The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(2^n, INF).

0 CpxTRS
↳1 DecreasingLoopProof (⇔, 293 ms)
↳2 BOUNDS(2^n, INF)
↳3 RenamingProof (⇔, 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), 325 ms)
    ↳10 BEST
        ↳11 typed CpxTrs
            ↳12 NoRewriteLemmaProof (LOWER BOUND(ID), 293 ms)
            ↳13 typed CpxTrs
            ↳14 NoRewriteLemmaProof (LOWER BOUND(ID), 116 ms)
            ↳15 typed CpxTrs
            ↳16 LowerBoundsProof (⇔, 0 ms)
            ↳17 BOUNDS(n^1, INF)
        ↳18 typed CpxTrs
            ↳19 LowerBoundsProof (⇔, 0 ms)
            ↳20 BOUNDS(n^1, INF)

(0) Obligation:

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

f(S(x'), S(x)) → h(g(x', S(x)), f(S(S(x')), x))
h(0, S(x)) → h(0, x)
h(0, 0) → 0
g(S(x), S(x')) → h(f(S(x), S(x')), g(x, S(S(x'))))
g(S(x), 0) → 0
f(S(x), 0) → 0
h(S(x), x2) → h(x, x2)
g(0, x2) → 0
f(0, x2) → 0

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
f(S(S(x15_1)), S(x)) →+ h(h(h(g(x15_1, S(x)), f(S(S(x15_1)), x)), g(x15_1, S(S(x)))), f(S(S(S(x15_1))), x))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,1].
The pumping substitution is [x / S(x)].
The result substitution is [ ].

The rewrite sequence
f(S(S(x15_1)), S(x)) →+ h(h(h(g(x15_1, S(x)), f(S(S(x15_1)), x)), g(x15_1, S(S(x)))), f(S(S(S(x15_1))), 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 [x15_1 / S(x15_1)].

(2) BOUNDS(2^n, 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:

f(S(x'), S(x)) → h(g(x', S(x)), f(S(S(x')), x))
h(0', S(x)) → h(0', x)
h(0', 0') → 0'
g(S(x), S(x')) → h(f(S(x), S(x')), g(x, S(S(x'))))
g(S(x), 0') → 0'
f(S(x), 0') → 0'
h(S(x), x2) → h(x, x2)
g(0', x2) → 0'
f(0', x2) → 0'

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
f(S(x'), S(x)) → h(g(x', S(x)), f(S(S(x')), x))
h(0', S(x)) → h(0', x)
h(0', 0') → 0'
g(S(x), S(x')) → h(f(S(x), S(x')), g(x, S(S(x'))))
g(S(x), 0') → 0'
f(S(x), 0') → 0'
h(S(x), x2) → h(x, x2)
g(0', x2) → 0'
f(0', x2) → 0'

Types:
f :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
h :: S:0' → S:0' → S:0'
g :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat → S:0'

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

Heuristically decided to analyse the following defined symbols:
f, h, g

They will be analysed ascendingly in the following order:
h < f
f = g
h < g

(8) Obligation:

Innermost TRS:
Rules:
f(S(x'), S(x)) → h(g(x', S(x)), f(S(S(x')), x))
h(0', S(x)) → h(0', x)
h(0', 0') → 0'
g(S(x), S(x')) → h(f(S(x), S(x')), g(x, S(S(x'))))
g(S(x), 0') → 0'
f(S(x), 0') → 0'
h(S(x), x2) → h(x, x2)
g(0', x2) → 0'
f(0', x2) → 0'

Types:
f :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
h :: S:0' → S:0' → S:0'
g :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat → S:0'

Generator Equations:
gen_S:0'2_0(0) ⇔ 0'
gen_S:0'2_0(+(x, 1)) ⇔ S(gen_S:0'2_0(x))

The following defined symbols remain to be analysed:
h, f, g

They will be analysed ascendingly in the following order:
h < f
f = g
h < g

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

Proved the following rewrite lemma:
h(gen_S:0'2_0(0), gen_S:0'2_0(n4_0)) → gen_S:0'2_0(0), rt ∈ Ω(1 + n40)

Induction Base:
h(gen_S:0'2_0(0), gen_S:0'2_0(0)) →RΩ(1)
0'

