Problem AG01 innermost 4.30

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputAG01 innermost 4.30

stdout:

MAYBE

Problem:
 minus(x,0()) -> x
 minus(s(x),s(y)) -> minus(x,y)
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
 if_quot(true(),x,y) -> s(quot(minus(x,y),y))
 if_quot(false(),x,y) -> 0()

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputAG01 innermost 4.30

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputAG01 innermost 4.30

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  minus(x, 0()) -> x
     , minus(s(x), s(y)) -> minus(x, y)
     , le(0(), y) -> true()
     , le(s(x), 0()) -> false()
     , le(s(x), s(y)) -> le(x, y)
     , quot(x, s(y)) -> if_quot(le(s(y), x), x, s(y))
     , if_quot(true(), x, y) -> s(quot(minus(x, y), y))
     , if_quot(false(), x, y) -> 0()}

Proof Output:    
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: minus^#(x, 0()) -> c_0()
              , 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
              , 3: le^#(0(), y) -> c_2()
              , 4: le^#(s(x), 0()) -> c_3()
              , 5: le^#(s(x), s(y)) -> c_4(le^#(x, y))
              , 6: quot^#(x, s(y)) -> c_5(if_quot^#(le(s(y), x), x, s(y)))
              , 7: if_quot^#(true(), x, y) -> c_6(quot^#(minus(x, y), y))
              , 8: if_quot^#(false(), x, y) -> c_7()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6,7}                                                     [         NA         ]
                |
                `->{8}                                                   [         NA         ]
             
             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                |->{3}                                                   [   YES(?,O(n^2))    ]
                |
                `->{4}                                                   [   YES(?,O(n^3))    ]
             
             ->{2}                                                       [   YES(?,O(n^1))    ]
                |
                `->{1}                                                   [   YES(?,O(n^3))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {2}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
                 Uargs(le^#) = {}, Uargs(c_4) = {}, Uargs(quot^#) = {},
                 Uargs(c_5) = {}, Uargs(if_quot^#) = {}, Uargs(c_6) = {}
               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]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                le(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]
                false() = [0]
                          [0]
                          [0]
                quot(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]
                if_quot(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]
                minus^#(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_0() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                le^#(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]
                quot^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                if_quot^#(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_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7() = [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: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {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]
                minus^#(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_1(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [7]
           
           * Path {2}->{1}: 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(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
                 Uargs(le^#) = {}, Uargs(c_4) = {}, Uargs(quot^#) = {},
                 Uargs(c_5) = {}, Uargs(if_quot^#) = {}, Uargs(c_6) = {}
               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]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                le(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]
                false() = [0]
                          [0]
                          [0]
                quot(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]
                if_quot(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]
                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() = [0]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                le^#(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]
                quot^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                if_quot^#(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_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [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: {minus^#(x, 0()) -> c_0()}
               Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                s(x1) = [1 2 0] x1 + [2]
                        [0 1 3]      [2]
                        [0 0 1]      [2]
                minus^#(x1, x2) = [0 0 0] x1 + [0 2 2] x2 + [0]
                                  [0 0 2]      [2 2 0]      [0]
                                  [0 0 0]      [0 2 2]      [0]
                c_0() = [1]
                        [0]
                        [0]
                c_1(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [0]
                          [0 0 0]      [7]
           
           * 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(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
                 Uargs(le^#) = {}, Uargs(c_4) = {1}, Uargs(quot^#) = {},
                 Uargs(c_5) = {}, Uargs(if_quot^#) = {}, Uargs(c_6) = {}
               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]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                le(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]
                false() = [0]
                          [0]
                          [0]
                quot(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]
                if_quot(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]
                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() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                le^#(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]
                quot^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                if_quot^#(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_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7() = [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: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {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]
                le^#(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_4(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [7]
           
           * 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(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
                 Uargs(le^#) = {}, Uargs(c_4) = {1}, Uargs(quot^#) = {},
                 Uargs(c_5) = {}, Uargs(if_quot^#) = {}, Uargs(c_6) = {}
               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]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                le(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]
                false() = [0]
                          [0]
                          [0]
                quot(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]
                if_quot(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]
                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() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                le^#(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]
                quot^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                if_quot^#(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_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7() = [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: {le^#(0(), y) -> c_2()}
               Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                s(x1) = [1 4 2] x1 + [0]
                        [0 0 2]      [0]
                        [0 0 1]      [0]
                le^#(x1, x2) = [2 2 2] x1 + [2 0 0] x2 + [0]
                               [2 2 2]      [0 0 4]      [0]
                               [2 2 2]      [0 0 0]      [0]
                c_2() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
           
