Problem Strategy outermost added 08 ExIntrod GM99

Tool CaT

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 ExIntrod GM99

stdout:

MAYBE

Problem:
 primes() -> sieve(from(s(s(0()))))
 from(X) -> cons(X,from(s(X)))
 head(cons(X,Y)) -> X
 tail(cons(X,Y)) -> Y
 if(true(),X,Y) -> X
 if(false(),X,Y) -> Y
 filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y))))
 sieve(cons(X,Y)) -> cons(X,filter(X,sieve(Y)))

Proof:
 Open

Tool IRC1

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 ExIntrod GM99

stdout:

MAYBE
 Warning when parsing problem:
                             
                               Unsupported strategy 'OUTERMOST'

Tool IRC2

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 ExIntrod GM99

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  primes() -> sieve(from(s(s(0()))))
     , from(X) -> cons(X, from(s(X)))
     , head(cons(X, Y)) -> X
     , tail(cons(X, Y)) -> Y
     , if(true(), X, Y) -> X
     , if(false(), X, Y) -> Y
     , filter(s(s(X)), cons(Y, Z)) ->
       if(divides(s(s(X)), Y),
          filter(s(s(X)), Z),
          cons(Y, filter(X, sieve(Y))))
     , sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))}

Proof Output:    
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: primes^#() -> c_0(sieve^#(from(s(s(0())))))
              , 2: from^#(X) -> c_1(from^#(s(X)))
              , 3: head^#(cons(X, Y)) -> c_2()
              , 4: tail^#(cons(X, Y)) -> c_3()
              , 5: if^#(true(), X, Y) -> c_4()
              , 6: if^#(false(), X, Y) -> c_5()
              , 7: filter^#(s(s(X)), cons(Y, Z)) ->
                   c_6(if^#(divides(s(s(X)), Y),
                            filter(s(s(X)), Z),
                            cons(Y, filter(X, sieve(Y)))))
              , 8: sieve^#(cons(X, Y)) -> c_7(filter^#(X, sieve(Y)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6}                                                       [    YES(?,O(1))     ]
             
             ->{5}                                                       [    YES(?,O(1))     ]
             
             ->{4}                                                       [    YES(?,O(1))     ]
             
             ->{3}                                                       [    YES(?,O(1))     ]
             
             ->{2}                                                       [       MAYBE        ]
             
             ->{1}                                                       [     inherited      ]
                |
                `->{8}                                                   [     inherited      ]
                    |
                    `->{7}                                               [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{8}->{7}.
           
           * Path {1}->{8}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {1}->{8}->{7}.
           
           * Path {1}->{8}->{7}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  from(X) -> cons(X, from(s(X)))
                , sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))
                , filter(s(s(X)), cons(Y, Z)) ->
                  if(divides(s(s(X)), Y),
                     filter(s(s(X)), Z),
                     cons(Y, filter(X, sieve(Y))))
                , if(true(), X, Y) -> X
                , if(false(), X, Y) -> 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 {2}: MAYBE
             ---------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
                 Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
                 Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {1}, Uargs(head^#) = {}, Uargs(tail^#) = {},
                 Uargs(if^#) = {}, Uargs(filter^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                primes() = [0]
                           [0]
                sieve(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                s(x1) = [1 0] x1 + [0]
                        [0 1]      [0]
                0() = [0]
                      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                filter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                primes^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sieve^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                from^#(x1) = [3 3] x1 + [0]
                             [3 3]      [0]
                c_1(x1) = [1 0] x1 + [0]
                          [0 1]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_3() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                filter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {from^#(X) -> c_1(from^#(s(X)))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {3}: 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(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
                 Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
                 Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
                 Uargs(if^#) = {}, Uargs(filter^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                primes() = [0]
                           [0]
                sieve(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                filter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                primes^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sieve^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_3() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                filter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             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: {head^#(cons(X, Y)) -> c_2()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(head^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
                               [0 0]      [0 0]      [2]
                head^#(x1) = [2 0] x1 + [7]
                             [2 2]      [7]
                c_2() = [0]
                        [1]
           
           * Path {4}: 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(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
                 Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
                 Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
                 Uargs(if^#) = {}, Uargs(filter^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                primes() = [0]
                           [0]
                sieve(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                filter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                primes^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sieve^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_3() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                filter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             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: {tail^#(cons(X, Y)) -> c_3()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(tail^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
                               [0 0]      [0 0]      [2]
                tail^#(x1) = [2 0] x1 + [7]
                             [2 2]      [7]
                c_3() = [0]
                        [1]
           
