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:

dbl(S(0), S(0)) → S(S(S(S(0))))
save(S(x)) → dbl(0, save(x))
save(0) → 0
dbl(0, y) → y

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
save(S(x)) →⁺ dbl(0, save(x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [x / S(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:

dbl(S(0'), S(0')) → S(S(S(S(0'))))
save(S(x)) → dbl(0', save(x))
save(0') → 0'
dbl(0', y) → y

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
dbl(S(0'), S(0')) → S(S(S(S(0'))))
save(S(x)) → dbl(0', save(x))
save(0') → 0'
dbl(0', y) → y

Types:
dbl :: 0':S → 0':S → 0':S
S :: 0':S → 0':S
0' :: 0':S
save :: 0':S → 0':S
hole_0':S1_0 :: 0':S
gen_0':S2_0 :: Nat → 0':S

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

Heuristically decided to analyse the following defined symbols:
save

(8) Obligation:

Innermost TRS:
Rules:
dbl(S(0'), S(0')) → S(S(S(S(0'))))
save(S(x)) → dbl(0', save(x))
save(0') → 0'
dbl(0', y) → y

Types:
dbl :: 0':S → 0':S → 0':S
S :: 0':S → 0':S
0' :: 0':S
save :: 0':S → 0':S
hole_0':S1_0 :: 0':S
gen_0':S2_0 :: Nat → 0':S

Generator Equations:
gen_0':S2_0(0) ⇔ 0'
gen_0':S2_0(+(x, 1)) ⇔ S(gen_0':S2_0(x))

The following defined symbols remain to be analysed:
save

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

Proved the following rewrite lemma:
save(gen_0':S2_0(n4_0)) → gen_0':S2_0(0), rt ∈ Ω(1 + n4₀)

Induction Base:
save(gen_0':S2_0(0)) →_R^Ω(1)
0'

Induction Step:
save(gen_0':S2_0(+(n4_0, 1))) →_R^Ω(1)
dbl(0', save(gen_0':S2_0(n4_0))) →_IH
dbl(0', gen_0':S2_0(0)) →_R^Ω(1)
gen_0':S2_0(0)

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

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
dbl(S(0'), S(0')) → S(S(S(S(0'))))
save(S(x)) → dbl(0', save(x))
save(0') → 0'
dbl(0', y) → y

Types:
dbl :: 0':S → 0':S → 0':S
S :: 0':S → 0':S
0' :: 0':S
save :: 0':S → 0':S
hole_0':S1_0 :: 0':S
gen_0':S2_0 :: Nat → 0':S

Lemmas:
save(gen_0':S2_0(n4_0)) → gen_0':S2_0(0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':S2_0(0) ⇔ 0'
gen_0':S2_0(+(x, 1)) ⇔ S(gen_0':S2_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
save(gen_0':S2_0(n4_0)) → gen_0':S2_0(0), rt ∈ Ω(1 + n4₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
dbl(S(0'), S(0')) → S(S(S(S(0'))))
save(S(x)) → dbl(0', save(x))
save(0') → 0'
dbl(0', y) → y

Types:
dbl :: 0':S → 0':S → 0':S
S :: 0':S → 0':S
0' :: 0':S
save :: 0':S → 0':S
hole_0':S1_0 :: 0':S
gen_0':S2_0 :: Nat → 0':S

Lemmas:
save(gen_0':S2_0(n4_0)) → gen_0':S2_0(0), rt ∈ Ω(1 + n4₀)

Generator Equations:
gen_0':S2_0(0) ⇔ 0'
gen_0':S2_0(+(x, 1)) ⇔ S(gen_0':S2_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
save(gen_0':S2_0(n4_0)) → gen_0':S2_0(0), rt ∈ Ω(1 + n4₀)

(16) BOUNDS(n^1, INF)