The Runtime Complexity (innermost) 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)), 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, first(X, Z))
half(0) → 0
half(s(0)) → 0
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

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

terms(N) → cons(recip(sqr(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, first(X, Z))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
terms(N) → cons(recip(sqr(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, first(X, Z))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: s:0' → cons:nil
cons :: recip → cons:nil → cons:nil
recip :: s:0' → recip
sqr :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → cons:nil → cons:nil
nil :: cons:nil
half :: s:0' → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_cons:nil4_0 :: Nat → cons:nil
gen_s:0'5_0 :: Nat → s:0'

(5) OrderProof (LOWER BOUND(ID) transformation)

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

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

(6) Obligation:

Innermost TRS:
Rules:
terms(N) → cons(recip(sqr(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, first(X, Z))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: s:0' → cons:nil
cons :: recip → cons:nil → cons:nil
recip :: s:0' → recip
sqr :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → cons:nil → cons:nil
nil :: cons:nil
half :: s:0' → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_cons:nil4_0 :: Nat → cons:nil
gen_s:0'5_0 :: Nat → s:0'

Generator Equations:
gen_cons:nil4_0(0) ⇔ nil
gen_cons:nil4_0(+(x, 1)) ⇔ cons(recip(0'), gen_cons:nil4_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, terms, sqr, dbl, first, half

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

(7) 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).

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
terms(N) → cons(recip(sqr(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, first(X, Z))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: s:0' → cons:nil
cons :: recip → cons:nil → cons:nil
recip :: s:0' → recip
sqr :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → cons:nil → cons:nil
nil :: cons:nil
half :: s:0' → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_cons:nil4_0 :: Nat → cons:nil
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_cons:nil4_0(0) ⇔ nil
gen_cons:nil4_0(+(x, 1)) ⇔ cons(recip(0'), gen_cons:nil4_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, terms, sqr, first, half

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

(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
dbl(gen_s:0'5_0(n870_0)) → gen_s:0'5_0(*(2, n870_0)), rt ∈ Ω(1 + n870₀)

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

Induction Step:
dbl(gen_s:0'5_0(+(n870_0, 1))) →_R^Ω(1)
s(s(dbl(gen_s:0'5_0(n870_0)))) →_IH
s(s(gen_s:0'5_0(*(2, c871_0))))

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

(11) Complex Obligation (BEST)

(12) Obligation:

Innermost TRS:
Rules:
terms(N) → cons(recip(sqr(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, first(X, Z))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: s:0' → cons:nil
cons :: recip → cons:nil → cons:nil
recip :: s:0' → recip
sqr :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → cons:nil → cons:nil
nil :: cons:nil
half :: s:0' → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_cons:nil4_0 :: Nat → cons:nil
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(n870_0)) → gen_s:0'5_0(*(2, n870_0)), rt ∈ Ω(1 + n870₀)

Generator Equations:
gen_cons:nil4_0(0) ⇔ nil
gen_cons:nil4_0(+(x, 1)) ⇔ cons(recip(0'), gen_cons:nil4_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, terms, first, half

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

(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sqr(gen_s:0'5_0(n1212_0)) → gen_s:0'5_0(*(n1212_0, n1212_0)), rt ∈ Ω(1 + n1212₀ + n1212₀² + n1212₀³)

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

Induction Step:
sqr(gen_s:0'5_0(+(n1212_0, 1))) →_R^Ω(1)
s(add(sqr(gen_s:0'5_0(n1212_0)), dbl(gen_s:0'5_0(n1212_0)))) →_IH
s(add(gen_s:0'5_0(*(c1213_0, c1213_0)), dbl(gen_s:0'5_0(n1212_0)))) →_L^{Ω(1 + n1212₀)}
s(add(gen_s:0'5_0(*(n1212_0, n1212_0)), gen_s:0'5_0(*(2, n1212_0)))) →_L^{Ω(1 + n1212₀2)}
s(gen_s:0'5_0(+(*(n1212_0, n1212_0), *(2, n1212_0))))

