Problem TCT 09 insertsort

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputTCT 09 insertsort

stdout:

MAYBE

Problem:
 ite(tt(),x,y) -> x
 ite(ff(),x,y) -> y
 lt(0(),s(y)) -> tt()
 lt(x,0()) -> ff()
 lt(s(x),s(y)) -> lt(x,y)
 insert(a,nil()) -> cons(a,nil())
 insert(a,cons(b,l)) -> ite(lt(a,b),cons(a,cons(b,l)),cons(b,insert(a,l)))
 sort(nil()) -> nil()
 sort(cons(a,l)) -> insert(a,sort(l))

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputTCT 09 insertsort

stdout:

MAYBE

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputTCT 09 insertsort

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  ite(tt(), x, y) -> x
     , ite(ff(), x, y) -> y
     , lt(0(), s(y)) -> tt()
     , lt(x, 0()) -> ff()
     , lt(s(x), s(y)) -> lt(x, y)
     , insert(a, nil()) -> cons(a, nil())
     , insert(a, cons(b, l)) ->
       ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
     , sort(nil()) -> nil()
     , sort(cons(a, l)) -> insert(a, sort(l))}

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: ite^#(tt(), x, y) -> c_0()
              , 2: ite^#(ff(), x, y) -> c_1()
              , 3: lt^#(0(), s(y)) -> c_2()
              , 4: lt^#(x, 0()) -> c_3()
              , 5: lt^#(s(x), s(y)) -> c_4(lt^#(x, y))
              , 6: insert^#(a, nil()) -> c_5()
              , 7: insert^#(a, cons(b, l)) ->
                   c_6(ite^#(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l))))
              , 8: sort^#(nil()) -> c_7()
              , 9: sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{9}                                                       [     inherited      ]
                |
                |->{6}                                                   [       MAYBE        ]
                |
                `->{7}                                                   [     inherited      ]
                    |
                    |->{1}                                               [         NA         ]
                    |
                    `->{2}                                               [         NA         ]
             
             ->{8}                                                       [    YES(?,O(1))     ]
             
             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                |->{3}                                                   [   YES(?,O(n^2))    ]
                |
                `->{4}                                                   [   YES(?,O(n^3))    ]
             
           
         
         Sub-problems:
         -------------
           * 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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(lt^#) = {}, Uargs(c_4) = {1}, Uargs(insert^#) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(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]
                tt() = [0]
                       [0]
                       [0]
                ff() = [0]
                       [0]
                       [0]
                lt(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]
                insert(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]
                nil() = [0]
                        [0]
                        [0]
                cons(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]
                sort(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                ite^#(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_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                lt^#(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]
                insert^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sort^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, 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]
                lt^#(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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(lt^#) = {}, Uargs(c_4) = {1}, Uargs(insert^#) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(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]
                tt() = [0]
                       [0]
                       [0]
                ff() = [0]
                       [0]
                       [0]
                lt(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]
                insert(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]
                nil() = [0]
                        [0]
                        [0]
                cons(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]
                sort(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                ite^#(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_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                lt^#(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]
                insert^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sort^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {lt^#(0(), s(y)) -> c_2()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [2]
                      [0]
                s(x1) = [1 1 2] x1 + [0]
                        [0 1 0]      [2]
                        [0 0 0]      [0]
                lt^#(x1, x2) = [0 2 0] x1 + [1 0 2] x2 + [0]
                               [7 1 0]      [4 0 0]      [0]
                               [4 2 0]      [4 0 0]      [0]
                c_2() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [3]
                          [2 0 0]      [2]
                          [0 0 0]      [2]
           
           * 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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(lt^#) = {}, Uargs(c_4) = {1}, Uargs(insert^#) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(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]
                tt() = [0]
                       [0]
                       [0]
                ff() = [0]
                       [0]
                       [0]
                lt(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]
                insert(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]
                nil() = [0]
                        [0]
                        [0]
                cons(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]
                sort(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                ite^#(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_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                lt^#(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]
                insert^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sort^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^3))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {lt^#(x, 0()) -> c_3()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {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]
                lt^#(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_3() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [0]
                          [0 0 0]      [7]
           
           * Path {8}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(lt^#) = {}, Uargs(c_4) = {}, Uargs(insert^#) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(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]
                tt() = [0]
                       [0]
                       [0]
                ff() = [0]
                       [0]
                       [0]
                lt(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]
                insert(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]
                nil() = [0]
                        [0]
                        [0]
                cons(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]
                sort(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                ite^#(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_0() = [0]
                        [0]
                        [0]
                c_1() = [0]
                        [0]
                        [0]
                lt^#(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]
                insert^#(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
                                   [0 0 0]      [0 0 0]      [0]
                                   [0 0 0]      [0 0 0]      [0]
                c_5() = [0]
                        [0]
                        [0]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sort^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {sort^#(nil()) -> c_7()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(sort^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nil() = [2]
                        [2]
                        [2]
                sort^#(x1) = [0 2 0] x1 + [7]
                             [2 2 0]      [3]
                             [2 2 2]      [3]
                c_7() = [0]
                        [1]
                        [1]
           
           * Path {9}: inherited
             -------------------
             
             This path is subsumed by the proof of path {9}->{7}->{1}.
           
