The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).

(0) Obligation:

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

terms(N) → cons(recip(sqr(N)), n__terms(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)))
half(0) → 0
half(s(0)) → 0
half(s(s(X))) → s(half(X))
half(dbl(X)) → X
terms(X) → n__terms(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__first(X1, X2)) → first(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
sqr(s(X)) →⁺ s(add(sqr(X), dbl(X)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
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:

terms(N) → cons(recip(sqr(N)), n__terms(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)))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X
terms(X) → n__terms(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__first(X1, X2)) → first(X1, 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(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)))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X
terms(X) → n__terms(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Types:
terms :: s:0' → n__terms:cons:nil:n__first
cons :: recip → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
recip :: s:0' → recip
sqr :: s:0' → s:0'
n__terms :: s:0' → n__terms:cons:nil:n__first
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
nil :: n__terms:cons:nil:n__first
n__first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
activate :: n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
half :: s:0' → s:0'
hole_n__terms:cons:nil:n__first1_0 :: n__terms:cons:nil:n__first
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_n__terms:cons:nil:n__first4_0 :: Nat → n__terms:cons:nil:n__first
gen_s:0'5_0 :: Nat → s:0'

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

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

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(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)))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X
terms(X) → n__terms(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Types:
terms :: s:0' → n__terms:cons:nil:n__first
cons :: recip → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
recip :: s:0' → recip
sqr :: s:0' → s:0'
n__terms :: s:0' → n__terms:cons:nil:n__first
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
nil :: n__terms:cons:nil:n__first
n__first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
activate :: n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
half :: s:0' → s:0'
hole_n__terms:cons:nil:n__first1_0 :: n__terms:cons:nil:n__first
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_n__terms:cons:nil:n__first4_0 :: Nat → n__terms:cons:nil:n__first
gen_s:0'5_0 :: Nat → s:0'

Generator Equations:
gen_n__terms:cons:nil:n__first4_0(0) ⇔ n__terms(0')
gen_n__terms:cons:nil:n__first4_0(+(x, 1)) ⇔ cons(recip(0'), gen_n__terms:cons:nil:n__first4_0(x))
gen_s:0'5_0(0) ⇔ 0'
gen_s:0'5_0(+(x, 1)) ⇔ s(gen_s:0'5_0(x))

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

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

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

Proved the following rewrite lemma:
add(gen_s:0'5_0(n7_0), gen_s:0'5_0(b)) → gen_s:0'5_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)

Induction Base:
add(gen_s:0'5_0(0), gen_s:0'5_0(b)) →_R^Ω(1)
gen_s:0'5_0(b)

Induction Step:
add(gen_s:0'5_0(+(n7_0, 1)), gen_s:0'5_0(b)) →_R^Ω(1)
s(add(gen_s:0'5_0(n7_0), gen_s:0'5_0(b))) →_IH
s(gen_s:0'5_0(+(b, c8_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:
terms(N) → cons(recip(sqr(N)), n__terms(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)))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X
terms(X) → n__terms(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Types:
terms :: s:0' → n__terms:cons:nil:n__first
cons :: recip → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
recip :: s:0' → recip
sqr :: s:0' → s:0'
n__terms :: s:0' → n__terms:cons:nil:n__first
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
nil :: n__terms:cons:nil:n__first
n__first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
activate :: n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
half :: s:0' → s:0'
hole_n__terms:cons:nil:n__first1_0 :: n__terms:cons:nil:n__first
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_n__terms:cons:nil:n__first4_0 :: Nat → n__terms:cons:nil:n__first
gen_s:0'5_0 :: Nat → s:0'

Lemmas:
add(gen_s:0'5_0(n7_0), gen_s:0'5_0(b)) → gen_s:0'5_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_n__terms:cons:nil:n__first4_0(0) ⇔ n__terms(0')
gen_n__terms:cons:nil:n__first4_0(+(x, 1)) ⇔ cons(recip(0'), gen_n__terms:cons:nil:n__first4_0(x))
gen_s:0'5_0(0) ⇔ 0'
gen_s:0'5_0(+(x, 1)) ⇔ s(gen_s:0'5_0(x))

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

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

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

Proved the following rewrite lemma:
dbl(gen_s:0'5_0(n860_0)) → gen_s:0'5_0(*(2, n860_0)), rt ∈ Ω(1 + n860₀)

Induction Base:
dbl(gen_s:0'5_0(0)) →_R^Ω(1)
0'

Induction Step:
dbl(gen_s:0'5_0(+(n860_0, 1))) →_R^Ω(1)
s(s(dbl(gen_s:0'5_0(n860_0)))) →_IH
s(s(gen_s:0'5_0(*(2, c861_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:
terms(N) → cons(recip(sqr(N)), n__terms(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)))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X
terms(X) → n__terms(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Types:
terms :: s:0' → n__terms:cons:nil:n__first
cons :: recip → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
recip :: s:0' → recip
sqr :: s:0' → s:0'
n__terms :: s:0' → n__terms:cons:nil:n__first
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
nil :: n__terms:cons:nil:n__first
n__first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
activate :: n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
half :: s:0' → s:0'
hole_n__terms:cons:nil:n__first1_0 :: n__terms:cons:nil:n__first
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_n__terms:cons:nil:n__first4_0 :: Nat → n__terms:cons:nil:n__first
gen_s:0'5_0 :: Nat → s:0'

Lemmas:
add(gen_s:0'5_0(n7_0), gen_s:0'5_0(b)) → gen_s:0'5_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)
dbl(gen_s:0'5_0(n860_0)) → gen_s:0'5_0(*(2, n860_0)), rt ∈ Ω(1 + n860₀)

Generator Equations:
gen_n__terms:cons:nil:n__first4_0(0) ⇔ n__terms(0')
gen_n__terms:cons:nil:n__first4_0(+(x, 1)) ⇔ cons(recip(0'), gen_n__terms:cons:nil:n__first4_0(x))
gen_s:0'5_0(0) ⇔ 0'
gen_s:0'5_0(+(x, 1)) ⇔ s(gen_s:0'5_0(x))

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

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sqr(gen_s:0'5_0(n1182_0)) → gen_s:0'5_0(*(n1182_0, n1182_0)), rt ∈ Ω(1 + n1182₀ + n1182₀² + n1182₀³)

Induction Base:
sqr(gen_s:0'5_0(0)) →_R^Ω(1)
0'

Induction Step:
sqr(gen_s:0'5_0(+(n1182_0, 1))) →_R^Ω(1)
s(add(sqr(gen_s:0'5_0(n1182_0)), dbl(gen_s:0'5_0(n1182_0)))) →_IH
s(add(gen_s:0'5_0(*(c1183_0, c1183_0)), dbl(gen_s:0'5_0(n1182_0)))) →_L^{Ω(1 + n1182₀)}
s(add(gen_s:0'5_0(*(n1182_0, n1182_0)), gen_s:0'5_0(*(2, n1182_0)))) →_L^{Ω(1 + n1182₀2)}
s(gen_s:0'5_0(+(*(n1182_0, n1182_0), *(2, n1182_0))))

We have rt ∈ Ω(n³) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n³).

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(N)), n__terms(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)))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X
terms(X) → n__terms(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Types:
terms :: s:0' → n__terms:cons:nil:n__first
cons :: recip → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
recip :: s:0' → recip
sqr :: s:0' → s:0'
n__terms :: s:0' → n__terms:cons:nil:n__first
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
nil :: n__terms:cons:nil:n__first
n__first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
activate :: n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
half :: s:0' → s:0'
hole_n__terms:cons:nil:n__first1_0 :: n__terms:cons:nil:n__first
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_n__terms:cons:nil:n__first4_0 :: Nat → n__terms:cons:nil:n__first
gen_s:0'5_0 :: Nat → s:0'

Lemmas:
add(gen_s:0'5_0(n7_0), gen_s:0'5_0(b)) → gen_s:0'5_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)
dbl(gen_s:0'5_0(n860_0)) → gen_s:0'5_0(*(2, n860_0)), rt ∈ Ω(1 + n860₀)
sqr(gen_s:0'5_0(n1182_0)) → gen_s:0'5_0(*(n1182_0, n1182_0)), rt ∈ Ω(1 + n1182₀ + n1182₀² + n1182₀³)

Generator Equations:
gen_n__terms:cons:nil:n__first4_0(0) ⇔ n__terms(0')
gen_n__terms:cons:nil:n__first4_0(+(x, 1)) ⇔ cons(recip(0'), gen_n__terms:cons:nil:n__first4_0(x))
gen_s:0'5_0(0) ⇔ 0'
gen_s:0'5_0(+(x, 1)) ⇔ s(gen_s:0'5_0(x))

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

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(19) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(N)), n__terms(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)))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X
terms(X) → n__terms(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Types:
terms :: s:0' → n__terms:cons:nil:n__first
cons :: recip → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
recip :: s:0' → recip
sqr :: s:0' → s:0'
n__terms :: s:0' → n__terms:cons:nil:n__first
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
nil :: n__terms:cons:nil:n__first
n__first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
activate :: n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
half :: s:0' → s:0'
hole_n__terms:cons:nil:n__first1_0 :: n__terms:cons:nil:n__first
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_n__terms:cons:nil:n__first4_0 :: Nat → n__terms:cons:nil:n__first
gen_s:0'5_0 :: Nat → s:0'

Lemmas:
add(gen_s:0'5_0(n7_0), gen_s:0'5_0(b)) → gen_s:0'5_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)
dbl(gen_s:0'5_0(n860_0)) → gen_s:0'5_0(*(2, n860_0)), rt ∈ Ω(1 + n860₀)
sqr(gen_s:0'5_0(n1182_0)) → gen_s:0'5_0(*(n1182_0, n1182_0)), rt ∈ Ω(1 + n1182₀ + n1182₀² + n1182₀³)

Generator Equations:
gen_n__terms:cons:nil:n__first4_0(0) ⇔ n__terms(0')
gen_n__terms:cons:nil:n__first4_0(+(x, 1)) ⇔ cons(recip(0'), gen_n__terms:cons:nil:n__first4_0(x))
gen_s:0'5_0(0) ⇔ 0'
gen_s:0'5_0(+(x, 1)) ⇔ s(gen_s:0'5_0(x))

The following defined symbols remain to be analysed:
half

(20) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
half(gen_s:0'5_0(*(2, n1601_0))) → gen_s:0'5_0(n1601_0), rt ∈ Ω(1 + n1601₀)

Induction Base:
half(gen_s:0'5_0(*(2, 0))) →_R^Ω(1)
0'

Induction Step:
half(gen_s:0'5_0(*(2, +(n1601_0, 1)))) →_R^Ω(1)
s(half(gen_s:0'5_0(*(2, n1601_0)))) →_IH
s(gen_s:0'5_0(c1602_0))

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

(21) Complex Obligation (BEST)

(22) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(N)), n__terms(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)))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X
terms(X) → n__terms(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Types:
terms :: s:0' → n__terms:cons:nil:n__first
cons :: recip → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
recip :: s:0' → recip
sqr :: s:0' → s:0'
n__terms :: s:0' → n__terms:cons:nil:n__first
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
nil :: n__terms:cons:nil:n__first
n__first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
activate :: n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
half :: s:0' → s:0'
hole_n__terms:cons:nil:n__first1_0 :: n__terms:cons:nil:n__first
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_n__terms:cons:nil:n__first4_0 :: Nat → n__terms:cons:nil:n__first
gen_s:0'5_0 :: Nat → s:0'

Lemmas:
add(gen_s:0'5_0(n7_0), gen_s:0'5_0(b)) → gen_s:0'5_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)
dbl(gen_s:0'5_0(n860_0)) → gen_s:0'5_0(*(2, n860_0)), rt ∈ Ω(1 + n860₀)
sqr(gen_s:0'5_0(n1182_0)) → gen_s:0'5_0(*(n1182_0, n1182_0)), rt ∈ Ω(1 + n1182₀ + n1182₀² + n1182₀³)
half(gen_s:0'5_0(*(2, n1601_0))) → gen_s:0'5_0(n1601_0), rt ∈ Ω(1 + n1601₀)

Generator Equations:
gen_n__terms:cons:nil:n__first4_0(0) ⇔ n__terms(0')
gen_n__terms:cons:nil:n__first4_0(+(x, 1)) ⇔ cons(recip(0'), gen_n__terms:cons:nil:n__first4_0(x))
gen_s:0'5_0(0) ⇔ 0'
gen_s:0'5_0(+(x, 1)) ⇔ s(gen_s:0'5_0(x))

No more defined symbols left to analyse.

(23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
sqr(gen_s:0'5_0(n1182_0)) → gen_s:0'5_0(*(n1182_0, n1182_0)), rt ∈ Ω(1 + n1182₀ + n1182₀² + n1182₀³)

(24) BOUNDS(n^3, INF)

(25) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(N)), n__terms(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)))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X
terms(X) → n__terms(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Types:
terms :: s:0' → n__terms:cons:nil:n__first
cons :: recip → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
recip :: s:0' → recip
sqr :: s:0' → s:0'
n__terms :: s:0' → n__terms:cons:nil:n__first
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
nil :: n__terms:cons:nil:n__first
n__first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
activate :: n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
half :: s:0' → s:0'
hole_n__terms:cons:nil:n__first1_0 :: n__terms:cons:nil:n__first
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_n__terms:cons:nil:n__first4_0 :: Nat → n__terms:cons:nil:n__first
gen_s:0'5_0 :: Nat → s:0'

Lemmas:
add(gen_s:0'5_0(n7_0), gen_s:0'5_0(b)) → gen_s:0'5_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)
dbl(gen_s:0'5_0(n860_0)) → gen_s:0'5_0(*(2, n860_0)), rt ∈ Ω(1 + n860₀)
sqr(gen_s:0'5_0(n1182_0)) → gen_s:0'5_0(*(n1182_0, n1182_0)), rt ∈ Ω(1 + n1182₀ + n1182₀² + n1182₀³)
half(gen_s:0'5_0(*(2, n1601_0))) → gen_s:0'5_0(n1601_0), rt ∈ Ω(1 + n1601₀)

Generator Equations:
gen_n__terms:cons:nil:n__first4_0(0) ⇔ n__terms(0')
gen_n__terms:cons:nil:n__first4_0(+(x, 1)) ⇔ cons(recip(0'), gen_n__terms:cons:nil:n__first4_0(x))
gen_s:0'5_0(0) ⇔ 0'
gen_s:0'5_0(+(x, 1)) ⇔ s(gen_s:0'5_0(x))

No more defined symbols left to analyse.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
sqr(gen_s:0'5_0(n1182_0)) → gen_s:0'5_0(*(n1182_0, n1182_0)), rt ∈ Ω(1 + n1182₀ + n1182₀² + n1182₀³)

(27) BOUNDS(n^3, INF)

(28) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(N)), n__terms(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)))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X
terms(X) → n__terms(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Types:
terms :: s:0' → n__terms:cons:nil:n__first
cons :: recip → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
recip :: s:0' → recip
sqr :: s:0' → s:0'
n__terms :: s:0' → n__terms:cons:nil:n__first
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
nil :: n__terms:cons:nil:n__first
n__first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
activate :: n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
half :: s:0' → s:0'
hole_n__terms:cons:nil:n__first1_0 :: n__terms:cons:nil:n__first
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_n__terms:cons:nil:n__first4_0 :: Nat → n__terms:cons:nil:n__first
gen_s:0'5_0 :: Nat → s:0'

Lemmas:
add(gen_s:0'5_0(n7_0), gen_s:0'5_0(b)) → gen_s:0'5_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)
dbl(gen_s:0'5_0(n860_0)) → gen_s:0'5_0(*(2, n860_0)), rt ∈ Ω(1 + n860₀)
sqr(gen_s:0'5_0(n1182_0)) → gen_s:0'5_0(*(n1182_0, n1182_0)), rt ∈ Ω(1 + n1182₀ + n1182₀² + n1182₀³)

Generator Equations:
gen_n__terms:cons:nil:n__first4_0(0) ⇔ n__terms(0')
gen_n__terms:cons:nil:n__first4_0(+(x, 1)) ⇔ cons(recip(0'), gen_n__terms:cons:nil:n__first4_0(x))
gen_s:0'5_0(0) ⇔ 0'
gen_s:0'5_0(+(x, 1)) ⇔ s(gen_s:0'5_0(x))

No more defined symbols left to analyse.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
sqr(gen_s:0'5_0(n1182_0)) → gen_s:0'5_0(*(n1182_0, n1182_0)), rt ∈ Ω(1 + n1182₀ + n1182₀² + n1182₀³)

(30) BOUNDS(n^3, INF)

(31) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(N)), n__terms(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)))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X
terms(X) → n__terms(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Types:
terms :: s:0' → n__terms:cons:nil:n__first
cons :: recip → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
recip :: s:0' → recip
sqr :: s:0' → s:0'
n__terms :: s:0' → n__terms:cons:nil:n__first
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
nil :: n__terms:cons:nil:n__first
n__first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
activate :: n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
half :: s:0' → s:0'
hole_n__terms:cons:nil:n__first1_0 :: n__terms:cons:nil:n__first
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_n__terms:cons:nil:n__first4_0 :: Nat → n__terms:cons:nil:n__first
gen_s:0'5_0 :: Nat → s:0'

Lemmas:
add(gen_s:0'5_0(n7_0), gen_s:0'5_0(b)) → gen_s:0'5_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)
dbl(gen_s:0'5_0(n860_0)) → gen_s:0'5_0(*(2, n860_0)), rt ∈ Ω(1 + n860₀)

Generator Equations:
gen_n__terms:cons:nil:n__first4_0(0) ⇔ n__terms(0')
gen_n__terms:cons:nil:n__first4_0(+(x, 1)) ⇔ cons(recip(0'), gen_n__terms:cons:nil:n__first4_0(x))
gen_s:0'5_0(0) ⇔ 0'
gen_s:0'5_0(+(x, 1)) ⇔ s(gen_s:0'5_0(x))

No more defined symbols left to analyse.

(32) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
add(gen_s:0'5_0(n7_0), gen_s:0'5_0(b)) → gen_s:0'5_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)

(33) BOUNDS(n^1, INF)

(34) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(N)), n__terms(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)))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X
terms(X) → n__terms(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Types:
terms :: s:0' → n__terms:cons:nil:n__first
cons :: recip → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
recip :: s:0' → recip
sqr :: s:0' → s:0'
n__terms :: s:0' → n__terms:cons:nil:n__first
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
nil :: n__terms:cons:nil:n__first
n__first :: s:0' → n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
activate :: n__terms:cons:nil:n__first → n__terms:cons:nil:n__first
half :: s:0' → s:0'
hole_n__terms:cons:nil:n__first1_0 :: n__terms:cons:nil:n__first
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_n__terms:cons:nil:n__first4_0 :: Nat → n__terms:cons:nil:n__first
gen_s:0'5_0 :: Nat → s:0'

Lemmas:
add(gen_s:0'5_0(n7_0), gen_s:0'5_0(b)) → gen_s:0'5_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)

Generator Equations:
gen_n__terms:cons:nil:n__first4_0(0) ⇔ n__terms(0')
gen_n__terms:cons:nil:n__first4_0(+(x, 1)) ⇔ cons(recip(0'), gen_n__terms:cons:nil:n__first4_0(x))
gen_s:0'5_0(0) ⇔ 0'
gen_s:0'5_0(+(x, 1)) ⇔ s(gen_s:0'5_0(x))

No more defined symbols left to analyse.

(35) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
add(gen_s:0'5_0(n7_0), gen_s:0'5_0(b)) → gen_s:0'5_0(+(n7_0, b)), rt ∈ Ω(1 + n7₀)

(36) BOUNDS(n^1, INF)