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

(14) Complex Obligation (BEST)

(15) Obligation:

Innermost TRS:
Rules:
terms(N) → cons(recip(sqr(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, first(X, Z))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: s:0' → cons:nil
cons :: recip → cons:nil → cons:nil
recip :: s:0' → recip
sqr :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → cons:nil → cons:nil
nil :: cons:nil
half :: s:0' → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_cons:nil4_0 :: Nat → cons:nil
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(n870_0)) → gen_s:0'5_0(*(2, n870_0)), rt ∈ Ω(1 + n870₀)
sqr(gen_s:0'5_0(n1212_0)) → gen_s:0'5_0(*(n1212_0, n1212_0)), rt ∈ Ω(1 + n1212₀ + n1212₀² + n1212₀³)

Generator Equations:
gen_cons:nil4_0(0) ⇔ nil
gen_cons:nil4_0(+(x, 1)) ⇔ cons(recip(0'), gen_cons:nil4_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:
terms, first, half

(16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(17) Obligation:

Innermost TRS:
Rules:
terms(N) → cons(recip(sqr(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, first(X, Z))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: s:0' → cons:nil
cons :: recip → cons:nil → cons:nil
recip :: s:0' → recip
sqr :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → cons:nil → cons:nil
nil :: cons:nil
half :: s:0' → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_cons:nil4_0 :: Nat → cons:nil
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(n870_0)) → gen_s:0'5_0(*(2, n870_0)), rt ∈ Ω(1 + n870₀)
sqr(gen_s:0'5_0(n1212_0)) → gen_s:0'5_0(*(n1212_0, n1212_0)), rt ∈ Ω(1 + n1212₀ + n1212₀² + n1212₀³)

Generator Equations:
gen_cons:nil4_0(0) ⇔ nil
gen_cons:nil4_0(+(x, 1)) ⇔ cons(recip(0'), gen_cons:nil4_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:
first, half

(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
first(gen_s:0'5_0(n1982_0), gen_cons:nil4_0(n1982_0)) → gen_cons:nil4_0(n1982_0), rt ∈ Ω(1 + n1982₀)

Induction Base:
first(gen_s:0'5_0(0), gen_cons:nil4_0(0)) →_R^Ω(1)
nil

Induction Step:
first(gen_s:0'5_0(+(n1982_0, 1)), gen_cons:nil4_0(+(n1982_0, 1))) →_R^Ω(1)
cons(recip(0'), first(gen_s:0'5_0(n1982_0), gen_cons:nil4_0(n1982_0))) →_IH
cons(recip(0'), gen_cons:nil4_0(c1983_0))

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

(19) Complex Obligation (BEST)

(20) Obligation:

Innermost TRS:
Rules:
terms(N) → cons(recip(sqr(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, first(X, Z))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: s:0' → cons:nil
cons :: recip → cons:nil → cons:nil
recip :: s:0' → recip
sqr :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → cons:nil → cons:nil
nil :: cons:nil
half :: s:0' → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_cons:nil4_0 :: Nat → cons:nil
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(n870_0)) → gen_s:0'5_0(*(2, n870_0)), rt ∈ Ω(1 + n870₀)
sqr(gen_s:0'5_0(n1212_0)) → gen_s:0'5_0(*(n1212_0, n1212_0)), rt ∈ Ω(1 + n1212₀ + n1212₀² + n1212₀³)
first(gen_s:0'5_0(n1982_0), gen_cons:nil4_0(n1982_0)) → gen_cons:nil4_0(n1982_0), rt ∈ Ω(1 + n1982₀)

Generator Equations:
gen_cons:nil4_0(0) ⇔ nil
gen_cons:nil4_0(+(x, 1)) ⇔ cons(recip(0'), gen_cons:nil4_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

(21) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
half(gen_s:0'5_0(*(2, n2365_0))) → gen_s:0'5_0(n2365_0), rt ∈ Ω(1 + n2365₀)

