### (0) Obligation:

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

app(nil, y) → y
reverse(nil) → nil
shuffle(nil) → nil

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [x / add(n, 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:

app(nil, y) → y
reverse(nil) → nil
shuffle(nil) → nil

S is empty.
Rewrite Strategy: FULL

### (5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:

### (6) Obligation:

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

app(nil, y) → y
reverse(nil) → nil
shuffle(nil) → nil

S is empty.
Rewrite Strategy: FULL

Infered types.

### (8) Obligation:

TRS:
Rules:
app(nil, y) → y
reverse(nil) → nil
shuffle(nil) → nil

Types:

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

Heuristically decided to analyse the following defined symbols:
app, reverse, shuffle

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

### (10) Obligation:

TRS:
Rules:
app(nil, y) → y
reverse(nil) → nil
shuffle(nil) → nil

Types:

Generator Equations:

The following defined symbols remain to be analysed:
app, reverse, shuffle

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

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

Proved the following rewrite lemma:

Induction Base:

Induction Step:

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

### (13) Obligation:

TRS:
Rules:
app(nil, y) → y
reverse(nil) → nil
shuffle(nil) → nil

Types:

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:
reverse, shuffle

They will be analysed ascendingly in the following order:
reverse < shuffle

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

Proved the following rewrite lemma:

Induction Base:
nil

Induction Step:

We have rt ∈ Ω(n2) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n2).

### (16) Obligation:

TRS:
Rules:
app(nil, y) → y
reverse(nil) → nil
shuffle(nil) → nil

Types:

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:
shuffle

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

Proved the following rewrite lemma:
shuffle(gen_nil:add2_0(n637_0)) → gen_nil:add2_0(n637_0), rt ∈ Ω(1 + n6370 + n63702 + n63703)

Induction Base:
nil

Induction Step:

We have rt ∈ Ω(n3) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n3).

### (19) Obligation:

TRS:
Rules:
app(nil, y) → y
reverse(nil) → nil
shuffle(nil) → nil

Types:

Lemmas:
shuffle(gen_nil:add2_0(n637_0)) → gen_nil:add2_0(n637_0), rt ∈ Ω(1 + n6370 + n63702 + n63703)

Generator Equations:

No more defined symbols left to analyse.

### (20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n3) was proven with the following lemma:
shuffle(gen_nil:add2_0(n637_0)) → gen_nil:add2_0(n637_0), rt ∈ Ω(1 + n6370 + n63702 + n63703)

### (22) Obligation:

TRS:
Rules:
app(nil, y) → y
reverse(nil) → nil
shuffle(nil) → nil

Types:

Lemmas:
shuffle(gen_nil:add2_0(n637_0)) → gen_nil:add2_0(n637_0), rt ∈ Ω(1 + n6370 + n63702 + n63703)

Generator Equations:

No more defined symbols left to analyse.

### (23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n3) was proven with the following lemma:
shuffle(gen_nil:add2_0(n637_0)) → gen_nil:add2_0(n637_0), rt ∈ Ω(1 + n6370 + n63702 + n63703)

### (25) Obligation:

TRS:
Rules:
app(nil, y) → y
reverse(nil) → nil
shuffle(nil) → nil

Types:

Lemmas:

Generator Equations:

No more defined symbols left to analyse.

### (26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:

### (28) Obligation:

TRS:
Rules:
app(nil, y) → y
reverse(nil) → nil
shuffle(nil) → nil

Types:

Lemmas: