### (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: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

### (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:

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: FULL

Infered types.

### (6) Obligation:

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

### (7) 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

### (8) Obligation:

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

### (9) 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)) →RΩ(1)
gen_0':s2_0(0)

Induction Step:
minus(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(n4_0, 1))) →RΩ(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 ircR ∈ Ω(n).

### (11) Obligation:

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

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

Proved the following rewrite lemma:
g(gen_0':s2_0(+(1, n228_0))) → *3_0, rt ∈ Ω(n2280)

Induction Base:
g(gen_0':s2_0(+(1, 0)))

Induction Step:
g(gen_0':s2_0(+(1, +(n228_0, 1)))) →RΩ(1)
minus(s(gen_0':s2_0(+(1, n228_0))), f(g(gen_0':s2_0(+(1, n228_0))))) →IH
minus(s(gen_0':s2_0(+(1, n228_0))), f(*3_0))

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

### (14) Obligation:

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)
g(gen_0':s2_0(+(1, n228_0))) → *3_0, rt ∈ Ω(n2280)

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

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

Proved the following rewrite lemma:
f(gen_0':s2_0(+(1, n4065_0))) → *3_0, rt ∈ Ω(n40650)

Induction Base:
f(gen_0':s2_0(+(1, 0)))

Induction Step:
f(gen_0':s2_0(+(1, +(n4065_0, 1)))) →RΩ(1)
minus(s(gen_0':s2_0(+(1, n4065_0))), g(f(gen_0':s2_0(+(1, n4065_0))))) →IH
minus(s(gen_0':s2_0(+(1, n4065_0))), g(*3_0))

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

### (17) Obligation:

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)
g(gen_0':s2_0(+(1, n228_0))) → *3_0, rt ∈ Ω(n2280)
f(gen_0':s2_0(+(1, n4065_0))) → *3_0, rt ∈ Ω(n40650)

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

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

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

Proved the following rewrite lemma:
g(gen_0':s2_0(+(1, n64234_0))) → *3_0, rt ∈ Ω(n642340)

Induction Base:
g(gen_0':s2_0(+(1, 0)))

Induction Step:
g(gen_0':s2_0(+(1, +(n64234_0, 1)))) →RΩ(1)
minus(s(gen_0':s2_0(+(1, n64234_0))), f(g(gen_0':s2_0(+(1, n64234_0))))) →IH
minus(s(gen_0':s2_0(+(1, n64234_0))), f(*3_0))

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

### (20) Obligation:

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)
g(gen_0':s2_0(+(1, n64234_0))) → *3_0, rt ∈ Ω(n642340)
f(gen_0':s2_0(+(1, n4065_0))) → *3_0, rt ∈ Ω(n40650)

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.

### (21) 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)

### (23) Obligation:

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)
g(gen_0':s2_0(+(1, n64234_0))) → *3_0, rt ∈ Ω(n642340)
f(gen_0':s2_0(+(1, n4065_0))) → *3_0, rt ∈ Ω(n40650)

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.

### (24) 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)

### (26) Obligation:

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)
g(gen_0':s2_0(+(1, n228_0))) → *3_0, rt ∈ Ω(n2280)
f(gen_0':s2_0(+(1, n4065_0))) → *3_0, rt ∈ Ω(n40650)

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.

### (27) 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)

### (29) Obligation:

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)
g(gen_0':s2_0(+(1, n228_0))) → *3_0, rt ∈ Ω(n2280)

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.

### (30) 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)

### (32) Obligation:

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.

### (33) 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)