           * Path {5}: 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(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
                 Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
                 Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
                 Uargs(if^#) = {}, Uargs(filter^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                primes() = [0]
                           [0]
                sieve(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                filter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                primes^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sieve^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_3() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                filter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             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: {if^#(true(), X, Y) -> c_4()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(if^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                true() = [2]
                         [2]
                if^#(x1, x2, x3) = [2 0] x1 + [0 0] x2 + [0 0] x3 + [7]
                                   [2 2]      [0 0]      [0 0]      [7]
                c_4() = [0]
                        [1]
           
           * Path {6}: 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(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
                 Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
                 Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
                 Uargs(if^#) = {}, Uargs(filter^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                primes() = [0]
                           [0]
                sieve(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                filter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                primes^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sieve^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_1(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2() = [0]
                        [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_3() = [0]
                        [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_4() = [0]
                        [0]
                c_5() = [0]
                        [0]
                filter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
             
             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: {if^#(false(), X, Y) -> c_5()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(if^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                false() = [2]
                          [2]
                if^#(x1, x2, x3) = [2 0] x1 + [0 0] x2 + [0 0] x3 + [7]
                                   [2 2]      [0 0]      [0 0]      [7]
                c_5() = [0]
                        [1]
    
    2) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: primes^#() -> c_0(sieve^#(from(s(s(0())))))
              , 2: from^#(X) -> c_1(from^#(s(X)))
              , 3: head^#(cons(X, Y)) -> c_2()
              , 4: tail^#(cons(X, Y)) -> c_3()
              , 5: if^#(true(), X, Y) -> c_4()
              , 6: if^#(false(), X, Y) -> c_5()
              , 7: filter^#(s(s(X)), cons(Y, Z)) ->
                   c_6(if^#(divides(s(s(X)), Y),
                            filter(s(s(X)), Z),
                            cons(Y, filter(X, sieve(Y)))))
              , 8: sieve^#(cons(X, Y)) -> c_7(filter^#(X, sieve(Y)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6}                                                       [    YES(?,O(1))     ]
             
             ->{5}                                                       [    YES(?,O(1))     ]
             
             ->{4}                                                       [    YES(?,O(1))     ]
             
             ->{3}                                                       [    YES(?,O(1))     ]
             
             ->{2}                                                       [       MAYBE        ]
             
             ->{1}                                                       [     inherited      ]
                |
                `->{8}                                                   [     inherited      ]
                    |
                    `->{7}                                               [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{8}->{7}.
           
           * Path {1}->{8}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {1}->{8}->{7}.
           
           * Path {1}->{8}->{7}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  from(X) -> cons(X, from(s(X)))
                , sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))
                , filter(s(s(X)), cons(Y, Z)) ->
                  if(divides(s(s(X)), Y),
                     filter(s(s(X)), Z),
                     cons(Y, filter(X, sieve(Y))))
                , if(true(), X, Y) -> X
                , if(false(), X, Y) -> 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 {2}: MAYBE
             ---------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
                 Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
                 Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {1}, Uargs(head^#) = {}, Uargs(tail^#) = {},
                 Uargs(if^#) = {}, Uargs(filter^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                primes() = [0]
                sieve(x1) = [0] x1 + [0]
                from(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                head(x1) = [0] x1 + [0]
                tail(x1) = [0] x1 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                false() = [0]
                filter(x1, x2) = [0] x1 + [0] x2 + [0]
                divides(x1, x2) = [0] x1 + [0] x2 + [0]
                primes^#() = [0]
                c_0(x1) = [0] x1 + [0]
                sieve^#(x1) = [0] x1 + [0]
                from^#(x1) = [3] x1 + [0]
                c_1(x1) = [1] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_2() = [0]
                tail^#(x1) = [0] x1 + [0]
                c_3() = [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_4() = [0]
                c_5() = [0]
                filter^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1) = [0] x1 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    innermost DP runtime-complexity with respect to
               Strict Rules: {from^#(X) -> c_1(from^#(s(X)))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {3}: 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(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
                 Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
                 Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
                 Uargs(if^#) = {}, Uargs(filter^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                primes() = [0]
                sieve(x1) = [0] x1 + [0]
                from(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                head(x1) = [0] x1 + [0]
                tail(x1) = [0] x1 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                false() = [0]
                filter(x1, x2) = [0] x1 + [0] x2 + [0]
                divides(x1, x2) = [0] x1 + [0] x2 + [0]
                primes^#() = [0]
                c_0(x1) = [0] x1 + [0]
                sieve^#(x1) = [0] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_2() = [0]
                tail^#(x1) = [0] x1 + [0]
                c_3() = [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_4() = [0]
                c_5() = [0]
                filter^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(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: {head^#(cons(X, Y)) -> c_2()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(head^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [0] x2 + [7]
                head^#(x1) = [1] x1 + [7]
                c_2() = [1]
           