           * Path {5}->{4}: 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(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
                 Uargs(le^#) = {}, Uargs(c_4) = {1}, Uargs(quot^#) = {},
                 Uargs(c_5) = {}, Uargs(if_quot^#) = {}, Uargs(c_6) = {}
               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]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                le(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]
                false() = [0]
                          [0]
                          [0]
                quot(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]
                if_quot(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]
                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() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                le^#(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]
                quot^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                if_quot^#(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_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [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: {le^#(s(x), 0()) -> c_3()}
               Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 3 2] x1 + [2]
                        [0 1 2]      [0]
                        [0 0 1]      [2]
                le^#(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_3() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [0]
                          [0 0 0]      [6]
           
           * Path {6,7}: NA
             --------------
             
             The usable rules for this path are:
             
               {  minus(x, 0()) -> x
                , minus(s(x), s(y)) -> minus(x, y)
                , le(0(), y) -> true()
                , le(s(x), 0()) -> false()
                , le(s(x), s(y)) -> le(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 {6,7}->{8}: NA
             -------------------
             
             The usable rules for this path are:
             
               {  minus(x, 0()) -> x
                , minus(s(x), s(y)) -> minus(x, y)
                , le(0(), y) -> true()
                , le(s(x), 0()) -> false()
                , le(s(x), s(y)) -> le(x, y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
                 Uargs(le^#) = {}, Uargs(c_4) = {}, Uargs(quot^#) = {1},
                 Uargs(c_5) = {1}, Uargs(if_quot^#) = {1}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [3 2 2] x1 + [0 0 2] x2 + [3]
                                [0 2 2]      [0 0 0]      [1]
                                [0 0 2]      [1 1 2]      [3]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 0 3] x1 + [0]
                        [0 1 3]      [0]
                        [0 0 1]      [2]
                le(x1, x2) = [0 0 1] x1 + [0 0 0] x2 + [2]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                true() = [1]
                         [0]
                         [0]
                false() = [1]
                          [0]
                          [0]
                quot(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]
                if_quot(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]
                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() = [0]
                        [0]
                        [0]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                le^#(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]
                quot^#(x1, x2) = [3 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                if_quot^#(x1, x2, x3) = [3 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_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_7() = [0]
                        [0]
                        [0]
             Complexity induced by the adequate RMI: YES(?,O(n^3))
             
             We have not generated a proof for the resulting sub-problem.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: minus^#(x, 0()) -> c_0()
              , 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
              , 3: le^#(0(), y) -> c_2()
              , 4: le^#(s(x), 0()) -> c_3()
              , 5: le^#(s(x), s(y)) -> c_4(le^#(x, y))
              , 6: quot^#(x, s(y)) -> c_5(if_quot^#(le(s(y), x), x, s(y)))
              , 7: if_quot^#(true(), x, y) -> c_6(quot^#(minus(x, y), y))
              , 8: if_quot^#(false(), x, y) -> c_7()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6,7}                                                     [       MAYBE        ]
                |
                `->{8}                                                   [         NA         ]
             
             ->{5}                                                       [   YES(?,O(n^2))    ]
                |
                |->{3}                                                   [   YES(?,O(n^2))    ]
                |
                `->{4}                                                   [   YES(?,O(n^1))    ]
             
             ->{2}                                                       [   YES(?,O(n^2))    ]
                |
                `->{1}                                                   [   YES(?,O(n^2))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {2}: YES(?,O(n^2))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
                 Uargs(le^#) = {}, Uargs(c_4) = {}, Uargs(quot^#) = {},
                 Uargs(c_5) = {}, Uargs(if_quot^#) = {}, Uargs(c_6) = {}
               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]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if_quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                      [0 0]      [0 0]      [0 0]      [0]
                minus^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                  [3 3]      [3 3]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                le^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if_quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                        [0 0]      [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [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: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                minus^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                                  [0 2]      [0 0]      [0]
                c_1(x1) = [1 2] x1 + [5]
                          [0 0]      [3]
           
           * Path {2}->{1}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
                 Uargs(le^#) = {}, Uargs(c_4) = {}, Uargs(quot^#) = {},
                 Uargs(c_5) = {}, Uargs(if_quot^#) = {}, Uargs(c_6) = {}
               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]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if_quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                      [0 0]      [0 0]      [0 0]      [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                le^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if_quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                        [0 0]      [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [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: {minus^#(x, 0()) -> c_0()}
               Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [0]
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [0]
                minus^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
                                  [0 0]      [4 1]      [0]
                c_0() = [1]
                        [0]
                c_1(x1) = [1 0] x1 + [6]
                          [0 0]      [7]
           
