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:

top(free(x)) → top(check(new(x)))
new(free(x)) → free(new(x))
old(free(x)) → free(old(x))
new(serve) → free(serve)
old(serve) → free(serve)
check(free(x)) → free(check(x))
check(new(x)) → new(check(x))
check(old(x)) → old(check(x))
check(old(x)) → old(x)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

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

top(free(x)) → top(check(new(x)))
new(free(x)) → free(new(x))
old(free(x)) → free(old(x))
new(serve) → free(serve)
old(serve) → free(serve)
check(free(x)) → free(check(x))
check(new(x)) → new(check(x))
check(old(x)) → old(check(x))
check(old(x)) → old(x)

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
top(free(x)) → top(check(new(x)))
new(free(x)) → free(new(x))
old(free(x)) → free(old(x))
new(serve) → free(serve)
old(serve) → free(serve)
check(free(x)) → free(check(x))
check(new(x)) → new(check(x))
check(old(x)) → old(check(x))
check(old(x)) → old(x)

Types:
top :: free:serve → top
free :: free:serve → free:serve
check :: free:serve → free:serve
new :: free:serve → free:serve
old :: free:serve → free:serve
serve :: free:serve
hole_top1_0 :: top
hole_free:serve2_0 :: free:serve
gen_free:serve3_0 :: Nat → free:serve

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

Heuristically decided to analyse the following defined symbols:
top, check, new, old

They will be analysed ascendingly in the following order:
check < top
new < top
new < check
old < check

(8) Obligation:

TRS:
Rules:
top(free(x)) → top(check(new(x)))
new(free(x)) → free(new(x))
old(free(x)) → free(old(x))
new(serve) → free(serve)
old(serve) → free(serve)
check(free(x)) → free(check(x))
check(new(x)) → new(check(x))
check(old(x)) → old(check(x))
check(old(x)) → old(x)

Types:
top :: free:serve → top
free :: free:serve → free:serve
check :: free:serve → free:serve
new :: free:serve → free:serve
old :: free:serve → free:serve
serve :: free:serve
hole_top1_0 :: top
hole_free:serve2_0 :: free:serve
gen_free:serve3_0 :: Nat → free:serve

Generator Equations:
gen_free:serve3_0(0) ⇔ serve
gen_free:serve3_0(+(x, 1)) ⇔ free(gen_free:serve3_0(x))

The following defined symbols remain to be analysed:
new, top, check, old

They will be analysed ascendingly in the following order:
check < top
new < top
new < check
old < check

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

Proved the following rewrite lemma:
new(gen_free:serve3_0(n5_0)) → gen_free:serve3_0(+(1, n5_0)), rt ∈ Ω(1 + n5₀)

Induction Base:
new(gen_free:serve3_0(0)) →_R^Ω(1)
free(serve)

Induction Step:
new(gen_free:serve3_0(+(n5_0, 1))) →_R^Ω(1)
free(new(gen_free:serve3_0(n5_0))) →_IH
free(gen_free:serve3_0(+(1, c6_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:
top(free(x)) → top(check(new(x)))
new(free(x)) → free(new(x))
old(free(x)) → free(old(x))
new(serve) → free(serve)
old(serve) → free(serve)
check(free(x)) → free(check(x))
check(new(x)) → new(check(x))
check(old(x)) → old(check(x))
check(old(x)) → old(x)

Types:
top :: free:serve → top
free :: free:serve → free:serve
check :: free:serve → free:serve
new :: free:serve → free:serve
old :: free:serve → free:serve
serve :: free:serve
hole_top1_0 :: top
hole_free:serve2_0 :: free:serve
gen_free:serve3_0 :: Nat → free:serve

Lemmas:
new(gen_free:serve3_0(n5_0)) → gen_free:serve3_0(+(1, n5_0)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_free:serve3_0(0) ⇔ serve
gen_free:serve3_0(+(x, 1)) ⇔ free(gen_free:serve3_0(x))

The following defined symbols remain to be analysed:
old, top, check

They will be analysed ascendingly in the following order:
check < top
old < check

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

Proved the following rewrite lemma:
old(gen_free:serve3_0(n194_0)) → gen_free:serve3_0(+(1, n194_0)), rt ∈ Ω(1 + n194₀)

Induction Base:
old(gen_free:serve3_0(0)) →_R^Ω(1)
free(serve)

Induction Step:
old(gen_free:serve3_0(+(n194_0, 1))) →_R^Ω(1)
free(old(gen_free:serve3_0(n194_0))) →_IH
free(gen_free:serve3_0(+(1, c195_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:
top(free(x)) → top(check(new(x)))
new(free(x)) → free(new(x))
old(free(x)) → free(old(x))
new(serve) → free(serve)
old(serve) → free(serve)
check(free(x)) → free(check(x))
check(new(x)) → new(check(x))
check(old(x)) → old(check(x))
check(old(x)) → old(x)

Types:
top :: free:serve → top
free :: free:serve → free:serve
check :: free:serve → free:serve
new :: free:serve → free:serve
old :: free:serve → free:serve
serve :: free:serve
hole_top1_0 :: top
hole_free:serve2_0 :: free:serve
gen_free:serve3_0 :: Nat → free:serve

Lemmas:
new(gen_free:serve3_0(n5_0)) → gen_free:serve3_0(+(1, n5_0)), rt ∈ Ω(1 + n5₀)
old(gen_free:serve3_0(n194_0)) → gen_free:serve3_0(+(1, n194_0)), rt ∈ Ω(1 + n194₀)

Generator Equations:
gen_free:serve3_0(0) ⇔ serve
gen_free:serve3_0(+(x, 1)) ⇔ free(gen_free:serve3_0(x))

The following defined symbols remain to be analysed:
check, top

They will be analysed ascendingly in the following order:
check < top

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

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

(16) Obligation:

TRS:
Rules:
top(free(x)) → top(check(new(x)))
new(free(x)) → free(new(x))
old(free(x)) → free(old(x))
new(serve) → free(serve)
old(serve) → free(serve)
check(free(x)) → free(check(x))
check(new(x)) → new(check(x))
check(old(x)) → old(check(x))
check(old(x)) → old(x)

Types:
top :: free:serve → top
free :: free:serve → free:serve
check :: free:serve → free:serve
new :: free:serve → free:serve
old :: free:serve → free:serve
serve :: free:serve
hole_top1_0 :: top
hole_free:serve2_0 :: free:serve
gen_free:serve3_0 :: Nat → free:serve

Lemmas:
new(gen_free:serve3_0(n5_0)) → gen_free:serve3_0(+(1, n5_0)), rt ∈ Ω(1 + n5₀)
old(gen_free:serve3_0(n194_0)) → gen_free:serve3_0(+(1, n194_0)), rt ∈ Ω(1 + n194₀)

Generator Equations:
gen_free:serve3_0(0) ⇔ serve
gen_free:serve3_0(+(x, 1)) ⇔ free(gen_free:serve3_0(x))

The following defined symbols remain to be analysed:
top

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(18) Obligation:

TRS:
Rules:
top(free(x)) → top(check(new(x)))
new(free(x)) → free(new(x))
old(free(x)) → free(old(x))
new(serve) → free(serve)
old(serve) → free(serve)
check(free(x)) → free(check(x))
check(new(x)) → new(check(x))
check(old(x)) → old(check(x))
check(old(x)) → old(x)

Types:
top :: free:serve → top
free :: free:serve → free:serve
check :: free:serve → free:serve
new :: free:serve → free:serve
old :: free:serve → free:serve
serve :: free:serve
hole_top1_0 :: top
hole_free:serve2_0 :: free:serve
gen_free:serve3_0 :: Nat → free:serve

Lemmas:
new(gen_free:serve3_0(n5_0)) → gen_free:serve3_0(+(1, n5_0)), rt ∈ Ω(1 + n5₀)
old(gen_free:serve3_0(n194_0)) → gen_free:serve3_0(+(1, n194_0)), rt ∈ Ω(1 + n194₀)

Generator Equations:
gen_free:serve3_0(0) ⇔ serve
gen_free:serve3_0(+(x, 1)) ⇔ free(gen_free:serve3_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
new(gen_free:serve3_0(n5_0)) → gen_free:serve3_0(+(1, n5_0)), rt ∈ Ω(1 + n5₀)

(20) BOUNDS(n^1, INF)

(21) Obligation:

TRS:
Rules:
top(free(x)) → top(check(new(x)))
new(free(x)) → free(new(x))
old(free(x)) → free(old(x))
new(serve) → free(serve)
old(serve) → free(serve)
check(free(x)) → free(check(x))
check(new(x)) → new(check(x))
check(old(x)) → old(check(x))
check(old(x)) → old(x)

Types:
top :: free:serve → top
free :: free:serve → free:serve
check :: free:serve → free:serve
new :: free:serve → free:serve
old :: free:serve → free:serve
serve :: free:serve
hole_top1_0 :: top
hole_free:serve2_0 :: free:serve
gen_free:serve3_0 :: Nat → free:serve

Lemmas:
new(gen_free:serve3_0(n5_0)) → gen_free:serve3_0(+(1, n5_0)), rt ∈ Ω(1 + n5₀)
old(gen_free:serve3_0(n194_0)) → gen_free:serve3_0(+(1, n194_0)), rt ∈ Ω(1 + n194₀)

Generator Equations:
gen_free:serve3_0(0) ⇔ serve
gen_free:serve3_0(+(x, 1)) ⇔ free(gen_free:serve3_0(x))

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
new(gen_free:serve3_0(n5_0)) → gen_free:serve3_0(+(1, n5_0)), rt ∈ Ω(1 + n5₀)

(23) BOUNDS(n^1, INF)

(24) Obligation:

TRS:
Rules:
top(free(x)) → top(check(new(x)))
new(free(x)) → free(new(x))
old(free(x)) → free(old(x))
new(serve) → free(serve)
old(serve) → free(serve)
check(free(x)) → free(check(x))
check(new(x)) → new(check(x))
check(old(x)) → old(check(x))
check(old(x)) → old(x)

Types:
top :: free:serve → top
free :: free:serve → free:serve
check :: free:serve → free:serve
new :: free:serve → free:serve
old :: free:serve → free:serve
serve :: free:serve
hole_top1_0 :: top
hole_free:serve2_0 :: free:serve
gen_free:serve3_0 :: Nat → free:serve

Lemmas:
new(gen_free:serve3_0(n5_0)) → gen_free:serve3_0(+(1, n5_0)), rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_free:serve3_0(0) ⇔ serve
gen_free:serve3_0(+(x, 1)) ⇔ free(gen_free:serve3_0(x))

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
new(gen_free:serve3_0(n5_0)) → gen_free:serve3_0(+(1, n5_0)), rt ∈ Ω(1 + n5₀)

(26) BOUNDS(n^1, INF)