The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

0 CpxTRS
↳1 RenamingProof (⇔, 0 ms)
↳2 CpxRelTRS
↳3 SlicingProof (LOWER BOUND(ID), 0 ms)
↳4 CpxRelTRS
↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 typed CpxTrs
↳7 OrderProof (LOWER BOUND(ID), 0 ms)
↳8 typed CpxTrs
↳9 RewriteLemmaProof (LOWER BOUND(ID), 402 ms)
↳10 BEST
    ↳11 typed CpxTrs
        ↳12 RewriteLemmaProof (LOWER BOUND(ID), 93 ms)
        ↳13 BEST
            ↳14 typed CpxTrs
                ↳15 LowerBoundsProof (⇔, 0 ms)
                ↳16 BOUNDS(n^1, INF)
            ↳17 typed CpxTrs
                ↳18 LowerBoundsProof (⇔, 0 ms)
                ↳19 BOUNDS(n^1, INF)
    ↳20 typed CpxTrs
        ↳21 LowerBoundsProof (⇔, 0 ms)
        ↳22 BOUNDS(n^1, INF)

(0) Obligation:

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

int(0, 0) → .(0, nil)
int(0, s(y)) → .(0, int(s(0), s(y)))
int(s(x), 0) → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(x, y)) → .(s(x), int_list(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:

int(0', 0') → .(0', nil)
int(0', s(y)) → .(0', int(s(0'), s(y)))
int(s(x), 0') → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(x, y)) → .(s(x), int_list(y))

S is empty.
Rewrite Strategy: INNERMOST

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
./0

(4) Obligation:

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

int(0', 0') → .(nil)
int(0', s(y)) → .(int(s(0'), s(y)))
int(s(x), 0') → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(y)) → .(int_list(y))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
int(0', 0') → .(nil)
int(0', s(y)) → .(int(s(0'), s(y)))
int(s(x), 0') → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(y)) → .(int_list(y))

Types:
int :: 0':s → 0':s → nil:.
0' :: 0':s
. :: nil:. → nil:.
nil :: nil:.
s :: 0':s → 0':s
int_list :: nil:. → nil:.
hole_nil:.1_0 :: nil:.
hole_0':s2_0 :: 0':s
gen_nil:.3_0 :: Nat → nil:.
gen_0':s4_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
int, int_list

They will be analysed ascendingly in the following order:
int_list < int

(8) Obligation:

Innermost TRS:
Rules:
int(0', 0') → .(nil)
int(0', s(y)) → .(int(s(0'), s(y)))
int(s(x), 0') → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(y)) → .(int_list(y))

Types:
int :: 0':s → 0':s → nil:.
0' :: 0':s
. :: nil:. → nil:.
nil :: nil:.
s :: 0':s → 0':s
int_list :: nil:. → nil:.
hole_nil:.1_0 :: nil:.
hole_0':s2_0 :: 0':s
gen_nil:.3_0 :: Nat → nil:.
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_nil:.3_0(0) ⇔ nil
gen_nil:.3_0(+(x, 1)) ⇔ .(gen_nil:.3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
int_list, int

They will be analysed ascendingly in the following order:
int_list < int

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

Proved the following rewrite lemma:
int_list(gen_nil:.3_0(n6_0)) → gen_nil:.3_0(n6_0), rt ∈ Ω(1 + n60)

Induction Base:
int_list(gen_nil:.3_0(0)) →RΩ(1)
nil

Induction Step:
int_list(gen_nil:.3_0(+(n6_0, 1))) →RΩ(1)
.(int_list(gen_nil:.3_0(n6_0))) →IH
.(gen_nil:.3_0(c7_0))

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

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
int(0', 0') → .(nil)
int(0', s(y)) → .(int(s(0'), s(y)))
int(s(x), 0') → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(y)) → .(int_list(y))

Types:
int :: 0':s → 0':s → nil:.
0' :: 0':s
. :: nil:. → nil:.
nil :: nil:.
s :: 0':s → 0':s
int_list :: nil:. → nil:.
hole_nil:.1_0 :: nil:.
hole_0':s2_0 :: 0':s
gen_nil:.3_0 :: Nat → nil:.
gen_0':s4_0 :: Nat → 0':s

Lemmas:
int_list(gen_nil:.3_0(n6_0)) → gen_nil:.3_0(n6_0), rt ∈ Ω(1 + n60)

Generator Equations:
gen_nil:.3_0(0) ⇔ nil
gen_nil:.3_0(+(x, 1)) ⇔ .(gen_nil:.3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
int

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

Proved the following rewrite lemma:
int(gen_0':s4_0(n208_0), gen_0':s4_0(n208_0)) → gen_nil:.3_0(1), rt ∈ Ω(1 + n2080)

Induction Base:
int(gen_0':s4_0(0), gen_0':s4_0(0)) →RΩ(1)
.(nil)

Induction Step:
int(gen_0':s4_0(+(n208_0, 1)), gen_0':s4_0(+(n208_0, 1))) →RΩ(1)
int_list(int(gen_0':s4_0(n208_0), gen_0':s4_0(n208_0))) →IH
int_list(gen_nil:.3_0(1)) →LΩ(2)
gen_nil:.3_0(1)

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

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
Rules:
int(0', 0') → .(nil)
int(0', s(y)) → .(int(s(0'), s(y)))
int(s(x), 0') → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(y)) → .(int_list(y))

Types:
int :: 0':s → 0':s → nil:.
0' :: 0':s
. :: nil:. → nil:.
nil :: nil:.
s :: 0':s → 0':s
int_list :: nil:. → nil:.
hole_nil:.1_0 :: nil:.
hole_0':s2_0 :: 0':s
gen_nil:.3_0 :: Nat → nil:.
gen_0':s4_0 :: Nat → 0':s

Lemmas:
int_list(gen_nil:.3_0(n6_0)) → gen_nil:.3_0(n6_0), rt ∈ Ω(1 + n60)
int(gen_0':s4_0(n208_0), gen_0':s4_0(n208_0)) → gen_nil:.3_0(1), rt ∈ Ω(1 + n2080)

Generator Equations:
gen_nil:.3_0(0) ⇔ nil
gen_nil:.3_0(+(x, 1)) ⇔ .(gen_nil:.3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
int_list(gen_nil:.3_0(n6_0)) → gen_nil:.3_0(n6_0), rt ∈ Ω(1 + n60)

(16) BOUNDS(n^1, INF)

(17) Obligation:

Innermost TRS:
Rules:
int(0', 0') → .(nil)
int(0', s(y)) → .(int(s(0'), s(y)))
int(s(x), 0') → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(y)) → .(int_list(y))

Types:
int :: 0':s → 0':s → nil:.
0' :: 0':s
. :: nil:. → nil:.
nil :: nil:.
s :: 0':s → 0':s
int_list :: nil:. → nil:.
hole_nil:.1_0 :: nil:.
hole_0':s2_0 :: 0':s
gen_nil:.3_0 :: Nat → nil:.
gen_0':s4_0 :: Nat → 0':s

Lemmas:
int_list(gen_nil:.3_0(n6_0)) → gen_nil:.3_0(n6_0), rt ∈ Ω(1 + n60)
int(gen_0':s4_0(n208_0), gen_0':s4_0(n208_0)) → gen_nil:.3_0(1), rt ∈ Ω(1 + n2080)

Generator Equations:
gen_nil:.3_0(0) ⇔ nil
gen_nil:.3_0(+(x, 1)) ⇔ .(gen_nil:.3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(18) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
int_list(gen_nil:.3_0(n6_0)) → gen_nil:.3_0(n6_0), rt ∈ Ω(1 + n60)

(19) BOUNDS(n^1, INF)

(20) Obligation:

Innermost TRS:
Rules:
int(0', 0') → .(nil)
int(0', s(y)) → .(int(s(0'), s(y)))
int(s(x), 0') → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(y)) → .(int_list(y))

Types:
int :: 0':s → 0':s → nil:.
0' :: 0':s
. :: nil:. → nil:.
nil :: nil:.
s :: 0':s → 0':s
int_list :: nil:. → nil:.
hole_nil:.1_0 :: nil:.
hole_0':s2_0 :: 0':s
gen_nil:.3_0 :: Nat → nil:.
gen_0':s4_0 :: Nat → 0':s

Lemmas:
int_list(gen_nil:.3_0(n6_0)) → gen_nil:.3_0(n6_0), rt ∈ Ω(1 + n60)

Generator Equations:
gen_nil:.3_0(0) ⇔ nil
gen_nil:.3_0(+(x, 1)) ⇔ .(gen_nil:.3_0(x))
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

(21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
int_list(gen_nil:.3_0(n6_0)) → gen_nil:.3_0(n6_0), rt ∈ Ω(1 + n60)

(22) BOUNDS(n^1, INF)