           * 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(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
                 Uargs(le^#) = {}, Uargs(c_4) = {1}, Uargs(quot^#) = {},
                 Uargs(c_5) = {}, Uargs(if_quot^#) = {}, Uargs(c_6) = {}
               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]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if_quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                      [0 0]      [0 0]      [0 0]      [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                le^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if_quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                        [0 0]      [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [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: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                le^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                               [0 2]      [0 0]      [0]
                c_4(x1) = [1 2] x1 + [5]
                          [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(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
                 Uargs(le^#) = {}, Uargs(c_4) = {1}, Uargs(quot^#) = {},
                 Uargs(c_5) = {}, Uargs(if_quot^#) = {}, Uargs(c_6) = {}
               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]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if_quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                      [0 0]      [0 0]      [0 0]      [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                le^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if_quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                        [0 0]      [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [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: {le^#(0(), y) -> c_2()}
               Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [0]
                le^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
                               [4 1]      [2 0]      [0]
                c_2() = [1]
                        [0]
                c_4(x1) = [1 0] x1 + [3]
                          [0 0]      [7]
           
           * 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(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
                 Uargs(le^#) = {}, Uargs(c_4) = {1}, Uargs(quot^#) = {},
                 Uargs(c_5) = {}, Uargs(if_quot^#) = {}, Uargs(c_6) = {}
               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]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if_quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                      [0 0]      [0 0]      [0 0]      [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                le^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if_quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                        [0 0]      [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {le^#(s(x), 0()) -> c_3()}
               Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [0]
                s(x1) = [1 4] x1 + [2]
                        [0 0]      [0]
                le^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
                               [0 0]      [0 2]      [0]
                c_3() = [1]
                        [0]
                c_4(x1) = [1 2] x1 + [3]
                          [0 0]      [0]
           
           * Path {6,7}: MAYBE
             -----------------
             
             The usable rules for this path are:
             
               {  minus(x, 0()) -> x
                , minus(s(x), s(y)) -> minus(x, y)
                , le(0(), y) -> true()
                , le(s(x), 0()) -> false()
                , le(s(x), s(y)) -> le(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:
                 {  quot^#(x, s(y)) -> c_5(if_quot^#(le(s(y), x), x, s(y)))
                  , if_quot^#(true(), x, y) -> c_6(quot^#(minus(x, y), y))
                  , minus(x, 0()) -> x
                  , minus(s(x), s(y)) -> minus(x, y)
                  , le(0(), y) -> true()
                  , le(s(x), 0()) -> false()
                  , le(s(x), s(y)) -> le(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {6,7}->{8}: NA
             -------------------
             
             The usable rules for this path are:
             
               {  minus(x, 0()) -> x
                , minus(s(x), s(y)) -> minus(x, y)
                , le(0(), y) -> true()
                , le(s(x), 0()) -> false()
                , le(s(x), s(y)) -> le(x, y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
                 Uargs(le^#) = {}, Uargs(c_4) = {}, Uargs(quot^#) = {1},
                 Uargs(c_5) = {1}, Uargs(if_quot^#) = {1}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [2 0] x1 + [0 0] x2 + [3]
                                [0 2]      [0 0]      [3]
                0() = [0]
                      [0]
                s(x1) = [1 0] x1 + [2]
                        [0 1]      [0]
                le(x1, x2) = [2 0] x1 + [0 0] x2 + [1]
                             [0 0]      [0 0]      [3]
                true() = [0]
                         [1]
                false() = [0]
                          [1]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if_quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                      [0 0]      [0 0]      [0 0]      [0]
                minus^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                le^#(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]
                quot^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                if_quot^#(x1, x2, x3) = [3 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                        [0 0]      [0 0]      [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_7() = [0]
                        [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: minus^#(x, 0()) -> c_0()
              , 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
              , 3: le^#(0(), y) -> c_2()
              , 4: le^#(s(x), 0()) -> c_3()
              , 5: le^#(s(x), s(y)) -> c_4(le^#(x, y))
              , 6: quot^#(x, s(y)) -> c_5(if_quot^#(le(s(y), x), x, s(y)))
              , 7: if_quot^#(true(), x, y) -> c_6(quot^#(minus(x, y), y))
              , 8: if_quot^#(false(), x, y) -> c_7()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6,7}                                                     [       MAYBE        ]
                |
                `->{8}                                                   [         NA         ]
             
             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                |->{3}                                                   [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [   YES(?,O(n^1))    ]
             
             ->{2}                                                       [   YES(?,O(n^1))    ]
                |
                `->{1}                                                   [   YES(?,O(n^1))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {2}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
                 Uargs(le^#) = {}, Uargs(c_4) = {}, Uargs(quot^#) = {},
                 Uargs(c_5) = {}, Uargs(if_quot^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                if_quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                minus^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_0() = [0]
                c_1(x1) = [1] x1 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5(x1) = [0] x1 + [0]
                if_quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [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: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                minus^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_1(x1) = [1] x1 + [7]
           
