The Runtime Complexity (full) 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)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
minus(minus(x, y), z) → minus(x, plus(y, z))
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))

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 [ ].

(2) BOUNDS(n^1, INF)

(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)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
minus(minus(x, y), z) → minus(x, plus(y, z))
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))

S is empty.
Rewrite Strategy: FULL

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

Infered types.

(6) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
minus(minus(x, y), z) → minus(x, plus(y, z))
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

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

Heuristically decided to analyse the following defined symbols:
minus, quot, plus, app, sum

They will be analysed ascendingly in the following order:
minus < quot
plus < minus
plus < sum
app < sum

(8) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
minus(minus(x, y), z) → minus(x, plus(y, z))
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
plus, minus, quot, app, sum

They will be analysed ascendingly in the following order:
minus < quot
plus < minus
plus < sum
app < sum

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

Proved the following rewrite lemma:
plus(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

Induction Base:
plus(gen_0':s3_0(0), gen_0':s3_0(b)) →_R^Ω(1)
gen_0':s3_0(b)

Induction Step:
plus(gen_0':s3_0(+(n6_0, 1)), gen_0':s3_0(b)) →_R^Ω(1)
s(plus(gen_0':s3_0(n6_0), gen_0':s3_0(b))) →_IH
s(gen_0':s3_0(+(b, c7_0)))

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

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
minus(minus(x, y), z) → minus(x, plus(y, z))
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
plus(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
minus, quot, app, sum

They will be analysed ascendingly in the following order:
minus < quot
app < sum

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

Proved the following rewrite lemma:
minus(gen_0':s3_0(+(1, n669_0)), gen_0':s3_0(+(1, n669_0))) → *5_0, rt ∈ Ω(n669₀)

Induction Base:
minus(gen_0':s3_0(+(1, 0)), gen_0':s3_0(+(1, 0)))

Induction Step:
minus(gen_0':s3_0(+(1, +(n669_0, 1))), gen_0':s3_0(+(1, +(n669_0, 1)))) →_R^Ω(1)
minus(gen_0':s3_0(+(1, n669_0)), gen_0':s3_0(+(1, n669_0))) →_IH
*5_0

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

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
minus(minus(x, y), z) → minus(x, plus(y, z))
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
plus(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
minus(gen_0':s3_0(+(1, n669_0)), gen_0':s3_0(+(1, n669_0))) → *5_0, rt ∈ Ω(n669₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
quot, app, sum

They will be analysed ascendingly in the following order:
app < sum

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

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

(16) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
minus(minus(x, y), z) → minus(x, plus(y, z))
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
plus(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
minus(gen_0':s3_0(+(1, n669_0)), gen_0':s3_0(+(1, n669_0))) → *5_0, rt ∈ Ω(n669₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
app, sum

They will be analysed ascendingly in the following order:
app < sum

(17) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
app(gen_nil:cons4_0(n3038_0), gen_nil:cons4_0(b)) → gen_nil:cons4_0(+(n3038_0, b)), rt ∈ Ω(1 + n3038₀)

Induction Base:
app(gen_nil:cons4_0(0), gen_nil:cons4_0(b)) →_R^Ω(1)
gen_nil:cons4_0(b)

Induction Step:
app(gen_nil:cons4_0(+(n3038_0, 1)), gen_nil:cons4_0(b)) →_R^Ω(1)
cons(0', app(gen_nil:cons4_0(n3038_0), gen_nil:cons4_0(b))) →_IH
cons(0', gen_nil:cons4_0(+(b, c3039_0)))

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

(18) Complex Obligation (BEST)

(19) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
minus(minus(x, y), z) → minus(x, plus(y, z))
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
plus(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
minus(gen_0':s3_0(+(1, n669_0)), gen_0':s3_0(+(1, n669_0))) → *5_0, rt ∈ Ω(n669₀)
app(gen_nil:cons4_0(n3038_0), gen_nil:cons4_0(b)) → gen_nil:cons4_0(+(n3038_0, b)), rt ∈ Ω(1 + n3038₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

