(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
fibs_2#1(zipwith_l, plus, tail_l, x3) → ConsL(S(0), zipwith_l#3(fibs, fibs_2(zipwith_l, plus, tail_l)))
cond_take_l_n_xs(ConsL(x16, x18), 0) → Nil
cond_take_l_n_xs(ConsL(x7, fibs_2(x4, x8, x12)), S(x16)) → Cons(x7, cond_take_l_n_xs(fibs_2#1(x4, x8, x12, bot[0]), x16))
cond_take_l_n_xs(ConsL(x7, zipwith_l_f_xs_ys(x4, x8, x12)), S(x16)) → Cons(x7, cond_take_l_n_xs(zipwith_l_f_xs_ys#1(x4, x8, x12, bot[0]), x16))
plus#2(0, x12) → x12
plus#2(S(x4), x2) → S(plus#2(x4, x2))
cond_zipwith_l_f_xs_ys_1(ConsL(x4, x3), x2, x1) → ConsL(plus#2(x2, x4), zipwith_l#3(x1, x3))
cond_zipwith_l_f_xs_ys(ConsL(x5, x4), zipwith_l_f_xs_ys(x1, x2, x3)) → cond_zipwith_l_f_xs_ys_1(zipwith_l_f_xs_ys#1(x1, x2, x3, bot[6]), x5, x4)
cond_zipwith_l_f_xs_ys(ConsL(x5, x4), fibs_2(x1, x2, x3)) → cond_zipwith_l_f_xs_ys_1(fibs_2#1(x1, x2, x3, bot[6]), x5, x4)
zipwith_l_f_xs_ys#1(plus, fibs, x5, x3) → cond_zipwith_l_f_xs_ys(ConsL(0, fibs_2(zipwith_l, plus, tail_l)), x5)
zipwith_l_f_xs_ys#1(plus, fibs_2(x3, x4, x5), x2, x1) → cond_zipwith_l_f_xs_ys(fibs_2#1(x3, x4, x5, bot[7]), x2)
zipwith_l_f_xs_ys#1(plus, zipwith_l_f_xs_ys(x3, x4, x5), x2, x1) → cond_zipwith_l_f_xs_ys(zipwith_l_f_xs_ys#1(x3, x4, x5, bot[7]), x2)
zipwith_l#3(x8, x4) → zipwith_l_f_xs_ys(plus, x8, x4)
main(x12) → cond_take_l_n_xs(ConsL(0, fibs_2(zipwith_l, plus, tail_l)), x12)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCpxRelTrsProof (BOTH BOUNDS(ID, ID) transformation)
Transformed TRS to relative TRS where S is empty.
(2) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
fibs_2#1(zipwith_l, plus, tail_l, x3) → ConsL(S(0), zipwith_l#3(fibs, fibs_2(zipwith_l, plus, tail_l)))
cond_take_l_n_xs(ConsL(x16, x18), 0) → Nil
cond_take_l_n_xs(ConsL(x7, fibs_2(x4, x8, x12)), S(x16)) → Cons(x7, cond_take_l_n_xs(fibs_2#1(x4, x8, x12, bot[0]), x16))
cond_take_l_n_xs(ConsL(x7, zipwith_l_f_xs_ys(x4, x8, x12)), S(x16)) → Cons(x7, cond_take_l_n_xs(zipwith_l_f_xs_ys#1(x4, x8, x12, bot[0]), x16))
plus#2(0, x12) → x12
plus#2(S(x4), x2) → S(plus#2(x4, x2))
cond_zipwith_l_f_xs_ys_1(ConsL(x4, x3), x2, x1) → ConsL(plus#2(x2, x4), zipwith_l#3(x1, x3))
cond_zipwith_l_f_xs_ys(ConsL(x5, x4), zipwith_l_f_xs_ys(x1, x2, x3)) → cond_zipwith_l_f_xs_ys_1(zipwith_l_f_xs_ys#1(x1, x2, x3, bot[6]), x5, x4)
cond_zipwith_l_f_xs_ys(ConsL(x5, x4), fibs_2(x1, x2, x3)) → cond_zipwith_l_f_xs_ys_1(fibs_2#1(x1, x2, x3, bot[6]), x5, x4)
zipwith_l_f_xs_ys#1(plus, fibs, x5, x3) → cond_zipwith_l_f_xs_ys(ConsL(0, fibs_2(zipwith_l, plus, tail_l)), x5)
zipwith_l_f_xs_ys#1(plus, fibs_2(x3, x4, x5), x2, x1) → cond_zipwith_l_f_xs_ys(fibs_2#1(x3, x4, x5, bot[7]), x2)
zipwith_l_f_xs_ys#1(plus, zipwith_l_f_xs_ys(x3, x4, x5), x2, x1) → cond_zipwith_l_f_xs_ys(zipwith_l_f_xs_ys#1(x3, x4, x5, bot[7]), x2)
zipwith_l#3(x8, x4) → zipwith_l_f_xs_ys(plus, x8, x4)
main(x12) → cond_take_l_n_xs(ConsL(0, fibs_2(zipwith_l, plus, tail_l)), x12)
S is empty.