           * Path {4}: 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(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
                 Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
                 Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
                 Uargs(if^#) = {}, Uargs(filter^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                primes() = [0]
                sieve(x1) = [0] x1 + [0]
                from(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                head(x1) = [0] x1 + [0]
                tail(x1) = [0] x1 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                false() = [0]
                filter(x1, x2) = [0] x1 + [0] x2 + [0]
                divides(x1, x2) = [0] x1 + [0] x2 + [0]
                primes^#() = [0]
                c_0(x1) = [0] x1 + [0]
                sieve^#(x1) = [0] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_2() = [0]
                tail^#(x1) = [0] x1 + [0]
                c_3() = [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_4() = [0]
                c_5() = [0]
                filter^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(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: {tail^#(cons(X, Y)) -> c_3()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(tail^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [0] x2 + [7]
                tail^#(x1) = [1] x1 + [7]
                c_3() = [1]
           
           * Path {5}: 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(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
                 Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
                 Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
                 Uargs(if^#) = {}, Uargs(filter^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                primes() = [0]
                sieve(x1) = [0] x1 + [0]
                from(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                head(x1) = [0] x1 + [0]
                tail(x1) = [0] x1 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                false() = [0]
                filter(x1, x2) = [0] x1 + [0] x2 + [0]
                divides(x1, x2) = [0] x1 + [0] x2 + [0]
                primes^#() = [0]
                c_0(x1) = [0] x1 + [0]
                sieve^#(x1) = [0] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_2() = [0]
                tail^#(x1) = [0] x1 + [0]
                c_3() = [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_4() = [0]
                c_5() = [0]
                filter^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(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: {if^#(true(), X, Y) -> c_4()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(if^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                true() = [7]
                if^#(x1, x2, x3) = [1] x1 + [0] x2 + [0] x3 + [7]
                c_4() = [1]
           
           * Path {6}: 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(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
                 Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
                 Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(tail^#) = {},
                 Uargs(if^#) = {}, Uargs(filter^#) = {}, Uargs(c_6) = {},
                 Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                primes() = [0]
                sieve(x1) = [0] x1 + [0]
                from(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                head(x1) = [0] x1 + [0]
                tail(x1) = [0] x1 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                false() = [0]
                filter(x1, x2) = [0] x1 + [0] x2 + [0]
                divides(x1, x2) = [0] x1 + [0] x2 + [0]
                primes^#() = [0]
                c_0(x1) = [0] x1 + [0]
                sieve^#(x1) = [0] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_1(x1) = [0] x1 + [0]
                head^#(x1) = [0] x1 + [0]
                c_2() = [0]
                tail^#(x1) = [0] x1 + [0]
                c_3() = [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_4() = [0]
                c_5() = [0]
                filter^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(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: {if^#(false(), X, Y) -> c_5()}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(if^#) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                false() = [7]
                if^#(x1, x2, x3) = [1] x1 + [0] x2 + [0] x3 + [7]
                c_5() = [1]
    
    4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    

Tool RC1

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 ExIntrod GM99

stdout:

MAYBE
 Warning when parsing problem:
                             
                               Unsupported strategy 'OUTERMOST'

Tool RC2

Execution TimeUnknown
Answer
MAYBE
InputStrategy outermost added 08 ExIntrod GM99

stdout:

MAYBE

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           MAYBE
Input Problem:    runtime-complexity with respect to
  Rules:
    {  primes() -> sieve(from(s(s(0()))))
     , from(X) -> cons(X, from(s(X)))
     , head(cons(X, Y)) -> X
     , tail(cons(X, Y)) -> Y
     , if(true(), X, Y) -> X
     , if(false(), X, Y) -> Y
     , filter(s(s(X)), cons(Y, Z)) ->
       if(divides(s(s(X)), Y),
          filter(s(s(X)), Z),
          cons(Y, filter(X, sieve(Y))))
     , sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))}

Proof Output:    
  None of the processors succeeded.
  
