(0) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
lt0(Cons(x', xs'), Cons(x, xs)) → lt0(xs', xs)
g(x, 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, 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, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
f(x, 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, 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, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
lt0(x, Nil) → False
g(x, Cons(x', xs)) → g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs))
f(x, Cons(x', xs)) → f(f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)), f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs)))
number42 → 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, 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, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
goal(x, y) → Cons(f(x, y), Cons(g(x, y), Nil))
The (relative) TRS S consists of the following rules:
g[Ite][False][Ite](False, Cons(x, xs), y) → g(xs, Cons(Cons(Nil, Nil), y))
g[Ite][False][Ite](True, x', Cons(x, xs)) → g(x', xs)
f[Ite][False][Ite](False, Cons(x, xs), y) → xs
f[Ite][False][Ite](True, x', Cons(x, xs)) → xs
f[Ite][False][Ite](False, x, y) → Cons(Cons(Nil, Nil), y)
f[Ite][False][Ite](True, x, y) → x
Rewrite Strategy: INNERMOST
(1) InfiniteLowerBoundProof (EQUIVALENT transformation)
The loop following loop proves infinite runtime complexity:
The rewrite sequence
f(Cons(x'14509_1, xs'14510_1), Cons(x', xs)) →+ f(Cons(Cons(Nil, Nil), Cons(x', xs)), Cons(Cons(Nil, Nil), Cons(x', xs)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [ ].
The result substitution is [x'14509_1 / Cons(Nil, Nil), xs'14510_1 / Cons(x', xs), x' / Cons(Nil, Nil), xs / Cons(x', xs)].
(2) BOUNDS(INF, INF)