The Runtime Complexity (full) of the given CpxTRS could be proven to be 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, 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)
activate(n__fib1(X1, X2)) → fib1(X1, X2)
activate(X) → X

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
add(s(X), Y) →⁺ s(add(X, Y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / s(X)].
The result substitution is [ ].

(2) BOUNDS(n^1, 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, 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)
activate(n__fib1(X1, X2)) → fib1(X1, 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, 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)
activate(n__fib1(X1, X2)) → fib1(X1, X2)
activate(X) → X

Types:
fib :: 0':s → 0':s
sel :: 0':s → n__fib1:cons → 0':s
fib1 :: 0':s → 0':s → n__fib1:cons
s :: 0':s → 0':s
0' :: 0':s
cons :: 0':s → n__fib1:cons → n__fib1:cons
n__fib1 :: 0':s → 0':s → n__fib1:cons
add :: 0':s → 0':s → 0':s
activate :: n__fib1:cons → n__fib1:cons
hole_0':s1_0 :: 0':s
hole_n__fib1:cons2_0 :: n__fib1:cons
gen_0':s3_0 :: Nat → 0':s
gen_n__fib1:cons4_0 :: Nat → n__fib1:cons

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

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

(8) Obligation:

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

Types:
fib :: 0':s → 0':s
sel :: 0':s → n__fib1:cons → 0':s
fib1 :: 0':s → 0':s → n__fib1:cons
s :: 0':s → 0':s
0' :: 0':s
cons :: 0':s → n__fib1:cons → n__fib1:cons
n__fib1 :: 0':s → 0':s → n__fib1:cons
add :: 0':s → 0':s → 0':s
activate :: n__fib1:cons → n__fib1:cons
hole_0':s1_0 :: 0':s
hole_n__fib1:cons2_0 :: n__fib1:cons
gen_0':s3_0 :: Nat → 0':s
gen_n__fib1:cons4_0 :: Nat → n__fib1:cons

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_n__fib1:cons4_0(0) ⇔ n__fib1(0', 0')
gen_n__fib1:cons4_0(+(x, 1)) ⇔ cons(0', gen_n__fib1:cons4_0(x))

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

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

Proved the following rewrite lemma:
sel(gen_0':s3_0(n6_0), gen_n__fib1:cons4_0(1)) → gen_0':s3_0(0), rt ∈ Ω(1 + n6₀)

Induction Base:
sel(gen_0':s3_0(0), gen_n__fib1:cons4_0(1)) →_R^Ω(1)
0'

Induction Step:
sel(gen_0':s3_0(+(n6_0, 1)), gen_n__fib1:cons4_0(1)) →_R^Ω(1)
sel(gen_0':s3_0(n6_0), activate(gen_n__fib1:cons4_0(0))) →_R^Ω(1)
sel(gen_0':s3_0(n6_0), fib1(0', 0')) →_R^Ω(1)
sel(gen_0':s3_0(n6_0), cons(0', n__fib1(0', add(0', 0')))) →_R^Ω(1)
sel(gen_0':s3_0(n6_0), cons(0', n__fib1(0', 0'))) →_IH
gen_0':s3_0(0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ 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, 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)
activate(n__fib1(X1, X2)) → fib1(X1, X2)
activate(X) → X

Types:
fib :: 0':s → 0':s
sel :: 0':s → n__fib1:cons → 0':s
fib1 :: 0':s → 0':s → n__fib1:cons
s :: 0':s → 0':s
0' :: 0':s
cons :: 0':s → n__fib1:cons → n__fib1:cons
n__fib1 :: 0':s → 0':s → n__fib1:cons
add :: 0':s → 0':s → 0':s
activate :: n__fib1:cons → n__fib1:cons
hole_0':s1_0 :: 0':s
hole_n__fib1:cons2_0 :: n__fib1:cons
gen_0':s3_0 :: Nat → 0':s
gen_n__fib1:cons4_0 :: Nat → n__fib1:cons

Lemmas:
sel(gen_0':s3_0(n6_0), gen_n__fib1:cons4_0(1)) → gen_0':s3_0(0), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_n__fib1:cons4_0(0) ⇔ n__fib1(0', 0')
gen_n__fib1:cons4_0(+(x, 1)) ⇔ cons(0', gen_n__fib1:cons4_0(x))

The following defined symbols remain to be analysed:
add

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
add(gen_0':s3_0(n326_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n326_0, b)), rt ∈ Ω(1 + n326₀)

Induction Base:
add(gen_0':s3_0(0), gen_0':s3_0(b)) →_R^Ω(1)
gen_0':s3_0(b)

