The Runtime Complexity (innermost) 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:

top1(free(x), y) → top2(check(new(x)), y)
top1(free(x), y) → top2(new(x), check(y))
top1(free(x), y) → top2(check(x), new(y))
top1(free(x), y) → top2(x, check(new(y)))
top2(x, free(y)) → top1(check(new(x)), y)
top2(x, free(y)) → top1(new(x), check(y))
top2(x, free(y)) → top1(check(x), new(y))
top2(x, free(y)) → top1(x, check(new(y)))
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: 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:

top1(free(x), y) → top2(check(new(x)), y)
top1(free(x), y) → top2(new(x), check(y))
top1(free(x), y) → top2(check(x), new(y))
top1(free(x), y) → top2(x, check(new(y)))
top2(x, free(y)) → top1(check(new(x)), y)
top2(x, free(y)) → top1(new(x), check(y))
top2(x, free(y)) → top1(check(x), new(y))
top2(x, free(y)) → top1(x, check(new(y)))
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: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
top1(free(x), y) → top2(check(new(x)), y)
top1(free(x), y) → top2(new(x), check(y))
top1(free(x), y) → top2(check(x), new(y))
top1(free(x), y) → top2(x, check(new(y)))
top2(x, free(y)) → top1(check(new(x)), y)
top2(x, free(y)) → top1(new(x), check(y))
top2(x, free(y)) → top1(check(x), new(y))
top2(x, free(y)) → top1(x, check(new(y)))
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:
top1 :: free:serve → free:serve → top1:top2
free :: free:serve → free:serve
top2 :: free:serve → free:serve → top1:top2
check :: free:serve → free:serve
new :: free:serve → free:serve
old :: free:serve → free:serve
serve :: free:serve
hole_top1:top21_0 :: top1:top2
hole_free:serve2_0 :: free:serve
gen_free:serve3_0 :: Nat → free:serve

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

Heuristically decided to analyse the following defined symbols:
top1, top2, check, new, old

They will be analysed ascendingly in the following order:
top1 = top2
check < top1
new < top1
check < top2
new < top2
new < check
old < check

(6) Obligation:

Innermost TRS:
Rules:
top1(free(x), y) → top2(check(new(x)), y)
top1(free(x), y) → top2(new(x), check(y))
top1(free(x), y) → top2(check(x), new(y))
top1(free(x), y) → top2(x, check(new(y)))
top2(x, free(y)) → top1(check(new(x)), y)
top2(x, free(y)) → top1(new(x), check(y))
top2(x, free(y)) → top1(check(x), new(y))
top2(x, free(y)) → top1(x, check(new(y)))
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:
top1 :: free:serve → free:serve → top1:top2
free :: free:serve → free:serve
top2 :: free:serve → free:serve → top1:top2
check :: free:serve → free:serve
new :: free:serve → free:serve
old :: free:serve → free:serve
serve :: free:serve
hole_top1:top21_0 :: top1:top2
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, top1, top2, check, old

They will be analysed ascendingly in the following order:
top1 = top2
check < top1
new < top1
check < top2
new < top2
new < check
old < check

(7) 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).

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
top1(free(x), y) → top2(check(new(x)), y)
top1(free(x), y) → top2(new(x), check(y))
top1(free(x), y) → top2(check(x), new(y))
top1(free(x), y) → top2(x, check(new(y)))
top2(x, free(y)) → top1(check(new(x)), y)
top2(x, free(y)) → top1(new(x), check(y))
top2(x, free(y)) → top1(check(x), new(y))
top2(x, free(y)) → top1(x, check(new(y)))
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:
top1 :: free:serve → free:serve → top1:top2
free :: free:serve → free:serve
top2 :: free:serve → free:serve → top1:top2
check :: free:serve → free:serve
new :: free:serve → free:serve
old :: free:serve → free:serve
serve :: free:serve
hole_top1:top21_0 :: top1:top2
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, top1, top2, check

They will be analysed ascendingly in the following order:
top1 = top2
check < top1
check < top2
old < check

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

Proved the following rewrite lemma:
old(gen_free:serve3_0(n254_0)) → gen_free:serve3_0(+(1, n254_0)), rt ∈ Ω(1 + n254₀)

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

Induction Step:
old(gen_free:serve3_0(+(n254_0, 1))) →_R^Ω(1)
free(old(gen_free:serve3_0(n254_0))) →_IH
free(gen_free:serve3_0(+(1, c255_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:
top1(free(x), y) → top2(check(new(x)), y)
top1(free(x), y) → top2(new(x), check(y))
top1(free(x), y) → top2(check(x), new(y))
top1(free(x), y) → top2(x, check(new(y)))
top2(x, free(y)) → top1(check(new(x)), y)
top2(x, free(y)) → top1(new(x), check(y))
top2(x, free(y)) → top1(check(x), new(y))
top2(x, free(y)) → top1(x, check(new(y)))
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:
top1 :: free:serve → free:serve → top1:top2
free :: free:serve → free:serve
top2 :: free:serve → free:serve → top1:top2
check :: free:serve → free:serve
new :: free:serve → free:serve
old :: free:serve → free:serve
serve :: free:serve
hole_top1:top21_0 :: top1:top2
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(n254_0)) → gen_free:serve3_0(+(1, n254_0)), rt ∈ Ω(1 + n254₀)

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, top1, top2

They will be analysed ascendingly in the following order:
top1 = top2
check < top1
check < top2

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

Proved the following rewrite lemma:
check(gen_free:serve3_0(+(1, n507_0))) → *4_0, rt ∈ Ω(n507₀)

Induction Base:
check(gen_free:serve3_0(+(1, 0)))

Induction Step:
check(gen_free:serve3_0(+(1, +(n507_0, 1)))) →_R^Ω(1)
free(check(gen_free:serve3_0(+(1, n507_0)))) →_IH
free(*4_0)

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

(14) Complex Obligation (BEST)

(15) Obligation:

Innermost TRS:
Rules:
top1(free(x), y) → top2(check(new(x)), y)
top1(free(x), y) → top2(new(x), check(y))
top1(free(x), y) → top2(check(x), new(y))
top1(free(x), y) → top2(x, check(new(y)))
top2(x, free(y)) → top1(check(new(x)), y)
top2(x, free(y)) → top1(new(x), check(y))
top2(x, free(y)) → top1(check(x), new(y))
top2(x, free(y)) → top1(x, check(new(y)))
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:
top1 :: free:serve → free:serve → top1:top2
free :: free:serve → free:serve
top2 :: free:serve → free:serve → top1:top2
check :: free:serve → free:serve
new :: free:serve → free:serve
old :: free:serve → free:serve
serve :: free:serve
hole_top1:top21_0 :: top1:top2
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(n254_0)) → gen_free:serve3_0(+(1, n254_0)), rt ∈ Ω(1 + n254₀)
check(gen_free:serve3_0(+(1, n507_0))) → *4_0, rt ∈ Ω(n507₀)

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:
top2, top1

They will be analysed ascendingly in the following order:
top1 = top2

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

Could not prove a rewrite lemma for the defined symbol top2.

(17) Obligation:

Innermost TRS:
Rules:
top1(free(x), y) → top2(check(new(x)), y)
top1(free(x), y) → top2(new(x), check(y))
top1(free(x), y) → top2(check(x), new(y))
top1(free(x), y) → top2(x, check(new(y)))
top2(x, free(y)) → top1(check(new(x)), y)
top2(x, free(y)) → top1(new(x), check(y))
top2(x, free(y)) → top1(check(x), new(y))
top2(x, free(y)) → top1(x, check(new(y)))
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:
top1 :: free:serve → free:serve → top1:top2
free :: free:serve → free:serve
top2 :: free:serve → free:serve → top1:top2
check :: free:serve → free:serve
new :: free:serve → free:serve
old :: free:serve → free:serve
serve :: free:serve
hole_top1:top21_0 :: top1:top2
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(n254_0)) → gen_free:serve3_0(+(1, n254_0)), rt ∈ Ω(1 + n254₀)
check(gen_free:serve3_0(+(1, n507_0))) → *4_0, rt ∈ Ω(n507₀)

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:
top1

They will be analysed ascendingly in the following order:
top1 = top2

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

Could not prove a rewrite lemma for the defined symbol top1.

(19) Obligation:

Innermost TRS:
Rules:
top1(free(x), y) → top2(check(new(x)), y)
top1(free(x), y) → top2(new(x), check(y))
top1(free(x), y) → top2(check(x), new(y))
top1(free(x), y) → top2(x, check(new(y)))
top2(x, free(y)) → top1(check(new(x)), y)
top2(x, free(y)) → top1(new(x), check(y))
top2(x, free(y)) → top1(check(x), new(y))
top2(x, free(y)) → top1(x, check(new(y)))
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:
top1 :: free:serve → free:serve → top1:top2
free :: free:serve → free:serve
top2 :: free:serve → free:serve → top1:top2
check :: free:serve → free:serve
new :: free:serve → free:serve
old :: free:serve → free:serve
serve :: free:serve
hole_top1:top21_0 :: top1:top2
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(n254_0)) → gen_free:serve3_0(+(1, n254_0)), rt ∈ Ω(1 + n254₀)
check(gen_free:serve3_0(+(1, n507_0))) → *4_0, rt ∈ Ω(n507₀)

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.

(20) 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)

(21) BOUNDS(n^1, INF)

(22) Obligation:

Innermost TRS:
Rules:
top1(free(x), y) → top2(check(new(x)), y)
top1(free(x), y) → top2(new(x), check(y))
top1(free(x), y) → top2(check(x), new(y))
top1(free(x), y) → top2(x, check(new(y)))
top2(x, free(y)) → top1(check(new(x)), y)
top2(x, free(y)) → top1(new(x), check(y))
top2(x, free(y)) → top1(check(x), new(y))
top2(x, free(y)) → top1(x, check(new(y)))
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:
top1 :: free:serve → free:serve → top1:top2
free :: free:serve → free:serve
top2 :: free:serve → free:serve → top1:top2
check :: free:serve → free:serve
new :: free:serve → free:serve
old :: free:serve → free:serve
serve :: free:serve
hole_top1:top21_0 :: top1:top2
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(n254_0)) → gen_free:serve3_0(+(1, n254_0)), rt ∈ Ω(1 + n254₀)
check(gen_free:serve3_0(+(1, n507_0))) → *4_0, rt ∈ Ω(n507₀)

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.

(23) 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)

(24) BOUNDS(n^1, INF)

(25) Obligation:

Innermost TRS:
Rules:
top1(free(x), y) → top2(check(new(x)), y)
top1(free(x), y) → top2(new(x), check(y))
top1(free(x), y) → top2(check(x), new(y))
top1(free(x), y) → top2(x, check(new(y)))
top2(x, free(y)) → top1(check(new(x)), y)
top2(x, free(y)) → top1(new(x), check(y))
top2(x, free(y)) → top1(check(x), new(y))
top2(x, free(y)) → top1(x, check(new(y)))
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:
top1 :: free:serve → free:serve → top1:top2
free :: free:serve → free:serve
top2 :: free:serve → free:serve → top1:top2
check :: free:serve → free:serve
new :: free:serve → free:serve
old :: free:serve → free:serve
serve :: free:serve
hole_top1:top21_0 :: top1:top2
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(n254_0)) → gen_free:serve3_0(+(1, n254_0)), rt ∈ Ω(1 + n254₀)

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.

(26) 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)

(27) BOUNDS(n^1, INF)

(28) Obligation:

Innermost TRS:
Rules:
top1(free(x), y) → top2(check(new(x)), y)
top1(free(x), y) → top2(new(x), check(y))
top1(free(x), y) → top2(check(x), new(y))
top1(free(x), y) → top2(x, check(new(y)))
top2(x, free(y)) → top1(check(new(x)), y)
top2(x, free(y)) → top1(new(x), check(y))
top2(x, free(y)) → top1(check(x), new(y))
top2(x, free(y)) → top1(x, check(new(y)))
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:
top1 :: free:serve → free:serve → top1:top2
free :: free:serve → free:serve
top2 :: free:serve → free:serve → top1:top2
check :: free:serve → free:serve
new :: free:serve → free:serve
old :: free:serve → free:serve
serve :: free:serve
hole_top1:top21_0 :: top1:top2
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.

(29) 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)

(30) BOUNDS(n^1, INF)