### (0) Obligation:

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

rev(nil) → nil
rev(rev(x)) → x
rev(++(x, y)) → ++(rev(y), rev(x))
++(nil, y) → y
++(x, nil) → x
++(.(x, y), z) → .(x, ++(y, z))
++(x, ++(y, z)) → ++(++(x, y), z)
make(x) → .(x, nil)

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

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

rev(nil) → nil
rev(rev(x)) → x
rev(++(x, y)) → ++(rev(y), rev(x))
++(nil, y) → y
++(x, nil) → x
++(.(x, y), z) → .(x, ++(y, z))
++(x, ++(y, z)) → ++(++(x, y), z)
make(x) → .(x, nil)

S is empty.
Rewrite Strategy: FULL

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

Sliced the following arguments:
./0
make/0

### (6) Obligation:

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

rev(nil) → nil
rev(rev(x)) → x
rev(++(x, y)) → ++(rev(y), rev(x))
++(nil, y) → y
++(x, nil) → x
++(.(y), z) → .(++(y, z))
++(x, ++(y, z)) → ++(++(x, y), z)
make.(nil)

S is empty.
Rewrite Strategy: FULL

Infered types.

### (8) Obligation:

TRS:
Rules:
rev(nil) → nil
rev(rev(x)) → x
rev(++(x, y)) → ++(rev(y), rev(x))
++(nil, y) → y
++(x, nil) → x
++(.(y), z) → .(++(y, z))
++(x, ++(y, z)) → ++(++(x, y), z)
make.(nil)

Types:
rev :: nil:. → nil:.
nil :: nil:.
++ :: nil:. → nil:. → nil:.
. :: nil:. → nil:.
make :: nil:.
hole_nil:.1_0 :: nil:.
gen_nil:.2_0 :: Nat → nil:.

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

Heuristically decided to analyse the following defined symbols:
rev, ++

They will be analysed ascendingly in the following order:
++ < rev

### (10) Obligation:

TRS:
Rules:
rev(nil) → nil
rev(rev(x)) → x
rev(++(x, y)) → ++(rev(y), rev(x))
++(nil, y) → y
++(x, nil) → x
++(.(y), z) → .(++(y, z))
++(x, ++(y, z)) → ++(++(x, y), z)
make.(nil)

Types:
rev :: nil:. → nil:.
nil :: nil:.
++ :: nil:. → nil:. → nil:.
. :: nil:. → nil:.
make :: nil:.
hole_nil:.1_0 :: nil:.
gen_nil:.2_0 :: Nat → nil:.

Generator Equations:
gen_nil:.2_0(0) ⇔ nil
gen_nil:.2_0(+(x, 1)) ⇔ .(gen_nil:.2_0(x))

The following defined symbols remain to be analysed:
++, rev

They will be analysed ascendingly in the following order:
++ < rev

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

Proved the following rewrite lemma:
++(gen_nil:.2_0(n4_0), gen_nil:.2_0(b)) → gen_nil:.2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)

Induction Base:
++(gen_nil:.2_0(0), gen_nil:.2_0(b)) →RΩ(1)
gen_nil:.2_0(b)

Induction Step:
++(gen_nil:.2_0(+(n4_0, 1)), gen_nil:.2_0(b)) →RΩ(1)
.(++(gen_nil:.2_0(n4_0), gen_nil:.2_0(b))) →IH
.(gen_nil:.2_0(+(b, c5_0)))

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

### (13) Obligation:

TRS:
Rules:
rev(nil) → nil
rev(rev(x)) → x
rev(++(x, y)) → ++(rev(y), rev(x))
++(nil, y) → y
++(x, nil) → x
++(.(y), z) → .(++(y, z))
++(x, ++(y, z)) → ++(++(x, y), z)
make.(nil)

Types:
rev :: nil:. → nil:.
nil :: nil:.
++ :: nil:. → nil:. → nil:.
. :: nil:. → nil:.
make :: nil:.
hole_nil:.1_0 :: nil:.
gen_nil:.2_0 :: Nat → nil:.

Lemmas:
++(gen_nil:.2_0(n4_0), gen_nil:.2_0(b)) → gen_nil:.2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)

Generator Equations:
gen_nil:.2_0(0) ⇔ nil
gen_nil:.2_0(+(x, 1)) ⇔ .(gen_nil:.2_0(x))

The following defined symbols remain to be analysed:
rev

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

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

### (15) Obligation:

TRS:
Rules:
rev(nil) → nil
rev(rev(x)) → x
rev(++(x, y)) → ++(rev(y), rev(x))
++(nil, y) → y
++(x, nil) → x
++(.(y), z) → .(++(y, z))
++(x, ++(y, z)) → ++(++(x, y), z)
make.(nil)

Types:
rev :: nil:. → nil:.
nil :: nil:.
++ :: nil:. → nil:. → nil:.
. :: nil:. → nil:.
make :: nil:.
hole_nil:.1_0 :: nil:.
gen_nil:.2_0 :: Nat → nil:.

Lemmas:
++(gen_nil:.2_0(n4_0), gen_nil:.2_0(b)) → gen_nil:.2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)

Generator Equations:
gen_nil:.2_0(0) ⇔ nil
gen_nil:.2_0(+(x, 1)) ⇔ .(gen_nil:.2_0(x))

No more defined symbols left to analyse.

### (16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
++(gen_nil:.2_0(n4_0), gen_nil:.2_0(b)) → gen_nil:.2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)

### (18) Obligation:

TRS:
Rules:
rev(nil) → nil
rev(rev(x)) → x
rev(++(x, y)) → ++(rev(y), rev(x))
++(nil, y) → y
++(x, nil) → x
++(.(y), z) → .(++(y, z))
++(x, ++(y, z)) → ++(++(x, y), z)
make.(nil)

Types:
rev :: nil:. → nil:.
nil :: nil:.
++ :: nil:. → nil:. → nil:.
. :: nil:. → nil:.
make :: nil:.
hole_nil:.1_0 :: nil:.
gen_nil:.2_0 :: Nat → nil:.

Lemmas:
++(gen_nil:.2_0(n4_0), gen_nil:.2_0(b)) → gen_nil:.2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)

Generator Equations:
gen_nil:.2_0(0) ⇔ nil
gen_nil:.2_0(+(x, 1)) ⇔ .(gen_nil:.2_0(x))

No more defined symbols left to analyse.

### (19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
++(gen_nil:.2_0(n4_0), gen_nil:.2_0(b)) → gen_nil:.2_0(+(n4_0, b)), rt ∈ Ω(1 + n40)