 3 2] x1 + [2]
                        [0 1 2]      [0]
                        [0 0 1]      [2]
                0() = [0]
                      [0]
                      [0]
                equal^#(x1, x2) = [2 2 0] x1 + [0 1 0] x2 + [0]
                                  [0 0 0]      [0 0 2]      [0]
                                  [0 1 0]      [0 2 4]      [0]
                c_6() = [1]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [0]
                          [0 0 0]      [6]
           
           * Path {9}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                cond(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0 0 0]      [0]
                equal(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                min(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                              [0 0 0]      [0 0 0]      [0]
                              [0 0 0]      [0 0 0]      [0]
                true() = [0]
                         [0]
                         [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                0() = [0]
                      [0]
                      [0]
                false() = [0]
                          [0]
                          [0]
                minus^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                cond^#(x1, x2, x3) = [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                                     [0 0 0]      [0 0 0]      [0 0 0]      [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                min^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                [0 0 0]      [0 0 0]      [0]
                                [0 0 0]      [0 0 0]      [0]
                c_2() = [0]
                        [0]
                        [0]
                c_3() = [0]
                        [0]
                        [0]
                c_4(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                equal^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6() = [0]
                        [0]
                        [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {equal^#(0(), s(y)) -> c_7()}
               Weak Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 1 2] x1 + [0]
                        [0 1 0]      [2]
                        [0 0 0]      [0]
                0() = [0]
                      [2]
                      [0]
                equal^#(x1, x2) = [0 2 0] x1 + [1 0 2] x2 + [0]
                                  [7 1 0]      [4 0 0]      [0]
                                  [4 2 0]      [4 0 0]      [0]
                c_7() = [1]
                        [0]
                        [0]
                c_8(x1) = [1 0 0] x1 + [3]
                          [2 0 0]      [2]
                          [0 0 0]      [2]
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: minus^#(x, y) -> c_0(cond^#(equal(min(x, y), y), x, y))
              , 2: cond^#(true(), x, y) -> c_1(minus^#(x, s(y)))
              , 3: min^#(0(), v) -> c_2()
              , 4: min^#(u, 0()) -> c_3()
              , 5: min^#(s(u), s(v)) -> c_4(min^#(u, v))
              , 6: equal^#(0(), 0()) -> c_5()
              , 7: equal^#(s(x), 0()) -> c_6()
              , 8: equal^#(0(), s(y)) -> c_7()
              , 9: equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{9}                                                       [   YES(?,O(n^2))    ]
                |
                |->{6}                                                   [   YES(?,O(n^1))    ]
                |
                |->{7}                                                   [   YES(?,O(n^1))    ]
                |
                `->{8}                                                   [   YES(?,O(n^2))    ]
             
             ->{5}                                                       [   YES(?,O(n^2))    ]
                |
                |->{3}                                                   [   YES(?,O(n^2))    ]
                |
                `->{4}                                                   [   YES(?,O(n^2))    ]
             
             ->{1,2}                                                     [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1,2}: MAYBE
             -----------------
             
             The usable rules for this path are:
             
               {  min(0(), v) -> 0()
                , min(u, 0()) -> 0()
                , min(s(u), s(v)) -> s(min(u, v))
                , equal(0(), 0()) -> true()
                , equal(s(x), 0()) -> false()
                , equal(0(), s(y)) -> false()
                , equal(s(x), s(y)) -> equal(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:
                 {  minus^#(x, y) -> c_0(cond^#(equal(min(x, y), y), x, y))
                  , cond^#(true(), x, y) -> c_1(minus^#(x, s(y)))
                  , min(0(), v) -> 0()
                  , min(u, 0()) -> 0()
                  , min(s(u), s(v)) -> s(min(u, v))
                  , equal(0(), 0()) -> true()
                  , equal(s(x), 0()) -> false()
                  , equal(0(), s(y)) -> false()
                  , equal(s(x), s(y)) -> equal(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {1}, Uargs(equal^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                equal(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]
                true() = [0]
                         [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                min^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                [3 3]      [3 3]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7() = [0]
                        [0]
                c_8(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(u), s(v)) -> c_4(min^#(u, v))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 6] x1 + [2]
                        [0 1]      [0]
                min^#(x1, x2) = [1 2] x1 + [1 0] x2 + [0]
                                [0 0]      [0 0]      [4]
                c_4(x1) = [1 0] x1 + [1]
                          [0 0]      [3]
           
