The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(2^n, INF).

0 CpxTRS
↳1 DecreasingLoopProof (⇔, 492 ms)
↳2 BOUNDS(2^n, INF)
↳3 RenamingProof (⇔, 0 ms)
↳4 CpxRelTRS
    ↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
    ↳6 typed CpxTrs
    ↳7 OrderProof (LOWER BOUND(ID), 0 ms)
    ↳8 typed CpxTrs
    ↳9 RewriteLemmaProof (LOWER BOUND(ID), 734 ms)
    ↳10 BEST
        ↳11 typed CpxTrs
            ↳12 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)
            ↳13 typed CpxTrs
            ↳14 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)
            ↳15 typed CpxTrs
            ↳16 LowerBoundsProof (⇔, 0 ms)
            ↳17 BOUNDS(n^1, INF)
        ↳18 typed CpxTrs
            ↳19 LowerBoundsProof (⇔, 0 ms)
            ↳20 BOUNDS(n^1, INF)

(0) Obligation:

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

fib(N) → sel(N, fib1(s(0), s(0)))
fib1(X, Y) → cons(X, n__fib1(Y, n__add(X, Y)))
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
fib1(X1, X2) → n__fib1(X1, X2)
add(X1, X2) → n__add(X1, X2)
activate(n__fib1(X1, X2)) → fib1(activate(X1), activate(X2))
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
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__fib1(X1, X2)) →+ cons(activate(X1), n__fib1(activate(X2), n__add(activate(X1), activate(X2))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / n__fib1(X1, X2)].
The result substitution is [ ].

The rewrite sequence
activate(n__fib1(X1, X2)) →+ cons(activate(X1), n__fib1(activate(X2), n__add(activate(X1), activate(X2))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1,0].
The pumping substitution is [X1 / n__fib1(X1, X2)].
The result substitution is [ ].

(2) BOUNDS(2^n, INF)

(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:

fib(N) → sel(N, fib1(s(0'), s(0')))
fib1(X, Y) → cons(X, n__fib1(Y, n__add(X, Y)))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
fib1(X1, X2) → n__fib1(X1, X2)
add(X1, X2) → n__add(X1, X2)
activate(n__fib1(X1, X2)) → fib1(activate(X1), activate(X2))
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(X) → X

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
Rules:
fib(N) → sel(N, fib1(s(0'), s(0')))
fib1(X, Y) → cons(X, n__fib1(Y, n__add(X, Y)))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
fib1(X1, X2) → n__fib1(X1, X2)
add(X1, X2) → n__add(X1, X2)
activate(n__fib1(X1, X2)) → fib1(activate(X1), activate(X2))
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(X) → X

Types:
fib :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
sel :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
fib1 :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
s :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
0' :: 0':s:n__add:n__fib1:cons
cons :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
n__fib1 :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
n__add :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
add :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
activate :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
hole_0':s:n__add:n__fib1:cons1_0 :: 0':s:n__add:n__fib1:cons
gen_0':s:n__add:n__fib1:cons2_0 :: Nat → 0':s:n__add:n__fib1:cons

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

Heuristically decided to analyse the following defined symbols:
sel, add, activate

They will be analysed ascendingly in the following order:
activate < sel
add < activate

(8) Obligation:

TRS:
Rules:
fib(N) → sel(N, fib1(s(0'), s(0')))
fib1(X, Y) → cons(X, n__fib1(Y, n__add(X, Y)))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
fib1(X1, X2) → n__fib1(X1, X2)
add(X1, X2) → n__add(X1, X2)
activate(n__fib1(X1, X2)) → fib1(activate(X1), activate(X2))
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(X) → X

Types:
fib :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
sel :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
fib1 :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
s :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
0' :: 0':s:n__add:n__fib1:cons
cons :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
n__fib1 :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
n__add :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
add :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
activate :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
hole_0':s:n__add:n__fib1:cons1_0 :: 0':s:n__add:n__fib1:cons
gen_0':s:n__add:n__fib1:cons2_0 :: Nat → 0':s:n__add:n__fib1:cons

Generator Equations:
gen_0':s:n__add:n__fib1:cons2_0(0) ⇔ 0'
gen_0':s:n__add:n__fib1:cons2_0(+(x, 1)) ⇔ s(gen_0':s:n__add:n__fib1:cons2_0(x))

The following defined symbols remain to be analysed:
add, sel, activate

They will be analysed ascendingly in the following order:
activate < sel
add < activate

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
add(gen_0':s:n__add:n__fib1:cons2_0(n4_0), gen_0':s:n__add:n__fib1:cons2_0(b)) → gen_0':s:n__add:n__fib1:cons2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)

Induction Base:
add(gen_0':s:n__add:n__fib1:cons2_0(0), gen_0':s:n__add:n__fib1:cons2_0(b)) →RΩ(1)
gen_0':s:n__add:n__fib1:cons2_0(b)

Induction Step:
add(gen_0':s:n__add:n__fib1:cons2_0(+(n4_0, 1)), gen_0':s:n__add:n__fib1:cons2_0(b)) →RΩ(1)
s(add(gen_0':s:n__add:n__fib1:cons2_0(n4_0), gen_0':s:n__add:n__fib1:cons2_0(b))) →IH
s(gen_0':s:n__add:n__fib1:cons2_0(+(b, c5_0)))

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
fib(N) → sel(N, fib1(s(0'), s(0')))
fib1(X, Y) → cons(X, n__fib1(Y, n__add(X, Y)))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
fib1(X1, X2) → n__fib1(X1, X2)
add(X1, X2) → n__add(X1, X2)
activate(n__fib1(X1, X2)) → fib1(activate(X1), activate(X2))
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(X) → X