           * Path {2}->{1}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1},
                 Uargs(le^#) = {}, Uargs(c_4) = {}, Uargs(quot^#) = {},
                 Uargs(c_5) = {}, Uargs(if_quot^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                if_quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [1] x1 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5(x1) = [0] x1 + [0]
                if_quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [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: {minus^#(x, 0()) -> c_0()}
               Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                minus^#(x1, x2) = [2] x1 + [2] x2 + [4]
                c_0() = [1]
                c_1(x1) = [1] x1 + [7]
           
           * 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(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
                 Uargs(le^#) = {}, Uargs(c_4) = {1}, Uargs(quot^#) = {},
                 Uargs(c_5) = {}, Uargs(if_quot^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                if_quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                le^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5(x1) = [0] x1 + [0]
                if_quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [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: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                le^#(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(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
                 Uargs(le^#) = {}, Uargs(c_4) = {1}, Uargs(quot^#) = {},
                 Uargs(c_5) = {}, Uargs(if_quot^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                if_quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5(x1) = [0] x1 + [0]
                if_quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [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: {le^#(0(), y) -> c_2()}
               Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                le^#(x1, x2) = [2] x1 + [0] x2 + [4]
                c_2() = [1]
                c_4(x1) = [1] x1 + [2]
           
           * 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(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
                 Uargs(le^#) = {}, Uargs(c_4) = {1}, Uargs(quot^#) = {},
                 Uargs(c_5) = {}, Uargs(if_quot^#) = {}, Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                if_quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5(x1) = [0] x1 + [0]
                if_quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [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: {le^#(s(x), 0()) -> c_3()}
               Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                le^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_3() = [1]
                c_4(x1) = [1] x1 + [7]
           
           * Path {6,7}: MAYBE
             -----------------
             
             The usable rules for this path are:
             
               {  minus(x, 0()) -> x
                , minus(s(x), s(y)) -> minus(x, y)
                , le(0(), y) -> true()
                , le(s(x), 0()) -> false()
                , le(s(x), s(y)) -> le(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:
                 {  quot^#(x, s(y)) -> c_5(if_quot^#(le(s(y), x), x, s(y)))
                  , if_quot^#(true(), x, y) -> c_6(quot^#(minus(x, y), y))
                  , minus(x, 0()) -> x
                  , minus(s(x), s(y)) -> minus(x, y)
                  , le(0(), y) -> true()
                  , le(s(x), 0()) -> false()
                  , le(s(x), s(y)) -> le(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {6,7}->{8}: NA
             -------------------
             
             The usable rules for this path are:
             
               {  minus(x, 0()) -> x
                , minus(s(x), s(y)) -> minus(x, y)
                , le(0(), y) -> true()
                , le(s(x), 0()) -> false()
                , le(s(x), s(y)) -> le(x, y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {},
                 Uargs(le^#) = {}, Uargs(c_4) = {}, Uargs(quot^#) = {1},
                 Uargs(c_5) = {1}, Uargs(if_quot^#) = {1}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [2] x1 + [0] x2 + [1]
                0() = [0]
                s(x1) = [1] x1 + [2]
                le(x1, x2) = [0] x1 + [2] x2 + [1]
                true() = [0]
                false() = [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                if_quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0() = [0]
                c_1(x1) = [0] x1 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [3] x1 + [0] x2 + [0]
                c_5(x1) = [1] x1 + [0]
                if_quot^#(x1, x2, x3) = [3] x1 + [0] x2 + [0] x3 + [0]
                c_6(x1) = [1] x1 + [0]
                c_7() = [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
    
    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
InputAG01 innermost 4.30

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputAG01 innermost 4.30

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  minus(x, 0()) -> x
     , minus(s(x), s(y)) -> minus(x, y)
     , le(0(), y) -> true()
     , le(s(x), 0()) -> false()
     , le(s(x), s(y)) -> le(x, y)
     , quot(x, s(y)) -> if_quot(le(s(y), x), x, s(y))
     , if_quot(true(), x, y) -> s(quot(minus(x, y), y))
     , if_quot(false(), x, y) -> 0()}

Proof Output:    
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: minus^#(x, 0()) -> c_0(x)
              , 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
              , 3: le^#(0(), y) -> c_2()
              , 4: le^#(s(x), 0()) -> c_3()
              , 5: le^#(s(x), s(y)) -> c_4(le^#(x, y))
              , 6: quot^#(x, s(y)) -> c_5(if_quot^#(le(s(y), x), x, s(y)))
              , 7: if_quot^#(true(), x, y) -> c_6(quot^#(minus(x, y), y))
              , 8: if_quot^#(false(), x, y) -> c_7()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6,7}                                                     [         NA         ]
                |
                `->{8}                                                   [         NA         ]
             
             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                |->{3}                                                   [   YES(?,O(n^2))    ]
                |
                `->{4}                                                   [   YES(?,O(n^3))    ]
             
             ->{2}                                                       [   YES(?,O(n^1))    ]
                |
                `->{1}                                                   [   YES(?,O(n^2))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {2}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {1}, Uargs(le^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_5) = {}, Uargs(if_quot^#) = {},
                 Uargs(c_6) = {}
               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]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                le(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]
                false() = [0]
                          [0]
                          [0]
                quot(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]
                if_quot(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]
                minus^#(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_0(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                le^#(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]
                quot^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                if_quot^#(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_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7() = [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: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {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]
                minus^#(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_1(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [7]
           
           * Path {2}->{1}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {1}, Uargs(le^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_5) = {}, Uargs(if_quot^#) = {},
                 Uargs(c_6) = {}
               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]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                le(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]
                false() = [0]
                          [0]
                          [0]
                quot(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]
                if_quot(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]
                minus^#(x1, x2) = [3 3 3] x1 + [0 0 0] x2 + [0]
                                  [0 0 0]      [0 0 0]      [0]
                                  [0 0 0]      [0 0 0]      [0]
                c_0(x1) = [1 1 1] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                le^#(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]
                quot^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                if_quot^#(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_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7() = [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: {minus^#(x, 0()) -> c_0(x)}
               Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                s(x1) = [1 2 1] x1 + [0]
                        [0 0 4]      [2]
                        [0 0 1]      [2]
                minus^#(x1, x2) = [2 0 2] x1 + [2 0 2] x2 + [0]
                                  [0 2 0]      [0 3 2]      [0]
                                  [4 2 2]      [5 2 0]      [0]
                c_0(x1) = [0 0 0] x1 + [1]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_1(x1) = [1 0 0] x1 + [6]
                          [0 0 0]      [3]
                          [2 3 0]      [7]
           
           * 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(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1},
                 Uargs(quot^#) = {}, Uargs(c_5) = {}, Uargs(if_quot^#) = {},
                 Uargs(c_6) = {}
               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]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                le(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]
                false() = [0]
                          [0]
                          [0]
                quot(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]
                if_quot(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]
                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]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                le^#(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]
                quot^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                if_quot^#(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_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7() = [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: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {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]
                le^#(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_4(x1) = [1 0 0] x1 + [7]
                          [0 0 0]      [7]
                          [0 0 0]      [7]
           
           * 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(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1},
                 Uargs(quot^#) = {}, Uargs(c_5) = {}, Uargs(if_quot^#) = {},
                 Uargs(c_6) = {}
               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]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                le(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]
                false() = [0]
                          [0]
                          [0]
                quot(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]
                if_quot(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]
                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]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                le^#(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]
                quot^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                if_quot^#(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_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7() = [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: {le^#(0(), y) -> c_2()}
               Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                      [2]
                s(x1) = [1 4 2] x1 + [0]
                        [0 0 2]      [0]
                        [0 0 1]      [0]
                le^#(x1, x2) = [2 2 2] x1 + [2 0 0] x2 + [0]
                               [2 2 2]      [0 0 4]      [0]
                               [2 2 2]      [0 0 0]      [0]
                c_2() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
           
           * Path {5}->{4}: 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(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1},
                 Uargs(quot^#) = {}, Uargs(c_5) = {}, Uargs(if_quot^#) = {},
                 Uargs(c_6) = {}
               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]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [0 0 0] x1 + [0]
                        [0 0 0]      [0]
                        [0 0 0]      [0]
                le(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]
                false() = [0]
                          [0]
                          [0]
                quot(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]
                if_quot(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]
                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]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                le^#(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]
                quot^#(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(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                if_quot^#(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_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [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: {le^#(s(x), 0()) -> c_3()}
               Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 3 2] x1 + [2]
                        [0 1 2]      [0]
                        [0 0 1]      [2]
                le^#(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_3() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [0]
                          [0 0 0]      [6]
           
           * Path {6,7}: NA
             --------------
             
             The usable rules for this path are:
             
               {  minus(x, 0()) -> x
                , minus(s(x), s(y)) -> minus(x, y)
                , le(0(), y) -> true()
                , le(s(x), 0()) -> false()
                , le(s(x), s(y)) -> le(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 {6,7}->{8}: NA
             -------------------
             
             The usable rules for this path are:
             