           * Path {5}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {1}, Uargs(equal^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                equal(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]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7() = [0]
                        [0]
                c_8(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(), v) -> c_2()}
               Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [0]
                        [0 1]      [2]
                0() = [2]
                      [2]
                min^#(x1, x2) = [3 2] x1 + [2 3] x2 + [0]
                                [3 3]      [0 1]      [0]
                c_2() = [1]
                        [0]
                c_4(x1) = [1 0] x1 + [3]
                          [0 0]      [7]
           
           * Path {5}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {1}, Uargs(equal^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                equal(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]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7() = [0]
                        [0]
                c_8(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^#(u, 0()) -> c_3()}
               Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 3] x1 + [2]
                        [0 1]      [2]
                0() = [2]
                      [0]
                min^#(x1, x2) = [2 0] x1 + [2 3] x2 + [0]
                                [0 2]      [3 2]      [0]
                c_3() = [1]
                        [0]
                c_4(x1) = [1 0] x1 + [7]
                          [0 0]      [7]
           
           * 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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                equal(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]
                true() = [0]
                         [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                equal^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                  [3 3]      [3 3]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                equal^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                                  [0 2]      [0 0]      [0]
                c_8(x1) = [1 2] x1 + [5]
                          [0 0]      [3]
           
           * Path {9}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                equal(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]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {equal^#(0(), 0()) -> c_5()}
               Weak Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 0] x1 + [2]
                        [0 1]      [0]
                0() = [2]
                      [2]
                equal^#(x1, x2) = [2 2] x1 + [2 0] x2 + [0]
                                  [2 0]      [0 0]      [4]
                c_5() = [1]
                        [0]
                c_8(x1) = [1 0] x1 + [7]
                          [0 0]      [7]
           
           * Path {9}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                equal(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]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {equal^#(s(x), 0()) -> c_6()}
               Weak Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 4] x1 + [2]
                        [0 0]      [0]
                0() = [0]
                      [0]
                equal^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
                                  [0 0]      [0 2]      [0]
                c_6() = [1]
                        [0]
                c_8(x1) = [1 2] x1 + [3]
                          [0 0]      [0]
           
           * Path {9}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                cond(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                equal(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]
                true() = [0]
                         [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                false() = [0]
                          [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                cond^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                     [0 0]      [0 0]      [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                min^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                [0 0]      [0 0]      [0]
                c_2() = [0]
                        [0]
                c_3() = [0]
                        [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                equal^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6() = [0]
                        [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {equal^#(0(), s(y)) -> c_7()}
               Weak Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 6] x1 + [2]
                        [0 1]      [2]
                0() = [2]
                      [2]
                equal^#(x1, x2) = [2 1] x1 + [0 1] x2 + [0]
                                  [1 2]      [2 0]      [0]
                c_7() = [1]
                        [0]
                c_8(x1) = [1 0] x1 + [7]
                          [0 0]      [7]
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: minus^#(x, y) -> c_0(cond^#(equal(min(x, y), y), x, y))
              , 2: cond^#(true(), x, y) -> c_1(minus^#(x, s(y)))
              , 3: min^#(0(), v) -> c_2()
              , 4: min^#(u, 0()) -> c_3()
              , 5: min^#(s(u), s(v)) -> c_4(min^#(u, v))
              , 6: equal^#(0(), 0()) -> c_5()
              , 7: equal^#(s(x), 0()) -> c_6()
              , 8: equal^#(0(), s(y)) -> c_7()
              , 9: equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{9}                                                       [   YES(?,O(n^1))    ]
                |
                |->{6}                                                   [   YES(?,O(n^1))    ]
                |
                |->{7}                                                   [   YES(?,O(n^1))    ]
                |
                `->{8}                                                   [   YES(?,O(n^1))    ]
             
             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                |->{3}                                                   [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [   YES(?,O(n^1))    ]
             