Types:
fib :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
sel :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
fib1 :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
s :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
0' :: 0':s:n__add:n__fib1:cons
cons :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
n__fib1 :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
n__add :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
add :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
activate :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
hole_0':s:n__add:n__fib1:cons1_0 :: 0':s:n__add:n__fib1:cons
gen_0':s:n__add:n__fib1:cons2_0 :: Nat → 0':s:n__add:n__fib1:cons

Lemmas:
add(gen_0':s:n__add:n__fib1:cons2_0(n4_0), gen_0':s:n__add:n__fib1:cons2_0(b)) → gen_0':s:n__add:n__fib1:cons2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)

Generator Equations:
gen_0':s:n__add:n__fib1:cons2_0(0) ⇔ 0'
gen_0':s:n__add:n__fib1:cons2_0(+(x, 1)) ⇔ s(gen_0':s:n__add:n__fib1:cons2_0(x))

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

They will be analysed ascendingly in the following order:
activate < sel

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

TRS:
Rules:
fib(N) → sel(N, fib1(s(0'), s(0')))
fib1(X, Y) → cons(X, n__fib1(Y, n__add(X, Y)))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
fib1(X1, X2) → n__fib1(X1, X2)
add(X1, X2) → n__add(X1, X2)
activate(n__fib1(X1, X2)) → fib1(activate(X1), activate(X2))
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(X) → X

Types:
fib :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
sel :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
fib1 :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
s :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
0' :: 0':s:n__add:n__fib1:cons
cons :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
n__fib1 :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
n__add :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
add :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
activate :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
hole_0':s:n__add:n__fib1:cons1_0 :: 0':s:n__add:n__fib1:cons
gen_0':s:n__add:n__fib1:cons2_0 :: Nat → 0':s:n__add:n__fib1:cons

Lemmas:
add(gen_0':s:n__add:n__fib1:cons2_0(n4_0), gen_0':s:n__add:n__fib1:cons2_0(b)) → gen_0':s:n__add:n__fib1:cons2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)

Generator Equations:
gen_0':s:n__add:n__fib1:cons2_0(0) ⇔ 0'
gen_0':s:n__add:n__fib1:cons2_0(+(x, 1)) ⇔ s(gen_0':s:n__add:n__fib1:cons2_0(x))

The following defined symbols remain to be analysed:
sel

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(15) Obligation:

TRS:
Rules:
fib(N) → sel(N, fib1(s(0'), s(0')))
fib1(X, Y) → cons(X, n__fib1(Y, n__add(X, Y)))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
fib1(X1, X2) → n__fib1(X1, X2)
add(X1, X2) → n__add(X1, X2)
activate(n__fib1(X1, X2)) → fib1(activate(X1), activate(X2))
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(X) → X

Types:
fib :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
sel :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
fib1 :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
s :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
0' :: 0':s:n__add:n__fib1:cons
cons :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
n__fib1 :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
n__add :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
add :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
activate :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
hole_0':s:n__add:n__fib1:cons1_0 :: 0':s:n__add:n__fib1:cons
gen_0':s:n__add:n__fib1:cons2_0 :: Nat → 0':s:n__add:n__fib1:cons

Lemmas:
add(gen_0':s:n__add:n__fib1:cons2_0(n4_0), gen_0':s:n__add:n__fib1:cons2_0(b)) → gen_0':s:n__add:n__fib1:cons2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)

Generator Equations:
gen_0':s:n__add:n__fib1:cons2_0(0) ⇔ 0'
gen_0':s:n__add:n__fib1:cons2_0(+(x, 1)) ⇔ s(gen_0':s:n__add:n__fib1:cons2_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
add(gen_0':s:n__add:n__fib1:cons2_0(n4_0), gen_0':s:n__add:n__fib1:cons2_0(b)) → gen_0':s:n__add:n__fib1:cons2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)

(17) BOUNDS(n^1, INF)

(18) Obligation:

TRS:
Rules:
fib(N) → sel(N, fib1(s(0'), s(0')))
fib1(X, Y) → cons(X, n__fib1(Y, n__add(X, Y)))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
sel(0', cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
fib1(X1, X2) → n__fib1(X1, X2)
add(X1, X2) → n__add(X1, X2)
activate(n__fib1(X1, X2)) → fib1(activate(X1), activate(X2))
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(X) → X

Types:
fib :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
sel :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
fib1 :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
s :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
0' :: 0':s:n__add:n__fib1:cons
cons :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
n__fib1 :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
n__add :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
add :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
activate :: 0':s:n__add:n__fib1:cons → 0':s:n__add:n__fib1:cons
hole_0':s:n__add:n__fib1:cons1_0 :: 0':s:n__add:n__fib1:cons
gen_0':s:n__add:n__fib1:cons2_0 :: Nat → 0':s:n__add:n__fib1:cons

Lemmas:
add(gen_0':s:n__add:n__fib1:cons2_0(n4_0), gen_0':s:n__add:n__fib1:cons2_0(b)) → gen_0':s:n__add:n__fib1:cons2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)

Generator Equations:
gen_0':s:n__add:n__fib1:cons2_0(0) ⇔ 0'
gen_0':s:n__add:n__fib1:cons2_0(+(x, 1)) ⇔ s(gen_0':s:n__add:n__fib1:cons2_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
add(gen_0':s:n__add:n__fib1:cons2_0(n4_0), gen_0':s:n__add:n__fib1:cons2_0(b)) → gen_0':s:n__add:n__fib1:cons2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)

(20) BOUNDS(n^1, INF)