           * Path {9}->{6}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  sort(nil()) -> nil()
                , sort(cons(a, l)) -> insert(a, sort(l))
                , insert(a, nil()) -> cons(a, nil())
                , insert(a, cons(b, l)) ->
                  ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                , ite(tt(), x, y) -> x
                , ite(ff(), x, y) -> y
                , lt(0(), s(y)) -> tt()
                , lt(x, 0()) -> ff()
                , lt(s(x), s(y)) -> lt(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 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost runtime-complexity with respect to
               Rules:
                 {  sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))
                  , insert^#(a, nil()) -> c_5()
                  , sort(nil()) -> nil()
                  , sort(cons(a, l)) -> insert(a, sort(l))
                  , insert(a, nil()) -> cons(a, nil())
                  , insert(a, cons(b, l)) ->
                    ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                  , ite(tt(), x, y) -> x
                  , ite(ff(), x, y) -> y
                  , lt(0(), s(y)) -> tt()
                  , lt(x, 0()) -> ff()
                  , lt(s(x), s(y)) -> lt(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {9}->{7}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {9}->{7}->{1}.
           
           * Path {9}->{7}->{1}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  sort(nil()) -> nil()
                , sort(cons(a, l)) -> insert(a, sort(l))
                , insert(a, nil()) -> cons(a, nil())
                , insert(a, cons(b, l)) ->
                  ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                , ite(tt(), x, y) -> x
                , ite(ff(), x, y) -> y
                , lt(0(), s(y)) -> tt()
                , lt(x, 0()) -> ff()
                , lt(s(x), s(y)) -> lt(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 {9}->{7}->{2}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  sort(nil()) -> nil()
                , sort(cons(a, l)) -> insert(a, sort(l))
                , insert(a, nil()) -> cons(a, nil())
                , insert(a, cons(b, l)) ->
                  ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                , ite(tt(), x, y) -> x
                , ite(ff(), x, y) -> y
                , lt(0(), s(y)) -> tt()
                , lt(x, 0()) -> ff()
                , lt(s(x), s(y)) -> lt(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.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: ite^#(tt(), x, y) -> c_0()
              , 2: ite^#(ff(), x, y) -> c_1()
              , 3: lt^#(0(), s(y)) -> c_2()
              , 4: lt^#(x, 0()) -> c_3()
              , 5: lt^#(s(x), s(y)) -> c_4(lt^#(x, y))
              , 6: insert^#(a, nil()) -> c_5()
              , 7: insert^#(a, cons(b, l)) ->
                   c_6(ite^#(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l))))
              , 8: sort^#(nil()) -> c_7()
              , 9: sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{9}                                                       [     inherited      ]
                |
                |->{6}                                                   [       MAYBE        ]
                |
                `->{7}                                                   [     inherited      ]
                    |
                    |->{1}                                               [         NA         ]
                    |
                    `->{2}                                               [         NA         ]
             
             ->{8}                                                       [    YES(?,O(1))     ]
             
             ->{5}                                                       [   YES(?,O(n^2))    ]
                |
                |->{3}                                                   [   YES(?,O(n^2))    ]
                |
                `->{4}                                                   [   YES(?,O(n^2))    ]
             
           
         
         Sub-problems:
         -------------
           * 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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(lt^#) = {}, Uargs(c_4) = {1}, Uargs(insert^#) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                ff() = [0]
                       [0]
                lt(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]
                insert(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                sort(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                ite^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                    [0 0]      [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                lt^#(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]
                insert^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sort^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                lt^#(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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(lt^#) = {}, Uargs(c_4) = {1}, Uargs(insert^#) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                ff() = [0]
                       [0]
                lt(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]
                insert(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                sort(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                ite^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                    [0 0]      [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                lt^#(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]
                insert^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sort^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {lt^#(0(), s(y)) -> c_2()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                s(x1) = [1 6] x1 + [2]
                        [0 1]      [2]
                lt^#(x1, x2) = [2 1] x1 + [0 1] x2 + [0]
                               [1 2]      [2 0]      [0]
                c_2() = [1]
                        [0]
                c_4(x1) = [1 0] x1 + [7]
                          [0 0]      [7]
           
           * Path {5}->{4}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(lt^#) = {}, Uargs(c_4) = {1}, Uargs(insert^#) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                ff() = [0]
                       [0]
                lt(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]
                insert(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                sort(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                ite^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                    [0 0]      [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                lt^#(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]
                insert^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sort^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {lt^#(x, 0()) -> c_3()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [0]
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [0]
                lt^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
                               [0 0]      [4 1]      [0]
                c_3() = [1]
                        [0]
                c_4(x1) = [1 0] x1 + [6]
                          [0 0]      [7]
           
           * Path {8}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(lt^#) = {}, Uargs(c_4) = {}, Uargs(insert^#) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                ff() = [0]
                       [0]
                lt(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]
                insert(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                sort(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                ite^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                    [0 0]      [0 0]      [0 0]      [0]
                c_0() = [0]
                        [0]
                c_1() = [0]
                        [0]
                lt^#(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]
                insert^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_5() = [0]
                        [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sort^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {sort^#(nil()) -> c_7()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(sort^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nil() = [2]
                        [2]
                sort^#(x1) = [2 0] x1 + [7]
                             [2 2]      [7]
                c_7() = [0]
                        [1]
           
           * Path {9}: inherited
             -------------------
             
             This path is subsumed by the proof of path {9}->{7}->{1}.
           