             ->{1,2}                                                     [       MAYBE        ]
             
           
         
         Sub-problems:
         -------------
           * Path {1,2}: MAYBE
             -----------------
             
             The usable rules for this path are:
             
               {  min(0(), v) -> 0()
                , min(u, 0()) -> 0()
                , min(s(u), s(v)) -> s(min(u, v))
                , equal(0(), 0()) -> true()
                , equal(s(x), 0()) -> false()
                , equal(0(), s(y)) -> false()
                , equal(s(x), s(y)) -> equal(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:
                 {  minus^#(x, y) -> c_0(cond^#(equal(min(x, y), y), x, y))
                  , cond^#(true(), x, y) -> c_1(minus^#(x, s(y)))
                  , min(0(), v) -> 0()
                  , min(u, 0()) -> 0()
                  , min(s(u), s(v)) -> s(min(u, v))
                  , equal(0(), 0()) -> true()
                  , equal(s(x), 0()) -> false()
                  , equal(0(), s(y)) -> false()
                  , equal(s(x), s(y)) -> equal(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {1}, Uargs(equal^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                equal(x1, x2) = [0] x1 + [0] x2 + [0]
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                false() = [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1(x1) = [0] x1 + [0]
                min^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7() = [0]
                c_8(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(u), s(v)) -> c_4(min^#(u, v))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {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_4(x1) = [1] x1 + [7]
           
           * Path {5}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {1}, Uargs(equal^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                equal(x1, x2) = [0] x1 + [0] x2 + [0]
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1(x1) = [0] x1 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7() = [0]
                c_8(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(), v) -> c_2()}
               Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                min^#(x1, x2) = [3] x1 + [3] x2 + [2]
                c_2() = [1]
                c_4(x1) = [1] x1 + [5]
           
           * Path {5}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {1}, Uargs(equal^#) = {},
                 Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                equal(x1, x2) = [0] x1 + [0] x2 + [0]
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1(x1) = [0] x1 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7() = [0]
                c_8(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^#(u, 0()) -> c_3()}
               Weak Rules: {min^#(s(u), s(v)) -> c_4(min^#(u, v))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(min^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                min^#(x1, x2) = [2] x1 + [2] x2 + [4]
                c_3() = [1]
                c_4(x1) = [1] x1 + [6]
           
           * 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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                equal(x1, x2) = [0] x1 + [0] x2 + [0]
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                s(x1) = [1] x1 + [0]
                0() = [0]
                false() = [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1(x1) = [0] x1 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                equal^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                equal^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_8(x1) = [1] x1 + [7]
           
           * Path {9}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                equal(x1, x2) = [0] x1 + [0] x2 + [0]
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1(x1) = [0] x1 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {equal^#(0(), 0()) -> c_5()}
               Weak Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [0]
                equal^#(x1, x2) = [2] x1 + [0] x2 + [4]
                c_5() = [1]
                c_8(x1) = [1] x1 + [3]
           
           * Path {9}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                equal(x1, x2) = [0] x1 + [0] x2 + [0]
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1(x1) = [0] x1 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {equal^#(s(x), 0()) -> c_6()}
               Weak Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                equal^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_6() = [1]
                c_8(x1) = [1] x1 + [7]
           
           * Path {9}->{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(minus) = {}, Uargs(cond) = {}, Uargs(equal) = {},
                 Uargs(min) = {}, Uargs(s) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(cond^#) = {}, Uargs(c_1) = {},
                 Uargs(min^#) = {}, Uargs(c_4) = {}, Uargs(equal^#) = {},
                 Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                cond(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                equal(x1, x2) = [0] x1 + [0] x2 + [0]
                min(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                false() = [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                cond^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_1(x1) = [0] x1 + [0]
                min^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                equal^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6() = [0]
                c_7() = [0]
                c_8(x1) = [1] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {equal^#(0(), s(y)) -> c_7()}
               Weak Rules: {equal^#(s(x), s(y)) -> c_8(equal^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(equal^#) = {}, Uargs(c_8) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                0() = [2]
                equal^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_7() = [1]
                c_8(x1) = [1] x1 + [7]
    
    4) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    5) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    6) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.