The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(2^n, 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(S(x'))), x))
g(S(x), S(x')) → h(f(S(x), S(x')), g(x, S(S(S(x')))))
h(0, S(x)) → h(0, x)
h(0, 0) → 0
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(x14_1)), S(x)) →⁺ h(h(h(g(x14_1, S(x)), f(S(S(S(x14_1))), x)), g(x14_1, S(S(S(x))))), f(S(S(S(S(x14_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 [x14_1 / S(x14_1)].

The rewrite sequence
f(S(S(x14_1)), S(x)) →⁺ h(h(h(g(x14_1, S(x)), f(S(S(S(x14_1))), x)), g(x14_1, S(S(S(x))))), f(S(S(S(S(x14_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 [x14_1 / S(S(x14_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(S(x'))), x))
g(S(x), S(x')) → h(f(S(x), S(x')), g(x, S(S(S(x')))))
h(0', S(x)) → h(0', x)
h(0', 0') → 0'
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(S(x'))), x))
g(S(x), S(x')) → h(f(S(x), S(x')), g(x, S(S(S(x')))))
h(0', S(x)) → h(0', x)
h(0', 0') → 0'
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(S(x'))), x))
g(S(x), S(x')) → h(f(S(x), S(x')), g(x, S(S(S(x')))))
h(0', S(x)) → h(0', x)
h(0', 0') → 0'
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)) →_R^Ω(1)
0'

Induction Step:
h(gen_S:0'2_0(0), gen_S:0'2_0(+(n4_0, 1))) →_R^Ω(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 irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
f(S(x'), S(x)) → h(g(x', S(x)), f(S(S(S(x'))), x))
g(S(x), S(x')) → h(f(S(x), S(x')), g(x, S(S(S(x')))))
h(0', S(x)) → h(0', x)
h(0', 0') → 0'
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) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
g(gen_S:0'2_0(+(1, n867_0)), gen_S:0'2_0(1)) → *3_0, rt ∈ Ω(n867₀)

Induction Base:
g(gen_S:0'2_0(+(1, 0)), gen_S:0'2_0(1))

Induction Step:
g(gen_S:0'2_0(+(1, +(n867_0, 1))), gen_S:0'2_0(1)) →_R^Ω(1)
h(f(S(gen_S:0'2_0(+(1, n867_0))), S(gen_S:0'2_0(0))), g(gen_S:0'2_0(+(1, n867_0)), S(S(S(gen_S:0'2_0(0)))))) →_R^Ω(1)
h(h(g(gen_S:0'2_0(+(1, n867_0)), S(gen_S:0'2_0(0))), f(S(S(S(gen_S:0'2_0(+(1, n867_0))))), gen_S:0'2_0(0))), g(gen_S:0'2_0(+(1, n867_0)), S(S(S(gen_S:0'2_0(0)))))) →_IH
h(h(*3_0, f(S(S(S(gen_S:0'2_0(+(1, n867_0))))), gen_S:0'2_0(0))), g(gen_S:0'2_0(+(1, n867_0)), S(S(S(gen_S:0'2_0(0)))))) →_R^Ω(1)
h(h(*3_0, 0'), g(gen_S:0'2_0(+(1, n867_0)), S(S(S(gen_S:0'2_0(0)))))) →_R^Ω(1)
h(h(*3_0, 0'), h(f(S(gen_S:0'2_0(n867_0)), S(S(S(gen_S:0'2_0(0))))), g(gen_S:0'2_0(n867_0), S(S(S(S(S(gen_S:0'2_0(0))))))))) →_R^Ω(1)
h(h(*3_0, 0'), h(h(g(gen_S:0'2_0(n867_0), S(S(S(gen_S:0'2_0(0))))), f(S(S(S(gen_S:0'2_0(n867_0)))), S(S(gen_S:0'2_0(0))))), g(gen_S:0'2_0(n867_0), S(S(S(S(S(gen_S:0'2_0(0)))))))))

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

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
Rules:
f(S(x'), S(x)) → h(g(x', S(x)), f(S(S(S(x'))), x))
g(S(x), S(x')) → h(f(S(x), S(x')), g(x, S(S(S(x')))))
h(0', S(x)) → h(0', x)
h(0', 0') → 0'
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)
g(gen_S:0'2_0(+(1, n867_0)), gen_S:0'2_0(1)) → *3_0, rt ∈ Ω(n867₀)

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

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
f(gen_S:0'2_0(+(1, n3791_0)), gen_S:0'2_0(1)) → *3_0, rt ∈ Ω(n3791₀)

Induction Base:
f(gen_S:0'2_0(+(1, 0)), gen_S:0'2_0(1))

Induction Step:
f(gen_S:0'2_0(+(1, +(n3791_0, 1))), gen_S:0'2_0(1)) →_R^Ω(1)
h(g(gen_S:0'2_0(+(1, n3791_0)), S(gen_S:0'2_0(0))), f(S(S(S(gen_S:0'2_0(+(1, n3791_0))))), gen_S:0'2_0(0))) →_R^Ω(1)
h(h(f(S(gen_S:0'2_0(n3791_0)), S(gen_S:0'2_0(0))), g(gen_S:0'2_0(n3791_0), S(S(S(gen_S:0'2_0(0)))))), f(S(S(S(gen_S:0'2_0(+(1, n3791_0))))), gen_S:0'2_0(0))) →_IH
h(h(*3_0, g(gen_S:0'2_0(n3791_0), S(S(S(gen_S:0'2_0(0)))))), f(S(S(S(gen_S:0'2_0(+(1, n3791_0))))), gen_S:0'2_0(0)))

