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

(0) Obligation:

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

terms(N) → cons(recip(sqr(N)), n__terms(n__s(N)))
sqr(0) → 0
sqr(s(X)) → s(add(sqr(X), dbl(X)))
dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
add(0, X) → X
add(s(X), Y) → s(add(X, Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
terms(X) → n__terms(X)
s(X) → n__s(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__first(X1, X2)) → first(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__terms(X)) →⁺ cons(recip(sqr(activate(X))), n__terms(n__s(activate(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].
The pumping substitution is [X / n__terms(X)].
The result substitution is [ ].

The rewrite sequence
activate(n__terms(X)) →⁺ cons(recip(sqr(activate(X))), n__terms(n__s(activate(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
The pumping substitution is [X / n__terms(X)].
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:

terms(N) → cons(recip(sqr(N)), n__terms(n__s(N)))
sqr(0') → 0'
sqr(s(X)) → s(add(sqr(X), dbl(X)))
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
first(0', X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
terms(X) → n__terms(X)
s(X) → n__s(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__first(X1, X2)) → first(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:
terms(N) → cons(recip(sqr(N)), n__terms(n__s(N)))
sqr(0') → 0'
sqr(s(X)) → s(add(sqr(X), dbl(X)))
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
first(0', X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
terms(X) → n__terms(X)
s(X) → n__s(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → X

Types:
terms :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
cons :: recip → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
recip :: n__s:n__terms:cons:0':nil:n__first → recip
sqr :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
n__terms :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
n__s :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
0' :: n__s:n__terms:cons:0':nil:n__first
s :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
add :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
dbl :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
first :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
nil :: n__s:n__terms:cons:0':nil:n__first
n__first :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
activate :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
hole_n__s:n__terms:cons:0':nil:n__first1_0 :: n__s:n__terms:cons:0':nil:n__first
hole_recip2_0 :: recip
gen_n__s:n__terms:cons:0':nil:n__first3_0 :: Nat → n__s:n__terms:cons:0':nil:n__first

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

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

They will be analysed ascendingly in the following order:
add < sqr
dbl < sqr

(8) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(N)), n__terms(n__s(N)))
sqr(0') → 0'
sqr(s(X)) → s(add(sqr(X), dbl(X)))
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
first(0', X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
terms(X) → n__terms(X)
s(X) → n__s(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → X

Types:
terms :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
cons :: recip → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
recip :: n__s:n__terms:cons:0':nil:n__first → recip
sqr :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
n__terms :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
n__s :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
0' :: n__s:n__terms:cons:0':nil:n__first
s :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
add :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
dbl :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
first :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
nil :: n__s:n__terms:cons:0':nil:n__first
n__first :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
activate :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
hole_n__s:n__terms:cons:0':nil:n__first1_0 :: n__s:n__terms:cons:0':nil:n__first
hole_recip2_0 :: recip
gen_n__s:n__terms:cons:0':nil:n__first3_0 :: Nat → n__s:n__terms:cons:0':nil:n__first

Generator Equations:
gen_n__s:n__terms:cons:0':nil:n__first3_0(0) ⇔ 0'
gen_n__s:n__terms:cons:0':nil:n__first3_0(+(x, 1)) ⇔ cons(recip(0'), gen_n__s:n__terms:cons:0':nil:n__first3_0(x))

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

They will be analysed ascendingly in the following order:
add < sqr
dbl < sqr

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

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

(10) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(N)), n__terms(n__s(N)))
sqr(0') → 0'
sqr(s(X)) → s(add(sqr(X), dbl(X)))
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
first(0', X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
terms(X) → n__terms(X)
s(X) → n__s(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → X

Types:
terms :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
cons :: recip → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
recip :: n__s:n__terms:cons:0':nil:n__first → recip
sqr :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
n__terms :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
n__s :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
0' :: n__s:n__terms:cons:0':nil:n__first
s :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
add :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
dbl :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
first :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
nil :: n__s:n__terms:cons:0':nil:n__first
n__first :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
activate :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
hole_n__s:n__terms:cons:0':nil:n__first1_0 :: n__s:n__terms:cons:0':nil:n__first
hole_recip2_0 :: recip
gen_n__s:n__terms:cons:0':nil:n__first3_0 :: Nat → n__s:n__terms:cons:0':nil:n__first

Generator Equations:
gen_n__s:n__terms:cons:0':nil:n__first3_0(0) ⇔ 0'
gen_n__s:n__terms:cons:0':nil:n__first3_0(+(x, 1)) ⇔ cons(recip(0'), gen_n__s:n__terms:cons:0':nil:n__first3_0(x))

The following defined symbols remain to be analysed:
dbl, sqr, activate

They will be analysed ascendingly in the following order:
dbl < sqr

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

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

(12) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(N)), n__terms(n__s(N)))
sqr(0') → 0'
sqr(s(X)) → s(add(sqr(X), dbl(X)))
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
first(0', X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
terms(X) → n__terms(X)
s(X) → n__s(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → X

Types:
terms :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
cons :: recip → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
recip :: n__s:n__terms:cons:0':nil:n__first → recip
sqr :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
n__terms :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
n__s :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
0' :: n__s:n__terms:cons:0':nil:n__first
s :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
add :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
dbl :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
first :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
nil :: n__s:n__terms:cons:0':nil:n__first
n__first :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
activate :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
hole_n__s:n__terms:cons:0':nil:n__first1_0 :: n__s:n__terms:cons:0':nil:n__first
hole_recip2_0 :: recip
gen_n__s:n__terms:cons:0':nil:n__first3_0 :: Nat → n__s:n__terms:cons:0':nil:n__first

Generator Equations:
gen_n__s:n__terms:cons:0':nil:n__first3_0(0) ⇔ 0'
gen_n__s:n__terms:cons:0':nil:n__first3_0(+(x, 1)) ⇔ cons(recip(0'), gen_n__s:n__terms:cons:0':nil:n__first3_0(x))

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

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(14) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(N)), n__terms(n__s(N)))
sqr(0') → 0'
sqr(s(X)) → s(add(sqr(X), dbl(X)))
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
first(0', X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
terms(X) → n__terms(X)
s(X) → n__s(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → X

Types:
terms :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
cons :: recip → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
recip :: n__s:n__terms:cons:0':nil:n__first → recip
sqr :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
n__terms :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
n__s :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
0' :: n__s:n__terms:cons:0':nil:n__first
s :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
add :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
dbl :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
first :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
nil :: n__s:n__terms:cons:0':nil:n__first
n__first :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
activate :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
hole_n__s:n__terms:cons:0':nil:n__first1_0 :: n__s:n__terms:cons:0':nil:n__first
hole_recip2_0 :: recip
gen_n__s:n__terms:cons:0':nil:n__first3_0 :: Nat → n__s:n__terms:cons:0':nil:n__first

Generator Equations:
gen_n__s:n__terms:cons:0':nil:n__first3_0(0) ⇔ 0'
gen_n__s:n__terms:cons:0':nil:n__first3_0(+(x, 1)) ⇔ cons(recip(0'), gen_n__s:n__terms:cons:0':nil:n__first3_0(x))

The following defined symbols remain to be analysed:
activate

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(16) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(N)), n__terms(n__s(N)))
sqr(0') → 0'
sqr(s(X)) → s(add(sqr(X), dbl(X)))
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
add(0', X) → X
add(s(X), Y) → s(add(X, Y))
first(0', X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
terms(X) → n__terms(X)
s(X) → n__s(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → X

Types:
terms :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
cons :: recip → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
recip :: n__s:n__terms:cons:0':nil:n__first → recip
sqr :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
n__terms :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
n__s :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
0' :: n__s:n__terms:cons:0':nil:n__first
s :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
add :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
dbl :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
first :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
nil :: n__s:n__terms:cons:0':nil:n__first
n__first :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
activate :: n__s:n__terms:cons:0':nil:n__first → n__s:n__terms:cons:0':nil:n__first
hole_n__s:n__terms:cons:0':nil:n__first1_0 :: n__s:n__terms:cons:0':nil:n__first
hole_recip2_0 :: recip
gen_n__s:n__terms:cons:0':nil:n__first3_0 :: Nat → n__s:n__terms:cons:0':nil:n__first

Generator Equations:
gen_n__s:n__terms:cons:0':nil:n__first3_0(0) ⇔ 0'
gen_n__s:n__terms:cons:0':nil:n__first3_0(+(x, 1)) ⇔ cons(recip(0'), gen_n__s:n__terms:cons:0':nil:n__first3_0(x))

No more defined symbols left to analyse.