           * Path {9}->{6}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  sort(nil()) -> nil()
                , sort(cons(a, l)) -> insert(a, sort(l))
                , insert(a, nil()) -> cons(a, nil())
                , insert(a, cons(b, l)) ->
                  ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                , ite(tt(), x, y) -> x
                , ite(ff(), x, y) -> y
                , lt(0(), s(y)) -> tt()
                , lt(x, 0()) -> ff()
                , lt(s(x), s(y)) -> lt(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:
                 {  sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))
                  , insert^#(a, nil()) -> c_5()
                  , sort(nil()) -> nil()
                  , sort(cons(a, l)) -> insert(a, sort(l))
                  , insert(a, nil()) -> cons(a, nil())
                  , insert(a, cons(b, l)) ->
                    ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                  , ite(tt(), x, y) -> x
                  , ite(ff(), x, y) -> y
                  , lt(0(), s(y)) -> tt()
                  , lt(x, 0()) -> ff()
                  , lt(s(x), s(y)) -> lt(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {9}->{7}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {9}->{7}->{1}.
           
           * Path {9}->{7}->{1}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  sort(nil()) -> nil()
                , sort(cons(a, l)) -> insert(a, sort(l))
                , insert(a, nil()) -> cons(a, nil())
                , insert(a, cons(b, l)) ->
                  ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                , ite(tt(), x, y) -> x
                , ite(ff(), x, y) -> y
                , lt(0(), s(y)) -> tt()
                , lt(x, 0()) -> ff()
                , lt(s(x), s(y)) -> lt(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 {9}->{7}->{2}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  sort(nil()) -> nil()
                , sort(cons(a, l)) -> insert(a, sort(l))
                , insert(a, nil()) -> cons(a, nil())
                , insert(a, cons(b, l)) ->
                  ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                , ite(tt(), x, y) -> x
                , ite(ff(), x, y) -> y
                , lt(0(), s(y)) -> tt()
                , lt(x, 0()) -> ff()
                , lt(s(x), s(y)) -> lt(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.
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: ite^#(tt(), x, y) -> c_0()
              , 2: ite^#(ff(), x, y) -> c_1()
              , 3: lt^#(0(), s(y)) -> c_2()
              , 4: lt^#(x, 0()) -> c_3()
              , 5: lt^#(s(x), s(y)) -> c_4(lt^#(x, y))
              , 6: insert^#(a, nil()) -> c_5()
              , 7: insert^#(a, cons(b, l)) ->
                   c_6(ite^#(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l))))
              , 8: sort^#(nil()) -> c_7()
              , 9: sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{9}                                                       [     inherited      ]
                |
                |->{6}                                                   [       MAYBE        ]
                |
                `->{7}                                                   [     inherited      ]
                    |
                    |->{1}                                               [         NA         ]
                    |
                    `->{2}                                               [         NA         ]
             
             ->{8}                                                       [    YES(?,O(1))     ]
             
             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                |->{3}                                                   [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [   YES(?,O(n^1))    ]
             
           
         
         Sub-problems:
         -------------
           * 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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(lt^#) = {}, Uargs(c_4) = {1}, Uargs(insert^#) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                tt() = [0]
                ff() = [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                insert(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sort(x1) = [0] x1 + [0]
                ite^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0() = [0]
                c_1() = [0]
                lt^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                insert^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                sort^#(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                lt^#(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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(lt^#) = {}, Uargs(c_4) = {1}, Uargs(insert^#) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                tt() = [0]
                ff() = [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                insert(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sort(x1) = [0] x1 + [0]
                ite^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0() = [0]
                c_1() = [0]
                lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                insert^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                sort^#(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {lt^#(0(), s(y)) -> c_2()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                lt^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_2() = [1]
                c_4(x1) = [1] x1 + [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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(lt^#) = {}, Uargs(c_4) = {1}, Uargs(insert^#) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                tt() = [0]
                ff() = [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                insert(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sort(x1) = [0] x1 + [0]
                ite^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0() = [0]
                c_1() = [0]
                lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                insert^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                sort^#(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {lt^#(x, 0()) -> c_3()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                lt^#(x1, x2) = [2] x1 + [2] x2 + [4]
                c_3() = [1]
                c_4(x1) = [1] x1 + [7]
           
           * Path {8}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(lt^#) = {}, Uargs(c_4) = {}, Uargs(insert^#) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                tt() = [0]
                ff() = [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                insert(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sort(x1) = [0] x1 + [0]
                ite^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0() = [0]
                c_1() = [0]
                lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                insert^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5() = [0]
                c_6(x1) = [0] x1 + [0]
                sort^#(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {sort^#(nil()) -> c_7()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(sort^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nil() = [7]
                sort^#(x1) = [1] x1 + [7]
                c_7() = [1]
           
           * Path {9}: inherited
             -------------------
             
             This path is subsumed by the proof of path {9}->{7}->{1}.
           