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

(16) Complex Obligation (BEST)

(17) Obligation:

Innermost TRS:
Rules:
f(S(x'), S(x)) → h(g(x', S(x)), f(S(S(S(x'))), x))
g(S(x), S(x')) → h(f(S(x), S(x')), g(x, S(S(S(x')))))
h(0', S(x)) → h(0', x)
h(0', 0') → 0'
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)
g(gen_S:0'2_0(+(1, n867_0)), gen_S:0'2_0(1)) → *3_0, rt ∈ Ω(n867₀)
f(gen_S:0'2_0(+(1, n3791_0)), gen_S:0'2_0(1)) → *3_0, rt ∈ Ω(n3791₀)

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

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

(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
g(gen_S:0'2_0(+(1, n9137_0)), gen_S:0'2_0(1)) → *3_0, rt ∈ Ω(n9137₀)

Induction Base:
g(gen_S:0'2_0(+(1, 0)), gen_S:0'2_0(1))

Induction Step:
g(gen_S:0'2_0(+(1, +(n9137_0, 1))), gen_S:0'2_0(1)) →_R^Ω(1)
h(f(S(gen_S:0'2_0(+(1, n9137_0))), S(gen_S:0'2_0(0))), g(gen_S:0'2_0(+(1, n9137_0)), S(S(S(gen_S:0'2_0(0)))))) →_R^Ω(1)
h(h(g(gen_S:0'2_0(+(1, n9137_0)), S(gen_S:0'2_0(0))), f(S(S(S(gen_S:0'2_0(+(1, n9137_0))))), gen_S:0'2_0(0))), g(gen_S:0'2_0(+(1, n9137_0)), S(S(S(gen_S:0'2_0(0)))))) →_IH
h(h(*3_0, f(S(S(S(gen_S:0'2_0(+(1, n9137_0))))), gen_S:0'2_0(0))), g(gen_S:0'2_0(+(1, n9137_0)), S(S(S(gen_S:0'2_0(0)))))) →_R^Ω(1)
h(h(*3_0, 0'), g(gen_S:0'2_0(+(1, n9137_0)), S(S(S(gen_S:0'2_0(0)))))) →_R^Ω(1)
h(h(*3_0, 0'), h(f(S(gen_S:0'2_0(n9137_0)), S(S(S(gen_S:0'2_0(0))))), g(gen_S:0'2_0(n9137_0), S(S(S(S(S(gen_S:0'2_0(0))))))))) →_R^Ω(1)
h(h(*3_0, 0'), h(h(g(gen_S:0'2_0(n9137_0), S(S(S(gen_S:0'2_0(0))))), f(S(S(S(gen_S:0'2_0(n9137_0)))), S(S(gen_S:0'2_0(0))))), g(gen_S:0'2_0(n9137_0), S(S(S(S(S(gen_S:0'2_0(0)))))))))

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

(19) Complex Obligation (BEST)

(20) Obligation:

Innermost TRS:
Rules:
f(S(x'), S(x)) → h(g(x', S(x)), f(S(S(S(x'))), x))
g(S(x), S(x')) → h(f(S(x), S(x')), g(x, S(S(S(x')))))
h(0', S(x)) → h(0', x)
h(0', 0') → 0'
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)
g(gen_S:0'2_0(+(1, n9137_0)), gen_S:0'2_0(1)) → *3_0, rt ∈ Ω(n9137₀)
f(gen_S:0'2_0(+(1, n3791_0)), gen_S:0'2_0(1)) → *3_0, rt ∈ Ω(n3791₀)

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.

(21) 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)

(22) BOUNDS(n^1, INF)

(23) Obligation:

Innermost TRS:
Rules:
f(S(x'), S(x)) → h(g(x', S(x)), f(S(S(S(x'))), x))
g(S(x), S(x')) → h(f(S(x), S(x')), g(x, S(S(S(x')))))
h(0', S(x)) → h(0', x)
h(0', 0') → 0'
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)
g(gen_S:0'2_0(+(1, n9137_0)), gen_S:0'2_0(1)) → *3_0, rt ∈ Ω(n9137₀)
f(gen_S:0'2_0(+(1, n3791_0)), gen_S:0'2_0(1)) → *3_0, rt ∈ Ω(n3791₀)

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.

(24) 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)

(25) BOUNDS(n^1, INF)

(26) Obligation:

Innermost TRS:
Rules:
f(S(x'), S(x)) → h(g(x', S(x)), f(S(S(S(x'))), x))
g(S(x), S(x')) → h(f(S(x), S(x')), g(x, S(S(S(x')))))
h(0', S(x)) → h(0', x)
h(0', 0') → 0'
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)
g(gen_S:0'2_0(+(1, n867_0)), gen_S:0'2_0(1)) → *3_0, rt ∈ Ω(n867₀)
f(gen_S:0'2_0(+(1, n3791_0)), gen_S:0'2_0(1)) → *3_0, rt ∈ Ω(n3791₀)

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.

(27) 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)

(28) BOUNDS(n^1, INF)

(29) Obligation:

Innermost TRS:
Rules:
f(S(x'), S(x)) → h(g(x', S(x)), f(S(S(S(x'))), x))
g(S(x), S(x')) → h(f(S(x), S(x')), g(x, S(S(S(x')))))
h(0', S(x)) → h(0', x)
h(0', 0') → 0'
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)
g(gen_S:0'2_0(+(1, n867_0)), gen_S:0'2_0(1)) → *3_0, rt ∈ Ω(n867₀)

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.

(30) 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)

(31) BOUNDS(n^1, INF)

(32) Obligation:

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

(33) 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)

(34) BOUNDS(n^1, INF)