Induction Step:
h(gen_S:0'2_0(0), gen_S:0'2_0(+(n4_0, 1))) →RΩ(1)
h(0', gen_S:0'2_0(n4_0)) →IH
gen_S:0'2_0(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:
f(S(x'), S(x)) → h(g(x', S(x)), f(S(S(x')), x))
h(0', S(x)) → h(0', x)
h(0', 0') → 0'
g(S(x), S(x')) → h(f(S(x), S(x')), g(x, S(S(x'))))
g(S(x), 0') → 0'
f(S(x), 0') → 0'
h(S(x), x2) → h(x, x2)
g(0', x2) → 0'
f(0', x2) → 0'

Types:
f :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
h :: S:0' → S:0' → S:0'
g :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat → S:0'

Lemmas:
h(gen_S:0'2_0(0), gen_S:0'2_0(n4_0)) → gen_S:0'2_0(0), rt ∈ Ω(1 + n40)

Generator Equations:
gen_S:0'2_0(0) ⇔ 0'
gen_S:0'2_0(+(x, 1)) ⇔ S(gen_S:0'2_0(x))

The following defined symbols remain to be analysed:
g, f

They will be analysed ascendingly in the following order:
f = g

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

Innermost TRS:
Rules:
f(S(x'), S(x)) → h(g(x', S(x)), f(S(S(x')), x))
h(0', S(x)) → h(0', x)
h(0', 0') → 0'
g(S(x), S(x')) → h(f(S(x), S(x')), g(x, S(S(x'))))
g(S(x), 0') → 0'
f(S(x), 0') → 0'
h(S(x), x2) → h(x, x2)
g(0', x2) → 0'
f(0', x2) → 0'

Types:
f :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
h :: S:0' → S:0' → S:0'
g :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat → S:0'

Lemmas:
h(gen_S:0'2_0(0), gen_S:0'2_0(n4_0)) → gen_S:0'2_0(0), rt ∈ Ω(1 + n40)

Generator Equations:
gen_S:0'2_0(0) ⇔ 0'
gen_S:0'2_0(+(x, 1)) ⇔ S(gen_S:0'2_0(x))

The following defined symbols remain to be analysed:
f

They will be analysed ascendingly in the following order:
f = g

(14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(15) Obligation:

Innermost TRS:
Rules:
f(S(x'), S(x)) → h(g(x', S(x)), f(S(S(x')), x))
h(0', S(x)) → h(0', x)
h(0', 0') → 0'
g(S(x), S(x')) → h(f(S(x), S(x')), g(x, S(S(x'))))
g(S(x), 0') → 0'
f(S(x), 0') → 0'
h(S(x), x2) → h(x, x2)
g(0', x2) → 0'
f(0', x2) → 0'

Types:
f :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
h :: S:0' → S:0' → S:0'
g :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat → S:0'

Lemmas:
h(gen_S:0'2_0(0), gen_S:0'2_0(n4_0)) → gen_S:0'2_0(0), rt ∈ Ω(1 + n40)

Generator Equations:
gen_S:0'2_0(0) ⇔ 0'
gen_S:0'2_0(+(x, 1)) ⇔ S(gen_S:0'2_0(x))

No more defined symbols left to analyse.

(16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
h(gen_S:0'2_0(0), gen_S:0'2_0(n4_0)) → gen_S:0'2_0(0), rt ∈ Ω(1 + n40)

(17) BOUNDS(n^1, INF)

(18) Obligation:

Innermost TRS:
Rules:
f(S(x'), S(x)) → h(g(x', S(x)), f(S(S(x')), x))
h(0', S(x)) → h(0', x)
h(0', 0') → 0'
g(S(x), S(x')) → h(f(S(x), S(x')), g(x, S(S(x'))))
g(S(x), 0') → 0'
f(S(x), 0') → 0'
h(S(x), x2) → h(x, x2)
g(0', x2) → 0'
f(0', x2) → 0'

Types:
f :: S:0' → S:0' → S:0'
S :: S:0' → S:0'
h :: S:0' → S:0' → S:0'
g :: S:0' → S:0' → S:0'
0' :: S:0'
hole_S:0'1_0 :: S:0'
gen_S:0'2_0 :: Nat → S:0'

Lemmas:
h(gen_S:0'2_0(0), gen_S:0'2_0(n4_0)) → gen_S:0'2_0(0), rt ∈ Ω(1 + n40)

Generator Equations:
gen_S:0'2_0(0) ⇔ 0'
gen_S:0'2_0(+(x, 1)) ⇔ S(gen_S:0'2_0(x))

No more defined symbols left to analyse.

(19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
h(gen_S:0'2_0(0), gen_S:0'2_0(n4_0)) → gen_S:0'2_0(0), rt ∈ Ω(1 + n40)

(20) BOUNDS(n^1, INF)