### (0) Obligation:

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

minus(X, 0) → X
minus(s(X), s(Y)) → p(minus(X, Y))
p(s(X)) → X
div(0, s(Y)) → 0
div(s(X), s(Y)) → s(div(minus(X, Y), s(Y)))

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)) →+ p(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)) → p(minus(X, Y))
p(s(X)) → X
div(0', s(Y)) → 0'
div(s(X), s(Y)) → s(div(minus(X, Y), s(Y)))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
minus(X, 0') → X
minus(s(X), s(Y)) → p(minus(X, Y))
p(s(X)) → X
div(0', s(Y)) → 0'
div(s(X), s(Y)) → s(div(minus(X, Y), s(Y)))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
div :: 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, div

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

### (8) Obligation:

TRS:
Rules:
minus(X, 0') → X
minus(s(X), s(Y)) → p(minus(X, Y))
p(s(X)) → X
div(0', s(Y)) → 0'
div(s(X), s(Y)) → s(div(minus(X, Y), s(Y)))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
div :: 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, div

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

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

Proved the following rewrite lemma:
minus(gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

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

Induction Step:
minus(gen_0':s2_0(+(1, +(n4_0, 1))), gen_0':s2_0(+(1, +(n4_0, 1)))) →RΩ(1)
p(minus(gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(+(1, n4_0)))) →IH
p(*3_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)) → p(minus(X, Y))
p(s(X)) → X
div(0', s(Y)) → 0'
div(s(X), s(Y)) → s(div(minus(X, Y), s(Y)))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
div :: 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(+(1, n4_0)), gen_0':s2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(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:
div

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

Proved the following rewrite lemma:
div(gen_0':s2_0(n1815_0), gen_0':s2_0(1)) → gen_0':s2_0(n1815_0), rt ∈ Ω(1 + n18150)

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

Induction Step:
div(gen_0':s2_0(+(n1815_0, 1)), gen_0':s2_0(1)) →RΩ(1)
s(div(minus(gen_0':s2_0(n1815_0), gen_0':s2_0(0)), s(gen_0':s2_0(0)))) →RΩ(1)
s(div(gen_0':s2_0(n1815_0), s(gen_0':s2_0(0)))) →IH
s(gen_0':s2_0(c1816_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)) → p(minus(X, Y))
p(s(X)) → X
div(0', s(Y)) → 0'
div(s(X), s(Y)) → s(div(minus(X, Y), s(Y)))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
div :: 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(+(1, n4_0)), gen_0':s2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)
div(gen_0':s2_0(n1815_0), gen_0':s2_0(1)) → gen_0':s2_0(n1815_0), rt ∈ Ω(1 + n18150)

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.

### (15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
minus(gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

### (17) Obligation:

TRS:
Rules:
minus(X, 0') → X
minus(s(X), s(Y)) → p(minus(X, Y))
p(s(X)) → X
div(0', s(Y)) → 0'
div(s(X), s(Y)) → s(div(minus(X, Y), s(Y)))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
div :: 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(+(1, n4_0)), gen_0':s2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)
div(gen_0':s2_0(n1815_0), gen_0':s2_0(1)) → gen_0':s2_0(n1815_0), rt ∈ Ω(1 + n18150)

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(+(1, n4_0)), gen_0':s2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

### (20) Obligation:

TRS:
Rules:
minus(X, 0') → X
minus(s(X), s(Y)) → p(minus(X, Y))
p(s(X)) → X
div(0', s(Y)) → 0'
div(s(X), s(Y)) → s(div(minus(X, Y), s(Y)))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
p :: 0':s → 0':s
div :: 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(+(1, n4_0)), gen_0':s2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(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.

### (21) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
minus(gen_0':s2_0(+(1, n4_0)), gen_0':s2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)