Induction Step:
add(gen_0':s3_0(+(n326_0, 1)), gen_0':s3_0(b)) →_R^Ω(1)
s(add(gen_0':s3_0(n326_0), gen_0':s3_0(b))) →_IH
s(gen_0':s3_0(+(b, c327_0)))

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

(13) Complex Obligation (BEST)

(14) Obligation:

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

Types:
fib :: 0':s → 0':s
sel :: 0':s → n__fib1:cons → 0':s
fib1 :: 0':s → 0':s → n__fib1:cons
s :: 0':s → 0':s
0' :: 0':s
cons :: 0':s → n__fib1:cons → n__fib1:cons
n__fib1 :: 0':s → 0':s → n__fib1:cons
add :: 0':s → 0':s → 0':s
activate :: n__fib1:cons → n__fib1:cons
hole_0':s1_0 :: 0':s
hole_n__fib1:cons2_0 :: n__fib1:cons
gen_0':s3_0 :: Nat → 0':s
gen_n__fib1:cons4_0 :: Nat → n__fib1:cons

Lemmas:
sel(gen_0':s3_0(n6_0), gen_n__fib1:cons4_0(1)) → gen_0':s3_0(0), rt ∈ Ω(1 + n6₀)
add(gen_0':s3_0(n326_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n326_0, b)), rt ∈ Ω(1 + n326₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_n__fib1:cons4_0(0) ⇔ n__fib1(0', 0')
gen_n__fib1:cons4_0(+(x, 1)) ⇔ cons(0', gen_n__fib1:cons4_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
sel(gen_0':s3_0(n6_0), gen_n__fib1:cons4_0(1)) → gen_0':s3_0(0), rt ∈ Ω(1 + n6₀)

(16) BOUNDS(n^1, INF)

(17) Obligation:

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

Types:
fib :: 0':s → 0':s
sel :: 0':s → n__fib1:cons → 0':s
fib1 :: 0':s → 0':s → n__fib1:cons
s :: 0':s → 0':s
0' :: 0':s
cons :: 0':s → n__fib1:cons → n__fib1:cons
n__fib1 :: 0':s → 0':s → n__fib1:cons
add :: 0':s → 0':s → 0':s
activate :: n__fib1:cons → n__fib1:cons
hole_0':s1_0 :: 0':s
hole_n__fib1:cons2_0 :: n__fib1:cons
gen_0':s3_0 :: Nat → 0':s
gen_n__fib1:cons4_0 :: Nat → n__fib1:cons

Lemmas:
sel(gen_0':s3_0(n6_0), gen_n__fib1:cons4_0(1)) → gen_0':s3_0(0), rt ∈ Ω(1 + n6₀)
add(gen_0':s3_0(n326_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n326_0, b)), rt ∈ Ω(1 + n326₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_n__fib1:cons4_0(0) ⇔ n__fib1(0', 0')
gen_n__fib1:cons4_0(+(x, 1)) ⇔ cons(0', gen_n__fib1:cons4_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
sel(gen_0':s3_0(n6_0), gen_n__fib1:cons4_0(1)) → gen_0':s3_0(0), rt ∈ Ω(1 + n6₀)

(19) BOUNDS(n^1, INF)

(20) Obligation:

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

Types:
fib :: 0':s → 0':s
sel :: 0':s → n__fib1:cons → 0':s
fib1 :: 0':s → 0':s → n__fib1:cons
s :: 0':s → 0':s
0' :: 0':s
cons :: 0':s → n__fib1:cons → n__fib1:cons
n__fib1 :: 0':s → 0':s → n__fib1:cons
add :: 0':s → 0':s → 0':s
activate :: n__fib1:cons → n__fib1:cons
hole_0':s1_0 :: 0':s
hole_n__fib1:cons2_0 :: n__fib1:cons
gen_0':s3_0 :: Nat → 0':s
gen_n__fib1:cons4_0 :: Nat → n__fib1:cons

Lemmas:
sel(gen_0':s3_0(n6_0), gen_n__fib1:cons4_0(1)) → gen_0':s3_0(0), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_n__fib1:cons4_0(0) ⇔ n__fib1(0', 0')
gen_n__fib1:cons4_0(+(x, 1)) ⇔ cons(0', gen_n__fib1:cons4_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
sel(gen_0':s3_0(n6_0), gen_n__fib1:cons4_0(1)) → gen_0':s3_0(0), rt ∈ Ω(1 + n6₀)

(22) BOUNDS(n^1, INF)