The following defined symbols remain to be analysed:
sum

(20) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sum(gen_nil:cons4_0(+(1, n4035_0))) → gen_nil:cons4_0(1), rt ∈ Ω(1 + n4035₀)

Induction Base:
sum(gen_nil:cons4_0(+(1, 0))) →_R^Ω(1)
cons(0', nil)

Induction Step:
sum(gen_nil:cons4_0(+(1, +(n4035_0, 1)))) →_R^Ω(1)
sum(cons(plus(0', 0'), gen_nil:cons4_0(n4035_0))) →_L^Ω(1)
sum(cons(gen_0':s3_0(+(0, 0)), gen_nil:cons4_0(n4035_0))) →_IH
gen_nil:cons4_0(1)

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

(21) Complex Obligation (BEST)

(22) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
minus(minus(x, y), z) → minus(x, plus(y, z))
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
plus(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
minus(gen_0':s3_0(+(1, n669_0)), gen_0':s3_0(+(1, n669_0))) → *5_0, rt ∈ Ω(n669₀)
app(gen_nil:cons4_0(n3038_0), gen_nil:cons4_0(b)) → gen_nil:cons4_0(+(n3038_0, b)), rt ∈ Ω(1 + n3038₀)
sum(gen_nil:cons4_0(+(1, n4035_0))) → gen_nil:cons4_0(1), rt ∈ Ω(1 + n4035₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_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':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(24) BOUNDS(n^1, INF)

(25) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
minus(minus(x, y), z) → minus(x, plus(y, z))
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
plus(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
minus(gen_0':s3_0(+(1, n669_0)), gen_0':s3_0(+(1, n669_0))) → *5_0, rt ∈ Ω(n669₀)
app(gen_nil:cons4_0(n3038_0), gen_nil:cons4_0(b)) → gen_nil:cons4_0(+(n3038_0, b)), rt ∈ Ω(1 + n3038₀)
sum(gen_nil:cons4_0(+(1, n4035_0))) → gen_nil:cons4_0(1), rt ∈ Ω(1 + n4035₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
plus(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(27) BOUNDS(n^1, INF)

(28) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
minus(minus(x, y), z) → minus(x, plus(y, z))
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
plus(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
minus(gen_0':s3_0(+(1, n669_0)), gen_0':s3_0(+(1, n669_0))) → *5_0, rt ∈ Ω(n669₀)
app(gen_nil:cons4_0(n3038_0), gen_nil:cons4_0(b)) → gen_nil:cons4_0(+(n3038_0, b)), rt ∈ Ω(1 + n3038₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
plus(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(30) BOUNDS(n^1, INF)

(31) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
minus(minus(x, y), z) → minus(x, plus(y, z))
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
plus(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)
minus(gen_0':s3_0(+(1, n669_0)), gen_0':s3_0(+(1, n669_0))) → *5_0, rt ∈ Ω(n669₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(32) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
plus(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(33) BOUNDS(n^1, INF)

(34) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
plus(0', y) → y
plus(s(x), y) → s(plus(x, y))
minus(minus(x, y), z) → minus(x, plus(y, z))
app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
plus :: 0':s → 0':s → 0':s
app :: nil:cons → nil:cons → nil:cons
nil :: nil:cons
cons :: 0':s → nil:cons → nil:cons
sum :: nil:cons → nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
gen_0':s3_0 :: Nat → 0':s
gen_nil:cons4_0 :: Nat → nil:cons

Lemmas:
plus(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

Generator Equations:
gen_0':s3_0(0) ⇔ 0'
gen_0':s3_0(+(x, 1)) ⇔ s(gen_0':s3_0(x))
gen_nil:cons4_0(0) ⇔ nil
gen_nil:cons4_0(+(x, 1)) ⇔ cons(0', gen_nil:cons4_0(x))

No more defined symbols left to analyse.

(35) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
plus(gen_0':s3_0(n6_0), gen_0':s3_0(b)) → gen_0':s3_0(+(n6_0, b)), rt ∈ Ω(1 + n6₀)

(36) BOUNDS(n^1, INF)