### (0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, n__app(activate(XS), YS))
from(X) → cons(X, n__from(n__s(X)))
app(X1, X2) → n__app(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
niln__nil
prefix(X) → n__prefix(X)
activate(n__app(X1, X2)) → app(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__nil) → nil
activate(n__prefix(X)) → prefix(activate(X))
activate(X) → X

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
activate(n__app(n__from(X13306_0), X2)) →+ app(cons(activate(X13306_0), n__from(n__s(activate(X13306_0)))), activate(X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [X13306_0 / n__app(n__from(X13306_0), X2)].
The result substitution is [ ].

The rewrite sequence
activate(n__app(n__from(X13306_0), X2)) →+ app(cons(activate(X13306_0), n__from(n__s(activate(X13306_0)))), activate(X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,0,0].
The pumping substitution is [X13306_0 / n__app(n__from(X13306_0), X2)].
The result substitution is [ ].

### (3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

### (4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, n__app(activate(XS), YS))
from(X) → cons(X, n__from(n__s(X)))
app(X1, X2) → n__app(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
niln__nil
prefix(X) → n__prefix(X)
activate(n__app(X1, X2)) → app(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__nil) → nil
activate(n__prefix(X)) → prefix(activate(X))
activate(X) → X

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, n__app(activate(XS), YS))
from(X) → cons(X, n__from(n__s(X)))
app(X1, X2) → n__app(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
niln__nil
prefix(X) → n__prefix(X)
activate(n__app(X1, X2)) → app(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__nil) → nil
activate(n__prefix(X)) → prefix(activate(X))
activate(X) → X

Types:

### (7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
app, activate

They will be analysed ascendingly in the following order:
app = activate

### (8) Obligation:

TRS:
Rules:
app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, n__app(activate(XS), YS))
from(X) → cons(X, n__from(n__s(X)))
app(X1, X2) → n__app(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
niln__nil
prefix(X) → n__prefix(X)
activate(n__app(X1, X2)) → app(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__nil) → nil
activate(n__prefix(X)) → prefix(activate(X))
activate(X) → X

Types:

Generator Equations:

The following defined symbols remain to be analysed:
activate, app

They will be analysed ascendingly in the following order:
app = activate

### (9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol activate.

### (10) Obligation:

TRS:
Rules:
app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, n__app(activate(XS), YS))
from(X) → cons(X, n__from(n__s(X)))
app(X1, X2) → n__app(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
niln__nil
prefix(X) → n__prefix(X)
activate(n__app(X1, X2)) → app(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__nil) → nil
activate(n__prefix(X)) → prefix(activate(X))
activate(X) → X

Types:

Generator Equations:

The following defined symbols remain to be analysed:
app

They will be analysed ascendingly in the following order:
app = activate

### (11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol app.

### (12) Obligation:

TRS:
Rules:
app(nil, YS) → YS
app(cons(X, XS), YS) → cons(X, n__app(activate(XS), YS))
from(X) → cons(X, n__from(n__s(X)))
app(X1, X2) → n__app(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
niln__nil
prefix(X) → n__prefix(X)
activate(n__app(X1, X2)) → app(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__nil) → nil
activate(n__prefix(X)) → prefix(activate(X))
activate(X) → X

Types: