(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__half(0) → 0
a__half(s(0)) → 0
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)
Rewrite Strategy: FULL
(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:
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__half(0) → 0
a__half(s(0)) → 0
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)
S is empty.
Rewrite Strategy: FULL
(3) SlicingProof (LOWER BOUND(ID) transformation)
Sliced the following arguments:
cons/1
(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
a__terms(N) → cons(recip(a__sqr(mark(N))))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y)) → cons(mark(Y))
a__half(0) → 0
a__half(s(0)) → 0
a__half(s(s(X))) → s(a__half(mark(X)))
a__half(dbl(X)) → mark(X)
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(half(X)) → a__half(mark(X))
mark(cons(X1)) → cons(mark(X1))
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)
a__half(X) → half(X)
S is empty.
Rewrite Strategy: FULL
(5) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
mark(sqr(s(X18682_3))) →+ s(a__add(a__sqr(mark(mark(X18682_3))), a__dbl(mark(mark(X18682_3)))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0,0].
The pumping substitution is [X18682_3 / sqr(s(X18682_3))].
The result substitution is [ ].
The rewrite sequence
mark(sqr(s(X18682_3))) →+ s(a__add(a__sqr(mark(mark(X18682_3))), a__dbl(mark(mark(X18682_3)))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0,0].
The pumping substitution is [X18682_3 / sqr(s(X18682_3))].
The result substitution is [ ].
(6) BOUNDS(2^n, INF)