           * Path {9}->{6}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  sort(nil()) -> nil()
                , sort(cons(a, l)) -> insert(a, sort(l))
                , insert(a, nil()) -> cons(a, nil())
                , insert(a, cons(b, l)) ->
                  ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                , ite(tt(), x, y) -> x
                , ite(ff(), x, y) -> y
                , lt(0(), s(y)) -> tt()
                , lt(x, 0()) -> ff()
                , lt(s(x), s(y)) -> lt(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:
                 {  sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))
                  , insert^#(a, nil()) -> c_5()
                  , sort(nil()) -> nil()
                  , sort(cons(a, l)) -> insert(a, sort(l))
                  , insert(a, nil()) -> cons(a, nil())
                  , insert(a, cons(b, l)) ->
                    ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                  , ite(tt(), x, y) -> x
                  , ite(ff(), x, y) -> y
                  , lt(0(), s(y)) -> tt()
                  , lt(x, 0()) -> ff()
                  , lt(s(x), s(y)) -> lt(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {9}->{7}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {9}->{7}->{1}.
           
           * Path {9}->{7}->{1}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  sort(nil()) -> nil()
                , sort(cons(a, l)) -> insert(a, sort(l))
                , insert(a, nil()) -> cons(a, nil())
                , insert(a, cons(b, l)) ->
                  ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                , ite(tt(), x, y) -> x
                , ite(ff(), x, y) -> y
                , lt(0(), s(y)) -> tt()
                , lt(x, 0()) -> ff()
                , lt(s(x), s(y)) -> lt(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 {9}->{7}->{2}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  sort(nil()) -> nil()
                , sort(cons(a, l)) -> insert(a, sort(l))
                , insert(a, nil()) -> cons(a, nil())
                , insert(a, cons(b, l)) ->
                  ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                , ite(tt(), x, y) -> x
                , ite(ff(), x, y) -> y
                , lt(0(), s(y)) -> tt()
                , lt(x, 0()) -> ff()
                , lt(s(x), s(y)) -> lt(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.
    
    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
InputTCT 09 insertsort

stdout:

MAYBE

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputTCT 09 insertsort

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  ite(tt(), x, y) -> x
     , ite(ff(), x, y) -> y
     , lt(0(), s(y)) -> tt()
     , lt(x, 0()) -> ff()
     , lt(s(x), s(y)) -> lt(x, y)
     , insert(a, nil()) -> cons(a, nil())
     , insert(a, cons(b, l)) ->
       ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
     , sort(nil()) -> nil()
     , sort(cons(a, l)) -> insert(a, sort(l))}

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: ite^#(tt(), x, y) -> c_0(x)
              , 2: ite^#(ff(), x, y) -> c_1(y)
              , 3: lt^#(0(), s(y)) -> c_2()
              , 4: lt^#(x, 0()) -> c_3()
              , 5: lt^#(s(x), s(y)) -> c_4(lt^#(x, y))
              , 6: insert^#(a, nil()) -> c_5(a)
              , 7: insert^#(a, cons(b, l)) ->
                   c_6(ite^#(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l))))
              , 8: sort^#(nil()) -> c_7()
              , 9: sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{9}                                                       [     inherited      ]
                |
                |->{6}                                                   [       MAYBE        ]
                |
                `->{7}                                                   [     inherited      ]
                    |
                    |->{1}                                               [         NA         ]
                    |
                    `->{2}                                               [         NA         ]
             
             ->{8}                                                       [    YES(?,O(1))     ]
             
             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                |->{3}                                                   [   YES(?,O(n^2))    ]
                |
                `->{4}                                                   [   YES(?,O(n^3))    ]
             
           
         
         Sub-problems:
         -------------
           * 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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
                 Uargs(c_4) = {1}, Uargs(insert^#) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(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]
                tt() = [0]
                       [0]
                       [0]
                ff() = [0]
                       [0]
                       [0]
                lt(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]
                insert(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]
                nil() = [0]
                        [0]
                        [0]
                cons(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]
                sort(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                ite^#(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_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]
                lt^#(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]
                insert^#(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]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sort^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, 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]
                lt^#(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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
                 Uargs(c_4) = {1}, Uargs(insert^#) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(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]
                tt() = [0]
                       [0]
                       [0]
                ff() = [0]
                       [0]
                       [0]
                lt(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]
                insert(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]
                nil() = [0]
                        [0]
                        [0]
                cons(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]
                sort(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                ite^#(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_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]
                lt^#(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]
                insert^#(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]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sort^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {lt^#(0(), s(y)) -> c_2()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [0]
                      [2]
                      [0]
                s(x1) = [1 1 2] x1 + [0]
                        [0 1 0]      [2]
                        [0 0 0]      [0]
                lt^#(x1, x2) = [0 2 0] x1 + [1 0 2] x2 + [0]
                               [7 1 0]      [4 0 0]      [0]
                               [4 2 0]      [4 0 0]      [0]
                c_2() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [3]
                          [2 0 0]      [2]
                          [0 0 0]      [2]
           
