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:

top(sent(x)) → top(check(rest(x)))
rest(nil) → sent(nil)
rest(cons(x, y)) → sent(y)
check(sent(x)) → sent(check(x))
check(rest(x)) → rest(check(x))
check(cons(x, y)) → cons(check(x), y)
check(cons(x, y)) → cons(x, check(y))
check(cons(x, y)) → cons(x, y)

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:

top(sent(x)) → top(check(rest(x)))
rest(nil) → sent(nil)
rest(cons(x, y)) → sent(y)
check(sent(x)) → sent(check(x))
check(rest(x)) → rest(check(x))
check(cons(x, y)) → cons(check(x), y)
check(cons(x, y)) → cons(x, check(y))
check(cons(x, y)) → cons(x, y)

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
top(sent(x)) → top(check(rest(x)))
rest(nil) → sent(nil)
rest(cons(x, y)) → sent(y)
check(sent(x)) → sent(check(x))
check(rest(x)) → rest(check(x))
check(cons(x, y)) → cons(check(x), y)
check(cons(x, y)) → cons(x, check(y))
check(cons(x, y)) → cons(x, y)

Types:
top :: sent:nil:cons → top
sent :: sent:nil:cons → sent:nil:cons
check :: sent:nil:cons → sent:nil:cons
rest :: sent:nil:cons → sent:nil:cons
nil :: sent:nil:cons
cons :: sent:nil:cons → sent:nil:cons → sent:nil:cons
hole_top1_0 :: top
hole_sent:nil:cons2_0 :: sent:nil:cons
gen_sent:nil:cons3_0 :: Nat → sent:nil:cons

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

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

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

(6) Obligation:

Innermost TRS:
Rules:
top(sent(x)) → top(check(rest(x)))
rest(nil) → sent(nil)
rest(cons(x, y)) → sent(y)
check(sent(x)) → sent(check(x))
check(rest(x)) → rest(check(x))
check(cons(x, y)) → cons(check(x), y)
check(cons(x, y)) → cons(x, check(y))
check(cons(x, y)) → cons(x, y)

Types:
top :: sent:nil:cons → top
sent :: sent:nil:cons → sent:nil:cons
check :: sent:nil:cons → sent:nil:cons
rest :: sent:nil:cons → sent:nil:cons
nil :: sent:nil:cons
cons :: sent:nil:cons → sent:nil:cons → sent:nil:cons
hole_top1_0 :: top
hole_sent:nil:cons2_0 :: sent:nil:cons
gen_sent:nil:cons3_0 :: Nat → sent:nil:cons

Generator Equations:
gen_sent:nil:cons3_0(0) ⇔ nil
gen_sent:nil:cons3_0(+(x, 1)) ⇔ sent(gen_sent:nil:cons3_0(x))

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

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

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
check(gen_sent:nil:cons3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Induction Base:
check(gen_sent:nil:cons3_0(+(1, 0)))

Induction Step:
check(gen_sent:nil:cons3_0(+(1, +(n5_0, 1)))) →_R^Ω(1)
sent(check(gen_sent:nil:cons3_0(+(1, n5_0)))) →_IH
sent(*4_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:
top(sent(x)) → top(check(rest(x)))
rest(nil) → sent(nil)
rest(cons(x, y)) → sent(y)
check(sent(x)) → sent(check(x))
check(rest(x)) → rest(check(x))
check(cons(x, y)) → cons(check(x), y)
check(cons(x, y)) → cons(x, check(y))
check(cons(x, y)) → cons(x, y)

Types:
top :: sent:nil:cons → top
sent :: sent:nil:cons → sent:nil:cons
check :: sent:nil:cons → sent:nil:cons
rest :: sent:nil:cons → sent:nil:cons
nil :: sent:nil:cons
cons :: sent:nil:cons → sent:nil:cons → sent:nil:cons
hole_top1_0 :: top
hole_sent:nil:cons2_0 :: sent:nil:cons
gen_sent:nil:cons3_0 :: Nat → sent:nil:cons

Lemmas:
check(gen_sent:nil:cons3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_sent:nil:cons3_0(0) ⇔ nil
gen_sent:nil:cons3_0(+(x, 1)) ⇔ sent(gen_sent:nil:cons3_0(x))

The following defined symbols remain to be analysed:
top

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(11) Obligation:

Innermost TRS:
Rules:
top(sent(x)) → top(check(rest(x)))
rest(nil) → sent(nil)
rest(cons(x, y)) → sent(y)
check(sent(x)) → sent(check(x))
check(rest(x)) → rest(check(x))
check(cons(x, y)) → cons(check(x), y)
check(cons(x, y)) → cons(x, check(y))
check(cons(x, y)) → cons(x, y)

Types:
top :: sent:nil:cons → top
sent :: sent:nil:cons → sent:nil:cons
check :: sent:nil:cons → sent:nil:cons
rest :: sent:nil:cons → sent:nil:cons
nil :: sent:nil:cons
cons :: sent:nil:cons → sent:nil:cons → sent:nil:cons
hole_top1_0 :: top
hole_sent:nil:cons2_0 :: sent:nil:cons
gen_sent:nil:cons3_0 :: Nat → sent:nil:cons

Lemmas:
check(gen_sent:nil:cons3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_sent:nil:cons3_0(0) ⇔ nil
gen_sent:nil:cons3_0(+(x, 1)) ⇔ sent(gen_sent:nil:cons3_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
check(gen_sent:nil:cons3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
top(sent(x)) → top(check(rest(x)))
rest(nil) → sent(nil)
rest(cons(x, y)) → sent(y)
check(sent(x)) → sent(check(x))
check(rest(x)) → rest(check(x))
check(cons(x, y)) → cons(check(x), y)
check(cons(x, y)) → cons(x, check(y))
check(cons(x, y)) → cons(x, y)

Types:
top :: sent:nil:cons → top
sent :: sent:nil:cons → sent:nil:cons
check :: sent:nil:cons → sent:nil:cons
rest :: sent:nil:cons → sent:nil:cons
nil :: sent:nil:cons
cons :: sent:nil:cons → sent:nil:cons → sent:nil:cons
hole_top1_0 :: top
hole_sent:nil:cons2_0 :: sent:nil:cons
gen_sent:nil:cons3_0 :: Nat → sent:nil:cons

Lemmas:
check(gen_sent:nil:cons3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_sent:nil:cons3_0(0) ⇔ nil
gen_sent:nil:cons3_0(+(x, 1)) ⇔ sent(gen_sent:nil:cons3_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
check(gen_sent:nil:cons3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(16) BOUNDS(n^1, INF)