  Details of failed attempt(s):
  -----------------------------
    1) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: primes^#() -> c_0(sieve^#(from(s(s(0())))))
              , 2: from^#(X) -> c_1(X, from^#(s(X)))
              , 3: head^#(cons(X, Y)) -> c_2(X)
              , 4: tail^#(cons(X, Y)) -> c_3(Y)
              , 5: if^#(true(), X, Y) -> c_4(X)
              , 6: if^#(false(), X, Y) -> c_5(Y)
              , 7: filter^#(s(s(X)), cons(Y, Z)) ->
                   c_6(if^#(divides(s(s(X)), Y),
                            filter(s(s(X)), Z),
                            cons(Y, filter(X, sieve(Y)))))
              , 8: sieve^#(cons(X, Y)) -> c_7(X, filter^#(X, sieve(Y)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6}                                                       [    YES(?,O(1))     ]
             
             ->{5}                                                       [    YES(?,O(1))     ]
             
             ->{4}                                                       [   YES(?,O(n^2))    ]
             
             ->{3}                                                       [   YES(?,O(n^2))    ]
             
             ->{2}                                                       [       MAYBE        ]
             
             ->{1}                                                       [     inherited      ]
                |
                `->{8}                                                   [     inherited      ]
                    |
                    `->{7}                                               [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{8}->{7}.
           
           * Path {1}->{8}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {1}->{8}->{7}.
           
           * Path {1}->{8}->{7}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  from(X) -> cons(X, from(s(X)))
                , sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))
                , filter(s(s(X)), cons(Y, Z)) ->
                  if(divides(s(s(X)), Y),
                     filter(s(s(X)), Z),
                     cons(Y, filter(X, sieve(Y))))
                , if(true(), X, Y) -> X
                , if(false(), X, Y) -> 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 {2}: MAYBE
             ---------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
                 Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
                 Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {2}, Uargs(head^#) = {}, Uargs(c_2) = {},
                 Uargs(tail^#) = {}, Uargs(c_3) = {}, Uargs(if^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(filter^#) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                primes() = [0]
                           [0]
                sieve(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                s(x1) = [1 1] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                filter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                primes^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sieve^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                from^#(x1) = [1 3] x1 + [0]
                             [3 3]      [0]
                c_1(x1, x2) = [0 1] x1 + [1 0] x2 + [0]
                              [0 0]      [0 1]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                filter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {from^#(X) -> c_1(X, from^#(s(X)))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {3}: YES(?,O(n^2))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
                 Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
                 Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
                 Uargs(tail^#) = {}, Uargs(c_3) = {}, Uargs(if^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(filter^#) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                primes() = [0]
                           [0]
                sieve(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                cons(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
                               [0 1]      [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                filter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                primes^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sieve^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                head^#(x1) = [3 3] x1 + [0]
                             [0 0]      [0]
                c_2(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                filter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {head^#(cons(X, Y)) -> c_2(X)}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(head^#) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1 2] x1 + [0 0] x2 + [2]
                               [0 0]      [0 0]      [2]
                head^#(x1) = [2 2] x1 + [7]
                             [2 0]      [7]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [1]
           
           * Path {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(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
                 Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
                 Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
                 Uargs(tail^#) = {}, Uargs(c_3) = {}, Uargs(if^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(filter^#) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                primes() = [0]
                           [0]
                sieve(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                cons(x1, x2) = [1 1] x1 + [0 0] x2 + [0]
                               [0 1]      [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                filter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                primes^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sieve^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                tail^#(x1) = [3 3] x1 + [0]
                             [0 0]      [0]
                c_3(x1) = [1 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                filter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 2'
             --------------------------------------
             Answer:           YES(?,O(n^1))
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {tail^#(cons(X, Y)) -> c_3(Y)}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0 0] x1 + [1 2] x2 + [2]
                               [0 0]      [0 0]      [2]
                tail^#(x1) = [2 2] x1 + [7]
                             [2 0]      [7]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [1]
           
           * Path {5}: 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(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
                 Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
                 Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
                 Uargs(tail^#) = {}, Uargs(c_3) = {}, Uargs(if^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(filter^#) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                primes() = [0]
                           [0]
                sieve(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                filter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                primes^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sieve^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2, x3) = [0 0] x1 + [3 3] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_4(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                filter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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: {if^#(true(), X, Y) -> c_4(X)}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(if^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                true() = [2]
                         [2]
                if^#(x1, x2, x3) = [2 2] x1 + [7 7] x2 + [0 0] x3 + [7]
                                   [2 2]      [7 7]      [0 0]      [3]
                c_4(x1) = [1 3] x1 + [0]
                          [1 1]      [1]
           
           * Path {6}: 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(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
                 Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
                 Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
                 Uargs(tail^#) = {}, Uargs(c_3) = {}, Uargs(if^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(filter^#) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                primes() = [0]
                           [0]
                sieve(x1) = [0 0] x1 + [0]
                            [0 0]      [0]
                from(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                s(x1) = [0 0] x1 + [0]
                        [0 0]      [0]
                0() = [0]
                      [0]
                cons(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                               [0 0]      [0 0]      [0]
                head(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                tail(x1) = [0 0] x1 + [0]
                           [0 0]      [0]
                if(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
                                 [0 0]      [0 0]      [0 0]      [0]
                true() = [0]
                         [0]
                false() = [0]
                          [0]
                filter(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                 [0 0]      [0 0]      [0]
                divides(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                  [0 0]      [0 0]      [0]
                primes^#() = [0]
                             [0]
                c_0(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                sieve^#(x1) = [0 0] x1 + [0]
                              [0 0]      [0]
                from^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_1(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 0]      [0 0]      [0]
                head^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_2(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                tail^#(x1) = [0 0] x1 + [0]
                             [0 0]      [0]
                c_3(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                if^#(x1, x2, x3) = [0 0] x1 + [3 3] x2 + [0 0] x3 + [0]
                                   [0 0]      [0 0]      [0 0]      [0]
                c_4(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_5(x1) = [1 1] x1 + [0]
                          [0 0]      [0]
                filter^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                                   [0 0]      [0 0]      [0]
                c_6(x1) = [0 0] x1 + [0]
                          [0 0]      [0]
                c_7(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
                              [0 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: {if^#(false(), X, Y) -> c_5(Y)}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(if^#) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                false() = [2]
                          [2]
                if^#(x1, x2, x3) = [2 2] x1 + [0 0] x2 + [7 7] x3 + [7]
                                   [2 2]      [0 0]      [7 7]      [3]
                c_5(x1) = [1 3] x1 + [0]
                          [1 1]      [1]
    
    2) 'matrix-interpretation of dimension 1' failed due to the following reason:
         The input cannot be shown compatible
    
    3) 'wdg' failed due to the following reason:
         Transformation Details:
         -----------------------
           We have computed the following set of weak (innermost) dependency pairs:
           
             {  1: primes^#() -> c_0(sieve^#(from(s(s(0())))))
              , 2: from^#(X) -> c_1(X, from^#(s(X)))
              , 3: head^#(cons(X, Y)) -> c_2(X)
              , 4: tail^#(cons(X, Y)) -> c_3(Y)
              , 5: if^#(true(), X, Y) -> c_4(X)
              , 6: if^#(false(), X, Y) -> c_5(Y)
              , 7: filter^#(s(s(X)), cons(Y, Z)) ->
                   c_6(if^#(divides(s(s(X)), Y),
                            filter(s(s(X)), Z),
                            cons(Y, filter(X, sieve(Y)))))
              , 8: sieve^#(cons(X, Y)) -> c_7(X, filter^#(X, sieve(Y)))}
           
           Following Dependency Graph (modulo SCCs) was computed. (Answers to
           subproofs are indicated to the right.)
           
             ->{6}                                                       [    YES(?,O(1))     ]
             
             ->{5}                                                       [    YES(?,O(1))     ]
             
             ->{4}                                                       [   YES(?,O(n^1))    ]
             
             ->{3}                                                       [   YES(?,O(n^1))    ]
             
             ->{2}                                                       [       MAYBE        ]
             
             ->{1}                                                       [     inherited      ]
                |
                `->{8}                                                   [     inherited      ]
                    |
                    `->{7}                                               [         NA         ]
             
           
         
         Sub-problems:
         -------------
           * Path {1}: inherited
             -------------------
             
             This path is subsumed by the proof of path {1}->{8}->{7}.
           
           * Path {1}->{8}: inherited
             ------------------------
             
             This path is subsumed by the proof of path {1}->{8}->{7}.
           
           * Path {1}->{8}->{7}: NA
             ----------------------
             
             The usable rules for this path are:
             
               {  from(X) -> cons(X, from(s(X)))
                , sieve(cons(X, Y)) -> cons(X, filter(X, sieve(Y)))
                , filter(s(s(X)), cons(Y, Z)) ->
                  if(divides(s(s(X)), Y),
                     filter(s(s(X)), Z),
                     cons(Y, filter(X, sieve(Y))))
                , if(true(), X, Y) -> X
                , if(false(), X, Y) -> 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 {2}: MAYBE
             ---------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
                 Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
                 Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {2}, Uargs(head^#) = {}, Uargs(c_2) = {},
                 Uargs(tail^#) = {}, Uargs(c_3) = {}, Uargs(if^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(filter^#) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                primes() = [0]
                sieve(x1) = [0] x1 + [0]
                from(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                head(x1) = [0] x1 + [0]
                tail(x1) = [0] x1 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                false() = [0]
                filter(x1, x2) = [0] x1 + [0] x2 + [0]
                divides(x1, x2) = [0] x1 + [0] x2 + [0]
                primes^#() = [0]
                c_0(x1) = [0] x1 + [0]
                sieve^#(x1) = [0] x1 + [0]
                from^#(x1) = [3] x1 + [0]
                c_1(x1, x2) = [2] x1 + [1] x2 + [0]
                head^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                filter^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [0]
             
             We apply the sub-processor on the resulting sub-problem:
             
             'matrix-interpretation of dimension 1'
             --------------------------------------
             Answer:           MAYBE
             Input Problem:    DP runtime-complexity with respect to
               Strict Rules: {from^#(X) -> c_1(X, from^#(s(X)))}
               Weak Rules: {}
             
             Proof Output:    
               The input cannot be shown compatible
           
           * Path {3}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
                 Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
                 Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
                 Uargs(tail^#) = {}, Uargs(c_3) = {}, Uargs(if^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(filter^#) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                primes() = [0]
                sieve(x1) = [0] x1 + [0]
                from(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                cons(x1, x2) = [1] x1 + [0] x2 + [0]
                head(x1) = [0] x1 + [0]
                tail(x1) = [0] x1 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                false() = [0]
                filter(x1, x2) = [0] x1 + [0] x2 + [0]
                divides(x1, x2) = [0] x1 + [0] x2 + [0]
                primes^#() = [0]
                c_0(x1) = [0] x1 + [0]
                sieve^#(x1) = [0] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_1(x1, x2) = [0] x1 + [0] x2 + [0]
                head^#(x1) = [3] x1 + [0]
                c_2(x1) = [1] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                filter^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [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: {head^#(cons(X, Y)) -> c_2(X)}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(head^#) = {}, Uargs(c_2) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [1] x1 + [0] x2 + [7]
                head^#(x1) = [1] x1 + [7]
                c_2(x1) = [1] x1 + [1]
           
           * Path {4}: YES(?,O(n^1))
             -----------------------
             
             The usable rules of this path are empty.
             
             The weightgap principle applies, using the following adequate RMI:
               The following argument positions are usable:
                 Uargs(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
                 Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
                 Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
                 Uargs(tail^#) = {}, Uargs(c_3) = {}, Uargs(if^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(filter^#) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                primes() = [0]
                sieve(x1) = [0] x1 + [0]
                from(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                cons(x1, x2) = [1] x1 + [0] x2 + [0]
                head(x1) = [0] x1 + [0]
                tail(x1) = [0] x1 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                false() = [0]
                filter(x1, x2) = [0] x1 + [0] x2 + [0]
                divides(x1, x2) = [0] x1 + [0] x2 + [0]
                primes^#() = [0]
                c_0(x1) = [0] x1 + [0]
                sieve^#(x1) = [0] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_1(x1, x2) = [0] x1 + [0] x2 + [0]
                head^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                tail^#(x1) = [3] x1 + [0]
                c_3(x1) = [1] x1 + [0]
                if^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                filter^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [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: {tail^#(cons(X, Y)) -> c_3(Y)}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(cons) = {}, Uargs(tail^#) = {}, Uargs(c_3) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                cons(x1, x2) = [0] x1 + [1] x2 + [7]
                tail^#(x1) = [1] x1 + [7]
                c_3(x1) = [1] x1 + [1]
           