           * 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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
                 Uargs(c_4) = {1}, Uargs(insert^#) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(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]
                tt() = [0]
                       [0]
                       [0]
                ff() = [0]
                       [0]
                       [0]
                lt(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]
                insert(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]
                nil() = [0]
                        [0]
                        [0]
                cons(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]
                sort(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                ite^#(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_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]
                lt^#(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]
                insert^#(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]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sort^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(n^3))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {lt^#(x, 0()) -> c_3()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {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]
                lt^#(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_3() = [1]
                        [0]
                        [0]
                c_4(x1) = [1 0 0] x1 + [3]
                          [0 0 0]      [0]
                          [0 0 0]      [7]
           
           * Path {8}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
                 Uargs(c_4) = {}, Uargs(insert^#) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(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]
                tt() = [0]
                       [0]
                       [0]
                ff() = [0]
                       [0]
                       [0]
                lt(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]
                insert(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]
                nil() = [0]
                        [0]
                        [0]
                cons(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]
                sort(x1) = [0 0 0] x1 + [0]
                           [0 0 0]      [0]
                           [0 0 0]      [0]
                ite^#(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_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]
                lt^#(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]
                insert^#(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]
                c_6(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
                sort^#(x1) = [0 0 0] x1 + [0]
                             [0 0 0]      [0]
                             [0 0 0]      [0]
                c_7() = [0]
                        [0]
                        [0]
                c_8(x1) = [0 0 0] x1 + [0]
                          [0 0 0]      [0]
                          [0 0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 3'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {sort^#(nil()) -> c_7()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(sort^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nil() = [2]
                        [2]
                        [2]
                sort^#(x1) = [0 2 0] x1 + [7]
                             [2 2 0]      [3]
                             [2 2 2]      [3]
                c_7() = [0]
                        [1]
                        [1]
           
           * Path {9}: inherited
             -------------------
             
             This path is subsumed by the proof of path {9}->{7}->{1}.
           
           * Path {9}->{6}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  sort(nil()) -> nil()
                , sort(cons(a, l)) -> insert(a, sort(l))
                , insert(a, nil()) -> cons(a, nil())
                , insert(a, cons(b, l)) ->
                  ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                , ite(tt(), x, y) -> x
                , ite(ff(), x, y) -> y
                , lt(0(), s(y)) -> tt()
                , lt(x, 0()) -> ff()
                , lt(s(x), s(y)) -> lt(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 3'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    runtime-complexity with respect to
               Rules:
                 {  sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))
                  , insert^#(a, nil()) -> c_5(a)
                  , sort(nil()) -> nil()
                  , sort(cons(a, l)) -> insert(a, sort(l))
                  , insert(a, nil()) -> cons(a, nil())
                  , insert(a, cons(b, l)) ->
                    ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                  , ite(tt(), x, y) -> x
                  , ite(ff(), x, y) -> y
                  , lt(0(), s(y)) -> tt()
                  , lt(x, 0()) -> ff()
                  , lt(s(x), s(y)) -> lt(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {9}->{7}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {9}->{7}->{1}.
           
           * Path {9}->{7}->{1}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  sort(nil()) -> nil()
                , sort(cons(a, l)) -> insert(a, sort(l))
                , insert(a, nil()) -> cons(a, nil())
                , insert(a, cons(b, l)) ->
                  ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                , ite(tt(), x, y) -> x
                , ite(ff(), x, y) -> y
                , lt(0(), s(y)) -> tt()
                , lt(x, 0()) -> ff()
                , lt(s(x), s(y)) -> lt(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 {9}->{7}->{2}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  sort(nil()) -> nil()
                , sort(cons(a, l)) -> insert(a, sort(l))
                , insert(a, nil()) -> cons(a, nil())
                , insert(a, cons(b, l)) ->
                  ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                , ite(tt(), x, y) -> x
                , ite(ff(), x, y) -> y
                , lt(0(), s(y)) -> tt()
                , lt(x, 0()) -> ff()
                , lt(s(x), s(y)) -> lt(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.
    
    2) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: ite^#(tt(), x, y) -> c_0(x)
              , 2: ite^#(ff(), x, y) -> c_1(y)
              , 3: lt^#(0(), s(y)) -> c_2()
              , 4: lt^#(x, 0()) -> c_3()
              , 5: lt^#(s(x), s(y)) -> c_4(lt^#(x, y))
              , 6: insert^#(a, nil()) -> c_5(a)
              , 7: insert^#(a, cons(b, l)) ->
                   c_6(ite^#(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l))))
              , 8: sort^#(nil()) -> c_7()
              , 9: sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{9}                                                       [     inherited      ]
                |
                |->{6}                                                   [       MAYBE        ]
                |
                `->{7}                                                   [     inherited      ]
                    |
                    |->{1}                                               [         NA         ]
                    |
                    `->{2}                                               [         NA         ]
             
             ->{8}                                                       [    YES(?,O(1))     ]
             
             ->{5}                                                       [   YES(?,O(n^2))    ]
                |
                |->{3}                                                   [   YES(?,O(n^2))    ]
                |
                `->{4}                                                   [   YES(?,O(n^2))    ]
             