Rewrite Strategy: INNERMOST
(3) SlicingProof (LOWER BOUND(ID) transformation)
Sliced the following arguments:
fibs_2#1/3
Cons/0
zipwith_l_f_xs_ys#1/3
(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
fibs_2#1(zipwith_l, plus, tail_l) → ConsL(S(0), zipwith_l#3(fibs, fibs_2(zipwith_l, plus, tail_l)))
cond_take_l_n_xs(ConsL(x16, x18), 0) → Nil
cond_take_l_n_xs(ConsL(x7, fibs_2(x4, x8, x12)), S(x16)) → Cons(cond_take_l_n_xs(fibs_2#1(x4, x8, x12), x16))
cond_take_l_n_xs(ConsL(x7, zipwith_l_f_xs_ys(x4, x8, x12)), S(x16)) → Cons(cond_take_l_n_xs(zipwith_l_f_xs_ys#1(x4, x8, x12), x16))
plus#2(0, x12) → x12
plus#2(S(x4), x2) → S(plus#2(x4, x2))
cond_zipwith_l_f_xs_ys_1(ConsL(x4, x3), x2, x1) → ConsL(plus#2(x2, x4), zipwith_l#3(x1, x3))
cond_zipwith_l_f_xs_ys(ConsL(x5, x4), zipwith_l_f_xs_ys(x1, x2, x3)) → cond_zipwith_l_f_xs_ys_1(zipwith_l_f_xs_ys#1(x1, x2, x3), x5, x4)
cond_zipwith_l_f_xs_ys(ConsL(x5, x4), fibs_2(x1, x2, x3)) → cond_zipwith_l_f_xs_ys_1(fibs_2#1(x1, x2, x3), x5, x4)
zipwith_l_f_xs_ys#1(plus, fibs, x5) → cond_zipwith_l_f_xs_ys(ConsL(0, fibs_2(zipwith_l, plus, tail_l)), x5)
zipwith_l_f_xs_ys#1(plus, fibs_2(x3, x4, x5), x2) → cond_zipwith_l_f_xs_ys(fibs_2#1(x3, x4, x5), x2)
zipwith_l_f_xs_ys#1(plus, zipwith_l_f_xs_ys(x3, x4, x5), x2) → cond_zipwith_l_f_xs_ys(zipwith_l_f_xs_ys#1(x3, x4, x5), x2)
zipwith_l#3(x8, x4) → zipwith_l_f_xs_ys(plus, x8, x4)
main(x12) → cond_take_l_n_xs(ConsL(0, fibs_2(zipwith_l, plus, tail_l)), x12)
S is empty.
Rewrite Strategy: INNERMOST
(5) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
plus#2(S(x4), x2) →+ S(plus#2(x4, x2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [x4 / S(x4)].
The result substitution is [ ].
(6) BOUNDS(n^1, INF)