           * Path {5}: 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(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
                 Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
                 Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
                 Uargs(tail^#) = {}, Uargs(c_3) = {}, Uargs(if^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(filter^#) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                primes() = [0]
                sieve(x1) = [0] x1 + [0]
                from(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                head(x1) = [0] x1 + [0]
                tail(x1) = [0] x1 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                false() = [0]
                filter(x1, x2) = [0] x1 + [0] x2 + [0]
                divides(x1, x2) = [0] x1 + [0] x2 + [0]
                primes^#() = [0]
                c_0(x1) = [0] x1 + [0]
                sieve^#(x1) = [0] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_1(x1, x2) = [0] x1 + [0] x2 + [0]
                head^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                if^#(x1, x2, x3) = [0] x1 + [3] x2 + [0] x3 + [0]
                c_4(x1) = [1] x1 + [0]
                c_5(x1) = [0] x1 + [0]
                filter^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [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: {if^#(true(), X, Y) -> c_4(X)}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(if^#) = {}, Uargs(c_4) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                true() = [5]
                if^#(x1, x2, x3) = [3] x1 + [7] x2 + [0] x3 + [0]
                c_4(x1) = [1] x1 + [0]
           
           * Path {6}: 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(sieve) = {}, Uargs(from) = {}, Uargs(s) = {},
                 Uargs(cons) = {}, Uargs(head) = {}, Uargs(tail) = {},
                 Uargs(if) = {}, Uargs(filter) = {}, Uargs(divides) = {},
                 Uargs(c_0) = {}, Uargs(sieve^#) = {}, Uargs(from^#) = {},
                 Uargs(c_1) = {}, Uargs(head^#) = {}, Uargs(c_2) = {},
                 Uargs(tail^#) = {}, Uargs(c_3) = {}, Uargs(if^#) = {},
                 Uargs(c_4) = {}, Uargs(c_5) = {}, Uargs(filter^#) = {},
                 Uargs(c_6) = {}, Uargs(c_7) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                primes() = [0]
                sieve(x1) = [0] x1 + [0]
                from(x1) = [0] x1 + [0]
                s(x1) = [0] x1 + [0]
                0() = [0]
                cons(x1, x2) = [0] x1 + [0] x2 + [0]
                head(x1) = [0] x1 + [0]
                tail(x1) = [0] x1 + [0]
                if(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
                true() = [0]
                false() = [0]
                filter(x1, x2) = [0] x1 + [0] x2 + [0]
                divides(x1, x2) = [0] x1 + [0] x2 + [0]
                primes^#() = [0]
                c_0(x1) = [0] x1 + [0]
                sieve^#(x1) = [0] x1 + [0]
                from^#(x1) = [0] x1 + [0]
                c_1(x1, x2) = [0] x1 + [0] x2 + [0]
                head^#(x1) = [0] x1 + [0]
                c_2(x1) = [0] x1 + [0]
                tail^#(x1) = [0] x1 + [0]
                c_3(x1) = [0] x1 + [0]
                if^#(x1, x2, x3) = [0] x1 + [3] x2 + [0] x3 + [0]
                c_4(x1) = [0] x1 + [0]
                c_5(x1) = [1] x1 + [0]
                filter^#(x1, x2) = [0] x1 + [0] x2 + [0]
                c_6(x1) = [0] x1 + [0]
                c_7(x1, x2) = [0] x1 + [0] x2 + [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: {if^#(false(), X, Y) -> c_5(Y)}
               Weak Rules: {}
             
             Proof Output:    
               The following argument positions are usable:
                 Uargs(if^#) = {}, Uargs(c_5) = {}
               We have the following constructor-restricted matrix interpretation:
               Interpretation Functions:
                false() = [5]
                if^#(x1, x2, x3) = [3] x1 + [0] x2 + [7] x3 + [0]
                c_5(x1) = [1] x1 + [0]
    
    4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.
    
    5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
         match-boundness of the problem could not be verified.