           
         
         Sub-problems:
         -------------
           * 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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
                 Uargs(c_4) = {1}, Uargs(insert^#) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                ff() = [0]
                       [0]
                lt(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]
                insert(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                sort(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                ite^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                    [0 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]
                lt^#(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]
                insert^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sort^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1 2] x1 + [1]
                        [0 1]      [2]
                lt^#(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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
                 Uargs(c_4) = {1}, Uargs(insert^#) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                ff() = [0]
                       [0]
                lt(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]
                insert(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                sort(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                ite^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                    [0 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]
                lt^#(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]
                insert^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sort^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {lt^#(0(), s(y)) -> c_2()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [2]
                s(x1) = [1 6] x1 + [2]
                        [0 1]      [2]
                lt^#(x1, x2) = [2 1] x1 + [0 1] x2 + [0]
                               [1 2]      [2 0]      [0]
                c_2() = [1]
                        [0]
                c_4(x1) = [1 0] x1 + [7]
                          [0 0]      [7]
           
           * Path {5}->{4}: YES(?,O(n^2))
             ----------------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
                 Uargs(c_4) = {1}, Uargs(insert^#) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                ff() = [0]
                       [0]
                lt(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]
                insert(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                sort(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                ite^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                    [0 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]
                lt^#(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]
                insert^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sort^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^2))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {lt^#(x, 0()) -> c_3()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                      [0]
                s(x1) = [1 2] x1 + [2]
                        [0 1]      [0]
                lt^#(x1, x2) = [2 1] x1 + [2 0] x2 + [4]
                               [0 0]      [4 1]      [0]
                c_3() = [1]
                        [0]
                c_4(x1) = [1 0] x1 + [6]
                          [0 0]      [7]
           
           * Path {8}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
                 Uargs(c_4) = {}, Uargs(insert^#) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                  [0 0]      [0 0]      [0 0]      [0]
                tt() = [0]
                       [0]
                ff() = [0]
                       [0]
                lt(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]
                insert(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                nil() = [0]
                        [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                sort(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                ite^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                    [0 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]
                lt^#(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]
                insert^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sort^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_7() = [0]
                        [0]
                c_8(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {sort^#(nil()) -> c_7()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(sort^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nil() = [2]
                        [2]
                sort^#(x1) = [2 0] x1 + [7]
                             [2 2]      [7]
                c_7() = [0]
                        [1]
           
           * Path {9}: inherited
             -------------------
             
             This path is subsumed by the proof of path {9}->{7}->{1}.
           
           * Path {9}->{6}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  sort(nil()) -> nil()
                , sort(cons(a, l)) -> insert(a, sort(l))
                , insert(a, nil()) -> cons(a, nil())
                , insert(a, cons(b, l)) ->
                  ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                , ite(tt(), x, y) -> x
                , ite(ff(), x, y) -> y
                , lt(0(), s(y)) -> tt()
                , lt(x, 0()) -> ff()
                , lt(s(x), s(y)) -> lt(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:
                 {  sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))
                  , insert^#(a, nil()) -> c_5(a)
                  , sort(nil()) -> nil()
                  , sort(cons(a, l)) -> insert(a, sort(l))
                  , insert(a, nil()) -> cons(a, nil())
                  , insert(a, cons(b, l)) ->
                    ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                  , ite(tt(), x, y) -> x
                  , ite(ff(), x, y) -> y
                  , lt(0(), s(y)) -> tt()
                  , lt(x, 0()) -> ff()
                  , lt(s(x), s(y)) -> lt(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {9}->{7}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {9}->{7}->{1}.
           
           * Path {9}->{7}->{1}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  sort(nil()) -> nil()
                , sort(cons(a, l)) -> insert(a, sort(l))
                , insert(a, nil()) -> cons(a, nil())
                , insert(a, cons(b, l)) ->
                  ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                , ite(tt(), x, y) -> x
                , ite(ff(), x, y) -> y
                , lt(0(), s(y)) -> tt()
                , lt(x, 0()) -> ff()
                , lt(s(x), s(y)) -> lt(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 {9}->{7}->{2}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  sort(nil()) -> nil()
                , sort(cons(a, l)) -> insert(a, sort(l))
                , insert(a, nil()) -> cons(a, nil())
                , insert(a, cons(b, l)) ->
                  ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                , ite(tt(), x, y) -> x
                , ite(ff(), x, y) -> y
                , lt(0(), s(y)) -> tt()
                , lt(x, 0()) -> ff()
                , lt(s(x), s(y)) -> lt(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.
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: ite^#(tt(), x, y) -> c_0(x)
              , 2: ite^#(ff(), x, y) -> c_1(y)
              , 3: lt^#(0(), s(y)) -> c_2()
              , 4: lt^#(x, 0()) -> c_3()
              , 5: lt^#(s(x), s(y)) -> c_4(lt^#(x, y))
              , 6: insert^#(a, nil()) -> c_5(a)
              , 7: insert^#(a, cons(b, l)) ->
                   c_6(ite^#(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l))))
              , 8: sort^#(nil()) -> c_7()
              , 9: sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{9}                                                       [     inherited      ]
                |
                |->{6}                                                   [       MAYBE        ]
                |
                `->{7}                                                   [     inherited      ]
                    |
                    |->{1}                                               [         NA         ]
                    |
                    `->{2}                                               [         NA         ]
             
