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

rev(nil) → nil
rev(.(x, y)) → ++(rev(y), .(x, nil))
car(.(x, y)) → x
cdr(.(x, y)) → y
null(nil) → true
null(.(x, y)) → false
++(nil, y) → y
++(.(x, y), z) → .(x, ++(y, z))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

rev'(nil') → nil'
rev'(.'(x, y)) → ++'(rev'(y), .'(x, nil'))
car'(.'(x, y)) → x
cdr'(.'(x, y)) → y
null'(nil') → true'
null'(.'(x, y)) → false'
++'(nil', y) → y
++'(.'(x, y), z) → .'(x, ++'(y, z))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
rev'(nil') → nil'
rev'(.'(x, y)) → ++'(rev'(y), .'(x, nil'))
car'(.'(x, y)) → x
cdr'(.'(x, y)) → y
null'(nil') → true'
null'(.'(x, y)) → false'
++'(nil', y) → y
++'(.'(x, y), z) → .'(x, ++'(y, z))

Types:
rev' :: nil':.' → nil':.'
nil' :: nil':.'
.' :: car' → nil':.' → nil':.'
++' :: nil':.' → nil':.' → nil':.'
car' :: nil':.' → car'
cdr' :: nil':.' → nil':.'
null' :: nil':.' → true':false'
true' :: true':false'
false' :: true':false'
_hole_nil':.'1 :: nil':.'
_hole_car'2 :: car'
_hole_true':false'3 :: true':false'
_gen_nil':.'4 :: Nat → nil':.'

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

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

Rules:
rev'(nil') → nil'
rev'(.'(x, y)) → ++'(rev'(y), .'(x, nil'))
car'(.'(x, y)) → x
cdr'(.'(x, y)) → y
null'(nil') → true'
null'(.'(x, y)) → false'
++'(nil', y) → y
++'(.'(x, y), z) → .'(x, ++'(y, z))

Types:
rev' :: nil':.' → nil':.'
nil' :: nil':.'
.' :: car' → nil':.' → nil':.'
++' :: nil':.' → nil':.' → nil':.'
car' :: nil':.' → car'
cdr' :: nil':.' → nil':.'
null' :: nil':.' → true':false'
true' :: true':false'
false' :: true':false'
_hole_nil':.'1 :: nil':.'
_hole_car'2 :: car'
_hole_true':false'3 :: true':false'
_gen_nil':.'4 :: Nat → nil':.'

Generator Equations:
_gen_nil':.'4(0) ⇔ nil'
_gen_nil':.'4(+(x, 1)) ⇔ .'(_hole_car'2, _gen_nil':.'4(x))

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

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

Proved the following rewrite lemma:
++'(_gen_nil':.'4(_n6), _gen_nil':.'4(b)) → _gen_nil':.'4(+(_n6, b)), rt ∈ Ω(1 + n6)

Induction Base:
++'(_gen_nil':.'4(0), _gen_nil':.'4(b)) →RΩ(1)
_gen_nil':.'4(b)

Induction Step:
++'(_gen_nil':.'4(+(_\$n7, 1)), _gen_nil':.'4(_b147)) →RΩ(1)
.'(_hole_car'2, ++'(_gen_nil':.'4(_\$n7), _gen_nil':.'4(_b147))) →IH
.'(_hole_car'2, _gen_nil':.'4(+(_\$n7, _b147)))

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

Rules:
rev'(nil') → nil'
rev'(.'(x, y)) → ++'(rev'(y), .'(x, nil'))
car'(.'(x, y)) → x
cdr'(.'(x, y)) → y
null'(nil') → true'
null'(.'(x, y)) → false'
++'(nil', y) → y
++'(.'(x, y), z) → .'(x, ++'(y, z))

Types:
rev' :: nil':.' → nil':.'
nil' :: nil':.'
.' :: car' → nil':.' → nil':.'
++' :: nil':.' → nil':.' → nil':.'
car' :: nil':.' → car'
cdr' :: nil':.' → nil':.'
null' :: nil':.' → true':false'
true' :: true':false'
false' :: true':false'
_hole_nil':.'1 :: nil':.'
_hole_car'2 :: car'
_hole_true':false'3 :: true':false'
_gen_nil':.'4 :: Nat → nil':.'

Lemmas:
++'(_gen_nil':.'4(_n6), _gen_nil':.'4(b)) → _gen_nil':.'4(+(_n6, b)), rt ∈ Ω(1 + n6)

Generator Equations:
_gen_nil':.'4(0) ⇔ nil'
_gen_nil':.'4(+(x, 1)) ⇔ .'(_hole_car'2, _gen_nil':.'4(x))

The following defined symbols remain to be analysed:
rev'

Proved the following rewrite lemma:
rev'(_gen_nil':.'4(_n597)) → _gen_nil':.'4(_n597), rt ∈ Ω(1 + n597 + n5972)

Induction Base:
rev'(_gen_nil':.'4(0)) →RΩ(1)
nil'

Induction Step:
rev'(_gen_nil':.'4(+(_\$n598, 1))) →RΩ(1)
++'(rev'(_gen_nil':.'4(_\$n598)), .'(_hole_car'2, nil')) →IH
++'(_gen_nil':.'4(_\$n598), .'(_hole_car'2, nil')) →LΩ(1 + \$n598)
_gen_nil':.'4(+(_\$n598, +(0, 1)))

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

Rules:
rev'(nil') → nil'
rev'(.'(x, y)) → ++'(rev'(y), .'(x, nil'))
car'(.'(x, y)) → x
cdr'(.'(x, y)) → y
null'(nil') → true'
null'(.'(x, y)) → false'
++'(nil', y) → y
++'(.'(x, y), z) → .'(x, ++'(y, z))

Types:
rev' :: nil':.' → nil':.'
nil' :: nil':.'
.' :: car' → nil':.' → nil':.'
++' :: nil':.' → nil':.' → nil':.'
car' :: nil':.' → car'
cdr' :: nil':.' → nil':.'
null' :: nil':.' → true':false'
true' :: true':false'
false' :: true':false'
_hole_nil':.'1 :: nil':.'
_hole_car'2 :: car'
_hole_true':false'3 :: true':false'
_gen_nil':.'4 :: Nat → nil':.'

Lemmas:
++'(_gen_nil':.'4(_n6), _gen_nil':.'4(b)) → _gen_nil':.'4(+(_n6, b)), rt ∈ Ω(1 + n6)
rev'(_gen_nil':.'4(_n597)) → _gen_nil':.'4(_n597), rt ∈ Ω(1 + n597 + n5972)

Generator Equations:
_gen_nil':.'4(0) ⇔ nil'
_gen_nil':.'4(+(x, 1)) ⇔ .'(_hole_car'2, _gen_nil':.'4(x))

No more defined symbols left to analyse.

The lowerbound Ω(n2) was proven with the following lemma:
rev'(_gen_nil':.'4(_n597)) → _gen_nil':.'4(_n597), rt ∈ Ω(1 + n597 + n5972)