(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
rev_l#2(x8, x10) → Cons(x10, x8)
step_x_f#1(rev_l, x5, step_x_f(x2, x3, x4), x1) → step_x_f#1(x2, x3, x4, rev_l#2(x1, x5))
step_x_f#1(rev_l, x5, fleft_op_e_xs_1, x3) → rev_l#2(x3, x5)
foldr#3(Nil) → fleft_op_e_xs_1
foldr#3(Cons(x16, x6)) → step_x_f(rev_l, x16, foldr#3(x6))
main(Nil) → Nil
main(Cons(x8, x9)) → step_x_f#1(rev_l, x8, foldr#3(x9), Nil)
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:
rev_l#2(x8, x10) → Cons(x10, x8)
step_x_f#1(rev_l, x5, step_x_f(x2, x3, x4), x1) → step_x_f#1(x2, x3, x4, rev_l#2(x1, x5))
step_x_f#1(rev_l, x5, fleft_op_e_xs_1, x3) → rev_l#2(x3, x5)
foldr#3(Nil) → fleft_op_e_xs_1
foldr#3(Cons(x16, x6)) → step_x_f(rev_l, x16, foldr#3(x6))
main(Nil) → Nil
main(Cons(x8, x9)) → step_x_f#1(rev_l, x8, foldr#3(x9), Nil)
S is empty.
Rewrite Strategy: INNERMOST
(3) SlicingProof (LOWER BOUND(ID) transformation)
Sliced the following arguments:
rev_l#2/1
Cons/0
step_x_f#1/1
step_x_f/1
(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
rev_l#2(x8) → Cons(x8)
step_x_f#1(rev_l, step_x_f(x2, x4), x1) → step_x_f#1(x2, x4, rev_l#2(x1))
step_x_f#1(rev_l, fleft_op_e_xs_1, x3) → rev_l#2(x3)
foldr#3(Nil) → fleft_op_e_xs_1
foldr#3(Cons(x6)) → step_x_f(rev_l, foldr#3(x6))
main(Nil) → Nil
main(Cons(x9)) → step_x_f#1(rev_l, foldr#3(x9), Nil)
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
foldr#3(Cons(x6)) →+ step_x_f(rev_l, foldr#3(x6))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [x6 / Cons(x6)].
The result substitution is [ ].
(6) BOUNDS(n^1, INF)