               {  minus(x, 0()) -> x
                , minus(s(x), s(y)) -> minus(x, y)
                , le(0(), y) -> true()
                , le(s(x), 0()) -> false()
                , le(s(x), s(y)) -> le(x, y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {1}, Uargs(s) = {}, Uargs(le) = {2},
                 Uargs(quot) = {}, Uargs(if_quot) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(le^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {1}, Uargs(c_5) = {1},
                 Uargs(if_quot^#) = {1, 2}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [2 0 3] x1 + [0 0 1] x2 + [3]
                                [0 2 0]      [0 0 0]      [1]
                                [0 0 1]      [0 0 3]      [3]
                0() = [0]
                      [0]
                      [0]
                s(x1) = [1 0 0] x1 + [0]
                        [0 1 0]      [0]
                        [0 0 1]      [2]
                le(x1, x2) = [0 0 1] x1 + [1 0 0] x2 + [2]
                             [0 0 0]      [0 0 0]      [0]
                             [0 0 0]      [0 0 0]      [0]
                true() = [1]
                         [0]
                         [0]
                false() = [1]
                          [0]
                          [0]
                quot(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]
                if_quot(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]
                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]
                c_1(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                le^#(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]
                quot^#(x1, x2) = [3 0 0] x1 + [0 0 0] x2 + [0]
                                 [0 0 0]      [0 0 0]      [0]
                                 [0 0 0]      [0 0 0]      [0]
                c_5(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                if_quot^#(x1, x2, x3) = [3 0 0] x1 + [3 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_6(x1) = [1 0 0] x1 + [0]
                          [0 1 0]      [0]
                          [0 0 1]      [0]
                c_7() = [0]
                        [0]
                        [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: minus^#(x, 0()) -> c_0(x)
              , 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
              , 3: le^#(0(), y) -> c_2()
              , 4: le^#(s(x), 0()) -> c_3()
              , 5: le^#(s(x), s(y)) -> c_4(le^#(x, y))
              , 6: quot^#(x, s(y)) -> c_5(if_quot^#(le(s(y), x), x, s(y)))
              , 7: if_quot^#(true(), x, y) -> c_6(quot^#(minus(x, y), y))
              , 8: if_quot^#(false(), x, y) -> c_7()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6,7}                                                     [       MAYBE        ]
                |
                `->{8}                                                   [         NA         ]
             
             ->{5}                                                       [   YES(?,O(n^2))    ]
                |
                |->{3}                                                   [   YES(?,O(n^2))    ]
                |
                `->{4}                                                   [   YES(?,O(n^1))    ]
             
             ->{2}                                                       [   YES(?,O(n^2))    ]
                |
                `->{1}                                                   [   YES(?,O(n^2))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {2}: YES(?,O(n^2))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {1}, Uargs(le^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_5) = {}, Uargs(if_quot^#) = {},
                 Uargs(c_6) = {}
               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]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if_quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                      [0 0]      [0 0]      [0 0]      [0]
                minus^#(x1, x2) = [3 3] x1 + [3 3] x2 + [0]
                                  [3 3]      [3 3]      [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                le^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if_quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                        [0 0]      [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [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: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                minus^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                                  [0 2]      [0 0]      [0]
                c_1(x1) = [1 2] x1 + [5]
                          [0 0]      [3]
           
           * Path {2}->{1}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {1}, Uargs(le^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_5) = {}, Uargs(if_quot^#) = {},
                 Uargs(c_6) = {}
               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]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if_quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                      [0 0]      [0 0]      [0 0]      [0]
                minus^#(x1, x2) = [3 3] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                c_0(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                le^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if_quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                        [0 0]      [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [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: {minus^#(x, 0()) -> c_0(x)}
               Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [2]
                s(x1) = [1 2] x1 + [0]
                        [0 1]      [0]
                minus^#(x1, x2) = [2 2] x1 + [0 2] x2 + [0]
                                  [4 1]      [3 2]      [0]
                c_0(x1) = [0 0] x1 + [1]
                          [0 1]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [2 0]      [0]
           
           * 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(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1},
                 Uargs(quot^#) = {}, Uargs(c_5) = {}, Uargs(if_quot^#) = {},
                 Uargs(c_6) = {}
               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]
                0() = [0]
                      [0]
                s(x1) = [1 2] x1 + [0]
                        [0 0]      [0]
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if_quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                      [0 0]      [0 0]      [0 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]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                le^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if_quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                        [0 0]      [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [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: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                le^#(x1, x2) = [4 1] x1 + [1 2] x2 + [0]
                               [0 2]      [0 0]      [0]
                c_4(x1) = [1 2] x1 + [5]
                          [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(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1},
                 Uargs(quot^#) = {}, Uargs(c_5) = {}, Uargs(if_quot^#) = {},
                 Uargs(c_6) = {}
               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]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if_quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                      [0 0]      [0 0]      [0 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]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                le^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if_quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                        [0 0]      [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [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: {le^#(0(), y) -> c_2()}
               Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [0]
                le^#(x1, x2) = [3 3] x1 + [4 0] x2 + [0]
                               [4 1]      [2 0]      [0]
                c_2() = [1]
                        [0]
                c_4(x1) = [1 0] x1 + [3]
                          [0 0]      [7]
           
