### (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
dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
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
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 [ ].

### (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'
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
first(0', X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
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

Infered types.

### (6) Obligation:

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

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

### (8) Obligation:

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

They will be analysed ascendingly in the following order:
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:
gen_s:0'5_0(b)

Induction Step:
s(gen_s:0'5_0(+(b, c8_0)))

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

### (11) Obligation:

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

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(n788_0)) → gen_s:0'5_0(*(2, n788_0)), rt ∈ Ω(1 + n7880)

Induction Base:
dbl(gen_s:0'5_0(0)) →RΩ(1)
0'

Induction Step:
dbl(gen_s:0'5_0(+(n788_0, 1))) →RΩ(1)
s(s(dbl(gen_s:0'5_0(n788_0)))) →IH
s(s(gen_s:0'5_0(*(2, c789_0))))

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

### (14) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0') → 0'
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
first(0', X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
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
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(n788_0)) → gen_s:0'5_0(*(2, n788_0)), rt ∈ Ω(1 + n7880)

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

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

Proved the following rewrite lemma:
sqr(gen_s:0'5_0(n1094_0)) → gen_s:0'5_0(*(n1094_0, n1094_0)), rt ∈ Ω(1 + n10940 + n109402 + n109403)

Induction Base:
sqr(gen_s:0'5_0(0)) →RΩ(1)
0'

Induction Step:
sqr(gen_s:0'5_0(+(n1094_0, 1))) →RΩ(1)
s(add(gen_s:0'5_0(*(c1095_0, c1095_0)), dbl(gen_s:0'5_0(n1094_0)))) →LΩ(1 + n10940)
s(add(gen_s:0'5_0(*(n1094_0, n1094_0)), gen_s:0'5_0(*(2, n1094_0)))) →LΩ(1 + n109402)
s(gen_s:0'5_0(+(*(n1094_0, n1094_0), *(2, n1094_0))))

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

### (17) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0') → 0'
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
first(0', X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
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
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(n788_0)) → gen_s:0'5_0(*(2, n788_0)), rt ∈ Ω(1 + n7880)
sqr(gen_s:0'5_0(n1094_0)) → gen_s:0'5_0(*(n1094_0, n1094_0)), rt ∈ Ω(1 + n10940 + n109402 + n109403)

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

### (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'
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
first(0', X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
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
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(n788_0)) → gen_s:0'5_0(*(2, n788_0)), rt ∈ Ω(1 + n7880)
sqr(gen_s:0'5_0(n1094_0)) → gen_s:0'5_0(*(n1094_0, n1094_0)), rt ∈ Ω(1 + n10940 + n109402 + n109403)

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.

### (20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n3) was proven with the following lemma:
sqr(gen_s:0'5_0(n1094_0)) → gen_s:0'5_0(*(n1094_0, n1094_0)), rt ∈ Ω(1 + n10940 + n109402 + n109403)

### (22) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0') → 0'
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
first(0', X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
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
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(n788_0)) → gen_s:0'5_0(*(2, n788_0)), rt ∈ Ω(1 + n7880)
sqr(gen_s:0'5_0(n1094_0)) → gen_s:0'5_0(*(n1094_0, n1094_0)), rt ∈ Ω(1 + n10940 + n109402 + n109403)

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(n1094_0)) → gen_s:0'5_0(*(n1094_0, n1094_0)), rt ∈ Ω(1 + n10940 + n109402 + n109403)

### (25) Obligation:

TRS:
Rules:
terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0') → 0'
dbl(0') → 0'
dbl(s(X)) → s(s(dbl(X)))
first(0', X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
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
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(n788_0)) → gen_s:0'5_0(*(2, n788_0)), rt ∈ Ω(1 + n7880)

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(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)

### (28) Obligation:

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

### (29) 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)