             ->{8}                                                       [    YES(?,O(1))     ]
             
             ->{5}                                                       [   YES(?,O(n^1))    ]
                |
                |->{3}                                                   [   YES(?,O(n^1))    ]
                |
                `->{4}                                                   [   YES(?,O(n^1))    ]
             
           
         
         Sub-problems:
         -------------
           * 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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
                 Uargs(c_4) = {1}, Uargs(insert^#) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                tt() = [0]
                ff() = [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [1] x1 + [0]
                insert(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sort(x1) = [0] x1 + [0]
                ite^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                lt^#(x1, x2) = [3] x1 + [3] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                insert^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                sort^#(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                s(x1) = [1] x1 + [2]
                lt^#(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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
                 Uargs(c_4) = {1}, Uargs(insert^#) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                tt() = [0]
                ff() = [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                insert(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sort(x1) = [0] x1 + [0]
                ite^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                insert^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                sort^#(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {lt^#(0(), s(y)) -> c_2()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                lt^#(x1, x2) = [2] x1 + [2] x2 + [0]
                c_2() = [1]
                c_4(x1) = [1] x1 + [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(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
                 Uargs(c_4) = {1}, Uargs(insert^#) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                tt() = [0]
                ff() = [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                insert(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sort(x1) = [0] x1 + [0]
                ite^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [1] x1 + [0]
                insert^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                sort^#(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {lt^#(x, 0()) -> c_3()}
               Weak Rules: {lt^#(s(x), s(y)) -> c_4(lt^#(x, y))}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(s) = {}, Uargs(lt^#) = {}, Uargs(c_4) = {1}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                0() = [2]
                s(x1) = [1] x1 + [2]
                lt^#(x1, x2) = [2] x1 + [2] x2 + [4]
                c_3() = [1]
                c_4(x1) = [1] x1 + [7]
           
           * Path {8}: YES(?,O(1))
             ---------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(ite) = {}, Uargs(lt) = {}, Uargs(s) = {}, Uargs(insert) = {},
                 Uargs(cons) = {}, Uargs(sort) = {}, Uargs(ite^#) = {},
                 Uargs(c_0) = {}, Uargs(c_1) = {}, Uargs(lt^#) = {},
                 Uargs(c_4) = {}, Uargs(insert^#) = {}, Uargs(c_5) = {},
                 Uargs(c_6) = {}, Uargs(sort^#) = {}, Uargs(c_8) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                ite(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                tt() = [0]
                ff() = [0]
                lt(x1, x2) = [0] x1 + [0] x2 + [0]
                0() = [0]
                s(x1) = [0] x1 + [0]
                insert(x1, x2) = [0] x1 + [0] x2 + [0]
                nil() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                sort(x1) = [0] x1 + [0]
                ite^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_0(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                lt^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_2() = [0]
                c_3() = [0]
                c_4(x1) = [0] x1 + [0]
                insert^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_5(x1) = [0] x1 + [0]
                c_6(x1) = [0] x1 + [0]
                sort^#(x1) = [0] x1 + [0]
                c_7() = [0]
                c_8(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           YES(?,O(1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {sort^#(nil()) -> c_7()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(sort^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                nil() = [7]
                sort^#(x1) = [1] x1 + [7]
                c_7() = [1]
           
           * Path {9}: inherited
             -------------------
             
             This path is subsumed by the proof of path {9}->{7}->{1}.
           
           * Path {9}->{6}: MAYBE
             --------------------
             
             The usable rules for this path are:
             
               {  sort(nil()) -> nil()
                , sort(cons(a, l)) -> insert(a, sort(l))
                , insert(a, nil()) -> cons(a, nil())
                , insert(a, cons(b, l)) ->
                  ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                , ite(tt(), x, y) -> x
                , ite(ff(), x, y) -> y
                , lt(0(), s(y)) -> tt()
                , lt(x, 0()) -> ff()
                , lt(s(x), s(y)) -> lt(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:
                 {  sort^#(cons(a, l)) -> c_8(insert^#(a, sort(l)))
                  , insert^#(a, nil()) -> c_5(a)
                  , sort(nil()) -> nil()
                  , sort(cons(a, l)) -> insert(a, sort(l))
                  , insert(a, nil()) -> cons(a, nil())
                  , insert(a, cons(b, l)) ->
                    ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                  , ite(tt(), x, y) -> x
                  , ite(ff(), x, y) -> y
                  , lt(0(), s(y)) -> tt()
                  , lt(x, 0()) -> ff()
                  , lt(s(x), s(y)) -> lt(x, y)}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {9}->{7}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {9}->{7}->{1}.
           
