(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
inc(Cons(x, xs)) → Cons(Cons(Nil, Nil), inc(xs))
nestinc(Nil) → number17(Nil)
nestinc(Cons(x, xs)) → nestinc(inc(Cons(x, xs)))
inc(Nil) → Cons(Nil, Nil)
number17(x) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
goal(x) → nestinc(x)
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:
inc(Cons(x, xs)) → Cons(Cons(Nil, Nil), inc(xs))
nestinc(Nil) → number17(Nil)
nestinc(Cons(x, xs)) → nestinc(inc(Cons(x, xs)))
inc(Nil) → Cons(Nil, Nil)
number17(x) → Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil)))))))))))))))))
goal(x) → nestinc(x)
S is empty.
Rewrite Strategy: INNERMOST
(3) SlicingProof (LOWER BOUND(ID) transformation)
Sliced the following arguments:
Cons/0
number17/0
(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
inc(Cons(xs)) → Cons(inc(xs))
nestinc(Nil) → number17
nestinc(Cons(xs)) → nestinc(inc(Cons(xs)))
inc(Nil) → Cons(Nil)
number17 → Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Cons(Nil)))))))))))))))))
goal(x) → nestinc(x)
S is empty.
Rewrite Strategy: INNERMOST
(5) InfiniteLowerBoundProof (EQUIVALENT transformation)
The loop following loop proves infinite runtime complexity:
The rewrite sequence
nestinc(Cons(xs)) →+ nestinc(Cons(inc(xs)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [ ].
The result substitution is [xs / inc(xs)].
(6) BOUNDS(INF, INF)