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

Induction Step:
half(gen_s:0'5_0(*(2, +(n2365_0, 1)))) →_R^Ω(1)
s(half(gen_s:0'5_0(*(2, n2365_0)))) →_IH
s(gen_s:0'5_0(c2366_0))

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

(22) Complex Obligation (BEST)

(23) Obligation:

Innermost TRS:
Rules:
terms(N) → cons(recip(sqr(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, first(X, Z))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: s:0' → cons:nil
cons :: recip → cons:nil → cons:nil
recip :: s:0' → recip
sqr :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → cons:nil → cons:nil
nil :: cons:nil
half :: s:0' → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_cons:nil4_0 :: Nat → cons:nil
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(n870_0)) → gen_s:0'5_0(*(2, n870_0)), rt ∈ Ω(1 + n870₀)
sqr(gen_s:0'5_0(n1212_0)) → gen_s:0'5_0(*(n1212_0, n1212_0)), rt ∈ Ω(1 + n1212₀ + n1212₀² + n1212₀³)
first(gen_s:0'5_0(n1982_0), gen_cons:nil4_0(n1982_0)) → gen_cons:nil4_0(n1982_0), rt ∈ Ω(1 + n1982₀)
half(gen_s:0'5_0(*(2, n2365_0))) → gen_s:0'5_0(n2365_0), rt ∈ Ω(1 + n2365₀)

Generator Equations:
gen_cons:nil4_0(0) ⇔ nil
gen_cons:nil4_0(+(x, 1)) ⇔ cons(recip(0'), gen_cons:nil4_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.

(24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
sqr(gen_s:0'5_0(n1212_0)) → gen_s:0'5_0(*(n1212_0, n1212_0)), rt ∈ Ω(1 + n1212₀ + n1212₀² + n1212₀³)

(25) BOUNDS(n^3, INF)

(26) Obligation:

Innermost TRS:
Rules:
terms(N) → cons(recip(sqr(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, first(X, Z))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: s:0' → cons:nil
cons :: recip → cons:nil → cons:nil
recip :: s:0' → recip
sqr :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → cons:nil → cons:nil
nil :: cons:nil
half :: s:0' → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_cons:nil4_0 :: Nat → cons:nil
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(n870_0)) → gen_s:0'5_0(*(2, n870_0)), rt ∈ Ω(1 + n870₀)
sqr(gen_s:0'5_0(n1212_0)) → gen_s:0'5_0(*(n1212_0, n1212_0)), rt ∈ Ω(1 + n1212₀ + n1212₀² + n1212₀³)
first(gen_s:0'5_0(n1982_0), gen_cons:nil4_0(n1982_0)) → gen_cons:nil4_0(n1982_0), rt ∈ Ω(1 + n1982₀)
half(gen_s:0'5_0(*(2, n2365_0))) → gen_s:0'5_0(n2365_0), rt ∈ Ω(1 + n2365₀)

Generator Equations:
gen_cons:nil4_0(0) ⇔ nil
gen_cons:nil4_0(+(x, 1)) ⇔ cons(recip(0'), gen_cons:nil4_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.

(27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
sqr(gen_s:0'5_0(n1212_0)) → gen_s:0'5_0(*(n1212_0, n1212_0)), rt ∈ Ω(1 + n1212₀ + n1212₀² + n1212₀³)

(28) BOUNDS(n^3, INF)

(29) Obligation:

Innermost TRS:
Rules:
terms(N) → cons(recip(sqr(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, first(X, Z))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: s:0' → cons:nil
cons :: recip → cons:nil → cons:nil
recip :: s:0' → recip
sqr :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → cons:nil → cons:nil
nil :: cons:nil
half :: s:0' → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_cons:nil4_0 :: Nat → cons:nil
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(n870_0)) → gen_s:0'5_0(*(2, n870_0)), rt ∈ Ω(1 + n870₀)
sqr(gen_s:0'5_0(n1212_0)) → gen_s:0'5_0(*(n1212_0, n1212_0)), rt ∈ Ω(1 + n1212₀ + n1212₀² + n1212₀³)
first(gen_s:0'5_0(n1982_0), gen_cons:nil4_0(n1982_0)) → gen_cons:nil4_0(n1982_0), rt ∈ Ω(1 + n1982₀)

Generator Equations:
gen_cons:nil4_0(0) ⇔ nil
gen_cons:nil4_0(+(x, 1)) ⇔ cons(recip(0'), gen_cons:nil4_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.

(30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
sqr(gen_s:0'5_0(n1212_0)) → gen_s:0'5_0(*(n1212_0, n1212_0)), rt ∈ Ω(1 + n1212₀ + n1212₀² + n1212₀³)

(31) BOUNDS(n^3, INF)

(32) Obligation:

Innermost TRS:
Rules:
terms(N) → cons(recip(sqr(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, first(X, Z))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: s:0' → cons:nil
cons :: recip → cons:nil → cons:nil
recip :: s:0' → recip
sqr :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → cons:nil → cons:nil
nil :: cons:nil
half :: s:0' → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_cons:nil4_0 :: Nat → cons:nil
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(n870_0)) → gen_s:0'5_0(*(2, n870_0)), rt ∈ Ω(1 + n870₀)
sqr(gen_s:0'5_0(n1212_0)) → gen_s:0'5_0(*(n1212_0, n1212_0)), rt ∈ Ω(1 + n1212₀ + n1212₀² + n1212₀³)

Generator Equations:
gen_cons:nil4_0(0) ⇔ nil
gen_cons:nil4_0(+(x, 1)) ⇔ cons(recip(0'), gen_cons:nil4_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.

(33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n³) was proven with the following lemma:
sqr(gen_s:0'5_0(n1212_0)) → gen_s:0'5_0(*(n1212_0, n1212_0)), rt ∈ Ω(1 + n1212₀ + n1212₀² + n1212₀³)

(34) BOUNDS(n^3, INF)

(35) Obligation:

Innermost TRS:
Rules:
terms(N) → cons(recip(sqr(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, first(X, Z))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: s:0' → cons:nil
cons :: recip → cons:nil → cons:nil
recip :: s:0' → recip
sqr :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → cons:nil → cons:nil
nil :: cons:nil
half :: s:0' → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_cons:nil4_0 :: Nat → cons:nil
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(n870_0)) → gen_s:0'5_0(*(2, n870_0)), rt ∈ Ω(1 + n870₀)

Generator Equations:
gen_cons:nil4_0(0) ⇔ nil
gen_cons:nil4_0(+(x, 1)) ⇔ cons(recip(0'), gen_cons:nil4_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.

(36) 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)

(37) BOUNDS(n^1, INF)

(38) Obligation:

Innermost TRS:
Rules:
terms(N) → cons(recip(sqr(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, first(X, Z))
half(0') → 0'
half(s(0')) → 0'
half(s(s(X))) → s(half(X))
half(dbl(X)) → X

Types:
terms :: s:0' → cons:nil
cons :: recip → cons:nil → cons:nil
recip :: s:0' → recip
sqr :: s:0' → s:0'
s :: s:0' → s:0'
0' :: s:0'
add :: s:0' → s:0' → s:0'
dbl :: s:0' → s:0'
first :: s:0' → cons:nil → cons:nil
nil :: cons:nil
half :: s:0' → s:0'
hole_cons:nil1_0 :: cons:nil
hole_s:0'2_0 :: s:0'
hole_recip3_0 :: recip
gen_cons:nil4_0 :: Nat → cons:nil
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_cons:nil4_0(0) ⇔ nil
gen_cons:nil4_0(+(x, 1)) ⇔ cons(recip(0'), gen_cons:nil4_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.

(39) 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)

(40) BOUNDS(n^1, INF)