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:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
f(0) → s(0)
f(s(x)) → minus(s(x), g(f(x)))
g(0) → 0
g(s(x)) → minus(s(x), f(g(x)))

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:

minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
f(0') → s(0')
f(s(x)) → minus(s(x), g(f(x)))
g(0') → 0'
g(s(x)) → minus(s(x), f(g(x)))

S is empty.
Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
f(0') → s(0')
f(s(x)) → minus(s(x), g(f(x)))
g(0') → 0'
g(s(x)) → minus(s(x), f(g(x)))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: 0':s → 0':s
g :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

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

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

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

(6) Obligation:

Innermost TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
f(0') → s(0')
f(s(x)) → minus(s(x), g(f(x)))
g(0') → 0'
g(s(x)) → minus(s(x), f(g(x)))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: 0':s → 0':s
g :: 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:
minus, f, g

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

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

Proved the following rewrite lemma:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n4₀)

Induction Base:
minus(gen_0':s2_0(0), gen_0':s2_0(0)) →_R^Ω(1)
gen_0':s2_0(0)

Induction Step:
minus(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(n4_0, 1))) →_R^Ω(1)
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) →_IH
gen_0':s2_0(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:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
f(0') → s(0')
f(s(x)) → minus(s(x), g(f(x)))
g(0') → 0'
g(s(x)) → minus(s(x), f(g(x)))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: 0':s → 0':s
g :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s2_0(n4_0), 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))

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

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

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

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

(11) Obligation:

Innermost TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
f(0') → s(0')
f(s(x)) → minus(s(x), g(f(x)))
g(0') → 0'
g(s(x)) → minus(s(x), f(g(x)))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: 0':s → 0':s
g :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s2_0(n4_0), 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))

The following defined symbols remain to be analysed:
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 f.

(13) Obligation:

Innermost TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
f(0') → s(0')
f(s(x)) → minus(s(x), g(f(x)))
g(0') → 0'
g(s(x)) → minus(s(x), f(g(x)))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: 0':s → 0':s
g :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s2_0(n4_0), 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.

(14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n4₀)

(15) BOUNDS(n^1, INF)

(16) Obligation:

Innermost TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
f(0') → s(0')
f(s(x)) → minus(s(x), g(f(x)))
g(0') → 0'
g(s(x)) → minus(s(x), f(g(x)))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
f :: 0':s → 0':s
g :: 0':s → 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s2_0(n4_0), 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.

(17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
minus(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) → gen_0':s2_0(0), rt ∈ Ω(1 + n4₀)

(18) BOUNDS(n^1, INF)