           * Path {9}->{7}->{1}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  sort(nil()) -> nil()
                , sort(cons(a, l)) -> insert(a, sort(l))
                , insert(a, nil()) -> cons(a, nil())
                , insert(a, cons(b, l)) ->
                  ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                , ite(tt(), x, y) -> x
                , ite(ff(), x, y) -> y
                , lt(0(), s(y)) -> tt()
                , lt(x, 0()) -> ff()
                , lt(s(x), s(y)) -> lt(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 {9}->{7}->{2}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  sort(nil()) -> nil()
                , sort(cons(a, l)) -> insert(a, sort(l))
                , insert(a, nil()) -> cons(a, nil())
                , insert(a, cons(b, l)) ->
                  ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
                , ite(tt(), x, y) -> x
                , ite(ff(), x, y) -> y
                , lt(0(), s(y)) -> tt()
                , lt(x, 0()) -> ff()
                , lt(s(x), s(y)) -> lt(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.
    
    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 pair2rc

Execution TimeUnknown
Answer
TIMEOUT
InputTCT 09 insertsort

stdout:

TIMEOUT

We consider the following Problem:

  Strict Trs:
    {  ite(tt(), x, y) -> x
     , ite(ff(), x, y) -> y
     , lt(0(), s(y)) -> tt()
     , lt(x, 0()) -> ff()
     , lt(s(x), s(y)) -> lt(x, y)
     , insert(a, nil()) -> cons(a, nil())
     , insert(a, cons(b, l)) ->
       ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
     , sort(nil()) -> nil()
     , sort(cons(a, l)) -> insert(a, sort(l))}
  StartTerms: basic terms
  Strategy: none

Certificate: TIMEOUT

Application of 'pair2 (timeout of 60.0 seconds)':
-------------------------------------------------
  Computation stopped due to timeout after 60.0 seconds

Arrrr..

Tool pair3irc

Execution TimeUnknown
Answer
TIMEOUT
InputTCT 09 insertsort

stdout:

TIMEOUT

We consider the following Problem:

  Strict Trs:
    {  ite(tt(), x, y) -> x
     , ite(ff(), x, y) -> y
     , lt(0(), s(y)) -> tt()
     , lt(x, 0()) -> ff()
     , lt(s(x), s(y)) -> lt(x, y)
     , insert(a, nil()) -> cons(a, nil())
     , insert(a, cons(b, l)) ->
       ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
     , sort(nil()) -> nil()
     , sort(cons(a, l)) -> insert(a, sort(l))}
  StartTerms: basic terms
  Strategy: innermost

Certificate: TIMEOUT

Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
  Computation stopped due to timeout after 60.0 seconds

Arrrr..

Tool pair3rc

Execution TimeUnknown
Answer
TIMEOUT
InputTCT 09 insertsort

stdout:

TIMEOUT

We consider the following Problem:

  Strict Trs:
    {  ite(tt(), x, y) -> x
     , ite(ff(), x, y) -> y
     , lt(0(), s(y)) -> tt()
     , lt(x, 0()) -> ff()
     , lt(s(x), s(y)) -> lt(x, y)
     , insert(a, nil()) -> cons(a, nil())
     , insert(a, cons(b, l)) ->
       ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
     , sort(nil()) -> nil()
     , sort(cons(a, l)) -> insert(a, sort(l))}
  StartTerms: basic terms
  Strategy: none

Certificate: TIMEOUT

Application of 'pair3 (timeout of 60.0 seconds)':
-------------------------------------------------
  Computation stopped due to timeout after 60.0 seconds

Arrrr..

Tool rc

Execution TimeUnknown
Answer
TIMEOUT
InputTCT 09 insertsort

stdout:

TIMEOUT

We consider the following Problem:

  Strict Trs:
    {  ite(tt(), x, y) -> x
     , ite(ff(), x, y) -> y
     , lt(0(), s(y)) -> tt()
     , lt(x, 0()) -> ff()
     , lt(s(x), s(y)) -> lt(x, y)
     , insert(a, nil()) -> cons(a, nil())
     , insert(a, cons(b, l)) ->
       ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
     , sort(nil()) -> nil()
     , sort(cons(a, l)) -> insert(a, sort(l))}
  StartTerms: basic terms
  Strategy: none

Certificate: TIMEOUT

Application of 'rc (timeout of 60.0 seconds)':
----------------------------------------------
  Computation stopped due to timeout after 60.0 seconds

Arrrr..

Tool tup3irc

Execution Time65.033646ms
Answer
TIMEOUT
InputTCT 09 insertsort

stdout:

TIMEOUT

We consider the following Problem:

  Strict Trs:
    {  ite(tt(), x, y) -> x
     , ite(ff(), x, y) -> y
     , lt(0(), s(y)) -> tt()
     , lt(x, 0()) -> ff()
     , lt(s(x), s(y)) -> lt(x, y)
     , insert(a, nil()) -> cons(a, nil())
     , insert(a, cons(b, l)) ->
       ite(lt(a, b), cons(a, cons(b, l)), cons(b, insert(a, l)))
     , sort(nil()) -> nil()
     , sort(cons(a, l)) -> insert(a, sort(l))}
  StartTerms: basic terms
  Strategy: innermost

Certificate: TIMEOUT

Application of 'tup3 (timeout of 60.0 seconds)':
------------------------------------------------
  Computation stopped due to timeout after 60.0 seconds

Arrrr..