### (0) Obligation:

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

plus(0, Y) → Y
plus(s(X), Y) → s(plus(X, Y))
min(X, 0) → X
min(s(X), s(Y)) → min(X, Y)
min(min(X, Y), Z) → min(X, plus(Y, Z))
quot(0, s(Y)) → 0
quot(s(X), s(Y)) → s(quot(min(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
plus(s(X), Y) →+ s(plus(X, Y))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / s(X)].
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:

plus(0', Y) → Y
plus(s(X), Y) → s(plus(X, Y))
min(X, 0') → X
min(s(X), s(Y)) → min(X, Y)
min(min(X, Y), Z) → min(X, plus(Y, Z))
quot(0', s(Y)) → 0'
quot(s(X), s(Y)) → s(quot(min(X, Y), s(Y)))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
plus(0', Y) → Y
plus(s(X), Y) → s(plus(X, Y))
min(X, 0') → X
min(s(X), s(Y)) → min(X, Y)
min(min(X, Y), Z) → min(X, plus(Y, Z))
quot(0', s(Y)) → 0'
quot(s(X), s(Y)) → s(quot(min(X, Y), s(Y)))

Types:
plus :: 0':s:Z → 0':s:Z → 0':s:Z
0' :: 0':s:Z
s :: 0':s:Z → 0':s:Z
min :: 0':s:Z → 0':s:Z → 0':s:Z
Z :: 0':s:Z
quot :: 0':s:Z → 0':s:Z → 0':s:Z
hole_0':s:Z1_0 :: 0':s:Z
gen_0':s:Z2_0 :: Nat → 0':s:Z

### (7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
plus, min, quot

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

### (8) Obligation:

TRS:
Rules:
plus(0', Y) → Y
plus(s(X), Y) → s(plus(X, Y))
min(X, 0') → X
min(s(X), s(Y)) → min(X, Y)
min(min(X, Y), Z) → min(X, plus(Y, Z))
quot(0', s(Y)) → 0'
quot(s(X), s(Y)) → s(quot(min(X, Y), s(Y)))

Types:
plus :: 0':s:Z → 0':s:Z → 0':s:Z
0' :: 0':s:Z
s :: 0':s:Z → 0':s:Z
min :: 0':s:Z → 0':s:Z → 0':s:Z
Z :: 0':s:Z
quot :: 0':s:Z → 0':s:Z → 0':s:Z
hole_0':s:Z1_0 :: 0':s:Z
gen_0':s:Z2_0 :: Nat → 0':s:Z

Generator Equations:
gen_0':s:Z2_0(0) ⇔ 0'
gen_0':s:Z2_0(+(x, 1)) ⇔ s(gen_0':s:Z2_0(x))

The following defined symbols remain to be analysed:
plus, min, quot

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

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

Proved the following rewrite lemma:
plus(gen_0':s:Z2_0(n4_0), gen_0':s:Z2_0(b)) → gen_0':s:Z2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)

Induction Base:
plus(gen_0':s:Z2_0(0), gen_0':s:Z2_0(b)) →RΩ(1)
gen_0':s:Z2_0(b)

Induction Step:
plus(gen_0':s:Z2_0(+(n4_0, 1)), gen_0':s:Z2_0(b)) →RΩ(1)
s(plus(gen_0':s:Z2_0(n4_0), gen_0':s:Z2_0(b))) →IH
s(gen_0':s:Z2_0(+(b, c5_0)))

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

### (11) Obligation:

TRS:
Rules:
plus(0', Y) → Y
plus(s(X), Y) → s(plus(X, Y))
min(X, 0') → X
min(s(X), s(Y)) → min(X, Y)
min(min(X, Y), Z) → min(X, plus(Y, Z))
quot(0', s(Y)) → 0'
quot(s(X), s(Y)) → s(quot(min(X, Y), s(Y)))

Types:
plus :: 0':s:Z → 0':s:Z → 0':s:Z
0' :: 0':s:Z
s :: 0':s:Z → 0':s:Z
min :: 0':s:Z → 0':s:Z → 0':s:Z
Z :: 0':s:Z
quot :: 0':s:Z → 0':s:Z → 0':s:Z
hole_0':s:Z1_0 :: 0':s:Z
gen_0':s:Z2_0 :: Nat → 0':s:Z

Lemmas:
plus(gen_0':s:Z2_0(n4_0), gen_0':s:Z2_0(b)) → gen_0':s:Z2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)

Generator Equations:
gen_0':s:Z2_0(0) ⇔ 0'
gen_0':s:Z2_0(+(x, 1)) ⇔ s(gen_0':s:Z2_0(x))

The following defined symbols remain to be analysed:
min, quot

They will be analysed ascendingly in the following order:
min < quot

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

Proved the following rewrite lemma:
min(gen_0':s:Z2_0(n457_0), gen_0':s:Z2_0(n457_0)) → gen_0':s:Z2_0(0), rt ∈ Ω(1 + n4570)

Induction Base:
min(gen_0':s:Z2_0(0), gen_0':s:Z2_0(0)) →RΩ(1)
gen_0':s:Z2_0(0)

Induction Step:
min(gen_0':s:Z2_0(+(n457_0, 1)), gen_0':s:Z2_0(+(n457_0, 1))) →RΩ(1)
min(gen_0':s:Z2_0(n457_0), gen_0':s:Z2_0(n457_0)) →IH
gen_0':s:Z2_0(0)

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

### (14) Obligation:

TRS:
Rules:
plus(0', Y) → Y
plus(s(X), Y) → s(plus(X, Y))
min(X, 0') → X
min(s(X), s(Y)) → min(X, Y)
min(min(X, Y), Z) → min(X, plus(Y, Z))
quot(0', s(Y)) → 0'
quot(s(X), s(Y)) → s(quot(min(X, Y), s(Y)))