           * 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(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1},
                 Uargs(quot^#) = {}, Uargs(c_5) = {}, Uargs(if_quot^#) = {},
                 Uargs(c_6) = {}
               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]
                0() = [0]
                      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                le(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                             [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if_quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                      [0 0]      [0 0]      [0 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]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                le^#(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]
                quot^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if_quot^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                        [0 0]      [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7() = [0]
                        [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {le^#(s(x), 0()) -> c_3()}
               Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [0]
                s(x1) = [1 4] x1 + [2]
                        [0 0]      [0]
                le^#(x1, x2) = [2 0] x1 + [2 0] x2 + [0]
                               [0 0]      [0 2]      [0]
                c_3() = [1]
                        [0]
                c_4(x1) = [1 2] x1 + [3]
                          [0 0]      [0]
           
           * Path {6,7}: MAYBE
             -----------------
             
             The usable rules for this path are:
             
               {  minus(x, 0()) -> x
                , minus(s(x), s(y)) -> minus(x, y)
                , le(0(), y) -> true()
                , le(s(x), 0()) -> false()
                , le(s(x), s(y)) -> le(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:
                 {  quot^#(x, s(y)) -> c_5(if_quot^#(le(s(y), x), x, s(y)))
                  , if_quot^#(true(), x, y) -> c_6(quot^#(minus(x, y), y))
                  , minus(x, 0()) -> x
                  , minus(s(x), s(y)) -> minus(x, y)
                  , le(0(), y) -> true()
                  , le(s(x), 0()) -> false()
                  , le(s(x), s(y)) -> le(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {6,7}->{8}: NA
             -------------------
             
             The usable rules for this path are:
             
               {  minus(x, 0()) -> x
                , minus(s(x), s(y)) -> minus(x, y)
                , le(0(), y) -> true()
                , le(s(x), 0()) -> false()
                , le(s(x), s(y)) -> le(x, y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {1}, Uargs(s) = {}, Uargs(le) = {2},
                 Uargs(quot) = {}, Uargs(if_quot) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(le^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {1}, Uargs(c_5) = {1},
                 Uargs(if_quot^#) = {1, 2}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [2 0] x1 + [0 0] x2 + [3]
                                [0 2]      [0 0]      [3]
                0() = [0]
                      [0]
                s(x1) = [1 0] x1 + [1]
                        [0 1]      [0]
                le(x1, x2) = [0 0] x1 + [2 0] x2 + [1]
                             [0 0]      [0 0]      [3]
                true() = [0]
                         [1]
                false() = [0]
                          [1]
                quot(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                if_quot(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                      [0 0]      [0 0]      [0 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]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                le^#(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]
                quot^#(x1, x2) = [3 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                c_5(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                if_quot^#(x1, x2, x3) = [3 0] x1 + [3 0] x2 + [0 0] x3 + [0]
                                        [0 0]      [0 0]      [0 0]      [0]
                c_6(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                c_7() = [0]
                        [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: minus^#(x, 0()) -> c_0(x)
              , 2: minus^#(s(x), s(y)) -> c_1(minus^#(x, y))
              , 3: le^#(0(), y) -> c_2()
              , 4: le^#(s(x), 0()) -> c_3()
              , 5: le^#(s(x), s(y)) -> c_4(le^#(x, y))
              , 6: quot^#(x, s(y)) -> c_5(if_quot^#(le(s(y), x), x, s(y)))
              , 7: if_quot^#(true(), x, y) -> c_6(quot^#(minus(x, y), y))
              , 8: if_quot^#(false(), x, y) -> c_7()}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6,7}                                                     [       MAYBE        ]
                |
                `->{8}                                                   [         NA         ]
             
             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                |->{3}                                                   [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [   YES(?,O(n^1))    ]
             
             ->{2}                                                       [   YES(?,O(n^1))    ]
                |
                `->{1}                                                   [   YES(?,O(n^1))    ]
             
           
         
         Sub-problems:
         -------------
           * Path {2}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {1}, Uargs(le^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_5) = {}, Uargs(if_quot^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                if_quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                minus^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1(x1) = [1] x1 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5(x1) = [0] x1 + [0]
                if_quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [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: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                minus^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_1(x1) = [1] x1 + [7]
           
           * Path {2}->{1}: YES(?,O(n^1))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {}, Uargs(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {1}, Uargs(le^#) = {}, Uargs(c_4) = {},
                 Uargs(quot^#) = {}, Uargs(c_5) = {}, Uargs(if_quot^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                if_quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                minus^#(x1, x2) = [3] x1 + [0] x2 + [0]
                c_0(x1) = [1] x1 + [0]
                c_1(x1) = [1] x1 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5(x1) = [0] x1 + [0]
                if_quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [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: {minus^#(x, 0()) -> c_0(x)}
               Weak Rules: {minus^#(s(x), s(y)) -> c_1(minus^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                s(x1) = [1] x1 + [2]
                minus^#(x1, x2) = [2] x1 + [2] x2 + [2]
                c_0(x1) = [0] x1 + [1]
                c_1(x1) = [1] x1 + [5]
           
           * 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(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1},
                 Uargs(quot^#) = {}, Uargs(c_5) = {}, Uargs(if_quot^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                if_quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                le^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5(x1) = [0] x1 + [0]
                if_quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [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: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                le^#(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(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1},
                 Uargs(quot^#) = {}, Uargs(c_5) = {}, Uargs(if_quot^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                if_quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5(x1) = [0] x1 + [0]
                if_quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [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: {le^#(0(), y) -> c_2()}
               Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                le^#(x1, x2) = [2] x1 + [0] x2 + [4]
                c_2() = [1]
                c_4(x1) = [1] x1 + [2]
           
           * 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(s) = {}, Uargs(le) = {}, Uargs(quot) = {},
                 Uargs(if_quot) = {}, Uargs(minus^#) = {}, Uargs(c_0) = {},
                 Uargs(c_1) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1},
                 Uargs(quot^#) = {}, Uargs(c_5) = {}, Uargs(if_quot^#) = {},
                 Uargs(c_6) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                le(x1, x2) = [0] x1 + [0] x2 + [0]
                true() = [0]
                false() = [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                if_quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                quot^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5(x1) = [0] x1 + [0]
                if_quot^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7() = [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: {le^#(s(x), 0()) -> c_3()}
               Weak Rules: {le^#(s(x), s(y)) -> c_4(le^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(le^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                le^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_3() = [1]
                c_4(x1) = [1] x1 + [7]
           
           * Path {6,7}: MAYBE
             -----------------
             
             The usable rules for this path are:
             
               {  minus(x, 0()) -> x
                , minus(s(x), s(y)) -> minus(x, y)
                , le(0(), y) -> true()
                , le(s(x), 0()) -> false()
                , le(s(x), s(y)) -> le(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:
                 {  quot^#(x, s(y)) -> c_5(if_quot^#(le(s(y), x), x, s(y)))
                  , if_quot^#(true(), x, y) -> c_6(quot^#(minus(x, y), y))
                  , minus(x, 0()) -> x
                  , minus(s(x), s(y)) -> minus(x, y)
                  , le(0(), y) -> true()
                  , le(s(x), 0()) -> false()
                  , le(s(x), s(y)) -> le(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {6,7}->{8}: NA
             -------------------
             
             The usable rules for this path are:
             
               {  minus(x, 0()) -> x
                , minus(s(x), s(y)) -> minus(x, y)
                , le(0(), y) -> true()
                , le(s(x), 0()) -> false()
                , le(s(x), s(y)) -> le(x, y)}
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(minus) = {1}, Uargs(s) = {}, Uargs(le) = {2},
                 Uargs(quot) = {}, Uargs(if_quot) = {}, Uargs(minus^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(le^#) = {},
                 Uargs(c_4) = {}, Uargs(quot^#) = {1}, Uargs(c_5) = {1},
                 Uargs(if_quot^#) = {1, 2}, Uargs(c_6) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                minus(x1, x2) = [1] x1 + [0] x2 + [1]
                0() = [0]
                s(x1) = [1] x1 + [2]
                le(x1, x2) = [0] x1 + [2] x2 + [1]
                true() = [0]
                false() = [0]
                quot(x1, x2) = [0] x1 + [0] x2 + [0]
                if_quot(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                minus^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                le^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                quot^#(x1, x2) = [3] x1 + [0] x2 + [0]
                c_5(x1) = [1] x1 + [0]
                if_quot^#(x1, x2, x3) = [3] x1 + [3] x2 + [0] x3 + [0]
                c_6(x1) = [1] x1 + [0]
                c_7() = [0]
             Complexity induced by the adequate RMI: YES(?,O(n^1))
             
             We have not generated a proof for the resulting sub-problem.
    
    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.