### (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)
double(0) → 0
double(s(x)) → s(s(double(x)))
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(minus(x, y), double(y)))
plus(s(plus(x, y)), z) → s(plus(plus(x, y), z))

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)
double(0') → 0'
double(s(x)) → s(s(double(x)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(minus(x, y), double(y)))
plus(s(plus(x, y)), z) → s(plus(plus(x, y), z))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(minus(x, y), double(y)))
plus(s(plus(x, y)), z) → s(plus(plus(x, y), z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
double :: 0':s → 0':s
plus :: 0':s → 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, double, plus

They will be analysed ascendingly in the following order:
minus < plus
double < plus

### (8) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(minus(x, y), double(y)))
plus(s(plus(x, y)), z) → s(plus(plus(x, y), z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
double :: 0':s → 0':s
plus :: 0':s → 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, double, plus

They will be analysed ascendingly in the following order:
minus < plus
double < plus

### (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)
double(0') → 0'
double(s(x)) → s(s(double(x)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(minus(x, y), double(y)))
plus(s(plus(x, y)), z) → s(plus(plus(x, y), z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
double :: 0':s → 0':s
plus :: 0':s → 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:
double, plus

They will be analysed ascendingly in the following order:
double < plus

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

Proved the following rewrite lemma:
double(gen_0':s2_0(n252_0)) → gen_0':s2_0(*(2, n252_0)), rt ∈ Ω(1 + n2520)

Induction Base:
double(gen_0':s2_0(0)) →RΩ(1)
0'

Induction Step:
double(gen_0':s2_0(+(n252_0, 1))) →RΩ(1)
s(s(double(gen_0':s2_0(n252_0)))) →IH
s(s(gen_0':s2_0(*(2, c253_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)
double(0') → 0'
double(s(x)) → s(s(double(x)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(minus(x, y), double(y)))
plus(s(plus(x, y)), z) → s(plus(plus(x, y), z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
double :: 0':s → 0':s
plus :: 0':s → 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)
double(gen_0':s2_0(n252_0)) → gen_0':s2_0(*(2, n252_0)), rt ∈ Ω(1 + n2520)

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:
plus

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

Proved the following rewrite lemma:
plus(gen_0':s2_0(n496_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n496_0, b)), rt ∈ Ω(1 + n4960)

Induction Base:
plus(gen_0':s2_0(0), gen_0':s2_0(b)) →RΩ(1)
gen_0':s2_0(b)

Induction Step:
plus(gen_0':s2_0(+(n496_0, 1)), gen_0':s2_0(b)) →RΩ(1)
s(plus(gen_0':s2_0(n496_0), gen_0':s2_0(b))) →IH
s(gen_0':s2_0(+(b, c497_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)
double(0') → 0'
double(s(x)) → s(s(double(x)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(minus(x, y), double(y)))
plus(s(plus(x, y)), z) → s(plus(plus(x, y), z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
double :: 0':s → 0':s
plus :: 0':s → 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)
double(gen_0':s2_0(n252_0)) → gen_0':s2_0(*(2, n252_0)), rt ∈ Ω(1 + n2520)
plus(gen_0':s2_0(n496_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n496_0, b)), rt ∈ Ω(1 + n4960)

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.

### (18) 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)

### (20) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
double(0') → 0'
double(s(x)) → s(s(double(x)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(minus(x, y), double(y)))
plus(s(plus(x, y)), z) → s(plus(plus(x, y), z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
double :: 0':s → 0':s
plus :: 0':s → 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)
double(gen_0':s2_0(n252_0)) → gen_0':s2_0(*(2, n252_0)), rt ∈ Ω(1 + n2520)
plus(gen_0':s2_0(n496_0), gen_0':s2_0(b)) → gen_0':s2_0(+(n496_0, b)), rt ∈ Ω(1 + n4960)

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)
double(0') → 0'
double(s(x)) → s(s(double(x)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(minus(x, y), double(y)))
plus(s(plus(x, y)), z) → s(plus(plus(x, y), z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
double :: 0':s → 0':s
plus :: 0':s → 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)
double(gen_0':s2_0(n252_0)) → gen_0':s2_0(*(2, n252_0)), rt ∈ Ω(1 + n2520)

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)
double(0') → 0'
double(s(x)) → s(s(double(x)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(minus(x, y), double(y)))
plus(s(plus(x, y)), z) → s(plus(plus(x, y), z))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
double :: 0':s → 0':s
plus :: 0':s → 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.

### (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)