Types:
plus :: 0':s:Z → 0':s:Z → 0':s:Z
0' :: 0':s:Z
s :: 0':s:Z → 0':s:Z
min :: 0':s:Z → 0':s:Z → 0':s:Z
Z :: 0':s:Z
quot :: 0':s:Z → 0':s:Z → 0':s:Z
hole_0':s:Z1_0 :: 0':s:Z
gen_0':s:Z2_0 :: Nat → 0':s:Z

Lemmas:
plus(gen_0':s:Z2_0(n4_0), gen_0':s:Z2_0(b)) → gen_0':s:Z2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)
min(gen_0':s:Z2_0(n457_0), gen_0':s:Z2_0(n457_0)) → gen_0':s:Z2_0(0), rt ∈ Ω(1 + n4570)

Generator Equations:
gen_0':s:Z2_0(0) ⇔ 0'
gen_0':s:Z2_0(+(x, 1)) ⇔ s(gen_0':s:Z2_0(x))

The following defined symbols remain to be analysed:
quot

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

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

### (16) Obligation:

TRS:
Rules:
plus(0', Y) → Y
plus(s(X), Y) → s(plus(X, Y))
min(X, 0') → X
min(s(X), s(Y)) → min(X, Y)
min(min(X, Y), Z) → min(X, plus(Y, Z))
quot(0', s(Y)) → 0'
quot(s(X), s(Y)) → s(quot(min(X, Y), s(Y)))

Types:
plus :: 0':s:Z → 0':s:Z → 0':s:Z
0' :: 0':s:Z
s :: 0':s:Z → 0':s:Z
min :: 0':s:Z → 0':s:Z → 0':s:Z
Z :: 0':s:Z
quot :: 0':s:Z → 0':s:Z → 0':s:Z
hole_0':s:Z1_0 :: 0':s:Z
gen_0':s:Z2_0 :: Nat → 0':s:Z

Lemmas:
plus(gen_0':s:Z2_0(n4_0), gen_0':s:Z2_0(b)) → gen_0':s:Z2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)
min(gen_0':s:Z2_0(n457_0), gen_0':s:Z2_0(n457_0)) → gen_0':s:Z2_0(0), rt ∈ Ω(1 + n4570)

Generator Equations:
gen_0':s:Z2_0(0) ⇔ 0'
gen_0':s:Z2_0(+(x, 1)) ⇔ s(gen_0':s:Z2_0(x))

No more defined symbols left to analyse.

### (17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
plus(gen_0':s:Z2_0(n4_0), gen_0':s:Z2_0(b)) → gen_0':s:Z2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)

### (19) Obligation:

TRS:
Rules:
plus(0', Y) → Y
plus(s(X), Y) → s(plus(X, Y))
min(X, 0') → X
min(s(X), s(Y)) → min(X, Y)
min(min(X, Y), Z) → min(X, plus(Y, Z))
quot(0', s(Y)) → 0'
quot(s(X), s(Y)) → s(quot(min(X, Y), s(Y)))

Types:
plus :: 0':s:Z → 0':s:Z → 0':s:Z
0' :: 0':s:Z
s :: 0':s:Z → 0':s:Z
min :: 0':s:Z → 0':s:Z → 0':s:Z
Z :: 0':s:Z
quot :: 0':s:Z → 0':s:Z → 0':s:Z
hole_0':s:Z1_0 :: 0':s:Z
gen_0':s:Z2_0 :: Nat → 0':s:Z

Lemmas:
plus(gen_0':s:Z2_0(n4_0), gen_0':s:Z2_0(b)) → gen_0':s:Z2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)
min(gen_0':s:Z2_0(n457_0), gen_0':s:Z2_0(n457_0)) → gen_0':s:Z2_0(0), rt ∈ Ω(1 + n4570)

Generator Equations:
gen_0':s:Z2_0(0) ⇔ 0'
gen_0':s:Z2_0(+(x, 1)) ⇔ s(gen_0':s:Z2_0(x))

No more defined symbols left to analyse.

### (20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
plus(gen_0':s:Z2_0(n4_0), gen_0':s:Z2_0(b)) → gen_0':s:Z2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)

### (22) Obligation:

TRS:
Rules:
plus(0', Y) → Y
plus(s(X), Y) → s(plus(X, Y))
min(X, 0') → X
min(s(X), s(Y)) → min(X, Y)
min(min(X, Y), Z) → min(X, plus(Y, Z))
quot(0', s(Y)) → 0'
quot(s(X), s(Y)) → s(quot(min(X, Y), s(Y)))

Types:
plus :: 0':s:Z → 0':s:Z → 0':s:Z
0' :: 0':s:Z
s :: 0':s:Z → 0':s:Z
min :: 0':s:Z → 0':s:Z → 0':s:Z
Z :: 0':s:Z
quot :: 0':s:Z → 0':s:Z → 0':s:Z
hole_0':s:Z1_0 :: 0':s:Z
gen_0':s:Z2_0 :: Nat → 0':s:Z

Lemmas:
plus(gen_0':s:Z2_0(n4_0), gen_0':s:Z2_0(b)) → gen_0':s:Z2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)

Generator Equations:
gen_0':s:Z2_0(0) ⇔ 0'
gen_0':s:Z2_0(+(x, 1)) ⇔ s(gen_0':s:Z2_0(x))

No more defined symbols left to analyse.

### (23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
plus(gen_0':s:Z2_0(n4_0), gen_0':s:Z2_0(b)) → gen_0':s:Z2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)