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

and(tt, X) → activate(X)
plus(N, 0) → N
plus(N, s(M)) → s(plus(N, M))
x(N, 0) → 0
x(N, s(M)) → plus(x(N, M), N)
activate(X) → X

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

and'(tt', X) → activate'(X)
plus'(N, 0') → N
plus'(N, s'(M)) → s'(plus'(N, M))
x'(N, 0') → 0'
x'(N, s'(M)) → plus'(x'(N, M), N)
activate'(X) → X

Rewrite Strategy: INNERMOST

Infered types.

Rules:
and'(tt', X) → activate'(X)
plus'(N, 0') → N
plus'(N, s'(M)) → s'(plus'(N, M))
x'(N, 0') → 0'
x'(N, s'(M)) → plus'(x'(N, M), N)
activate'(X) → X

Types:
and' :: tt' → and':activate' → and':activate'
tt' :: tt'
activate' :: and':activate' → and':activate'
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
x' :: 0':s' → 0':s' → 0':s'
_hole_and':activate'1 :: and':activate'
_hole_tt'2 :: tt'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Heuristically decided to analyse the following defined symbols:
plus', x'

They will be analysed ascendingly in the following order:
plus' < x'

Rules:
and'(tt', X) → activate'(X)
plus'(N, 0') → N
plus'(N, s'(M)) → s'(plus'(N, M))
x'(N, 0') → 0'
x'(N, s'(M)) → plus'(x'(N, M), N)
activate'(X) → X

Types:
and' :: tt' → and':activate' → and':activate'
tt' :: tt'
activate' :: and':activate' → and':activate'
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
x' :: 0':s' → 0':s' → 0':s'
_hole_and':activate'1 :: and':activate'
_hole_tt'2 :: tt'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))

The following defined symbols remain to be analysed:
plus', x'

They will be analysed ascendingly in the following order:
plus' < x'

Proved the following rewrite lemma:
plus'(_gen_0':s'4(a), _gen_0':s'4(_n6)) → _gen_0':s'4(+(_n6, a)), rt ∈ Ω(1 + n6)

Induction Base:
plus'(_gen_0':s'4(a), _gen_0':s'4(0)) →RΩ(1)
_gen_0':s'4(a)

Induction Step:
plus'(_gen_0':s'4(_a139), _gen_0':s'4(+(_\$n7, 1))) →RΩ(1)
s'(plus'(_gen_0':s'4(_a139), _gen_0':s'4(_\$n7))) →IH
s'(_gen_0':s'4(+(_\$n7, _a139)))

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

Rules:
and'(tt', X) → activate'(X)
plus'(N, 0') → N
plus'(N, s'(M)) → s'(plus'(N, M))
x'(N, 0') → 0'
x'(N, s'(M)) → plus'(x'(N, M), N)
activate'(X) → X

Types:
and' :: tt' → and':activate' → and':activate'
tt' :: tt'
activate' :: and':activate' → and':activate'
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
x' :: 0':s' → 0':s' → 0':s'
_hole_and':activate'1 :: and':activate'
_hole_tt'2 :: tt'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
plus'(_gen_0':s'4(a), _gen_0':s'4(_n6)) → _gen_0':s'4(+(_n6, a)), rt ∈ Ω(1 + n6)

Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))

The following defined symbols remain to be analysed:
x'

Proved the following rewrite lemma:
x'(_gen_0':s'4(a), _gen_0':s'4(_n447)) → _gen_0':s'4(*(_n447, a)), rt ∈ Ω(1 + a707·n447 + n447)

Induction Base:
x'(_gen_0':s'4(a), _gen_0':s'4(0)) →RΩ(1)
0'

Induction Step:
x'(_gen_0':s'4(_a707), _gen_0':s'4(+(_\$n448, 1))) →RΩ(1)
plus'(x'(_gen_0':s'4(_a707), _gen_0':s'4(_\$n448)), _gen_0':s'4(_a707)) →IH
plus'(_gen_0':s'4(*(_\$n448, _a707)), _gen_0':s'4(_a707)) →LΩ(1 + a707)
_gen_0':s'4(+(_a707, *(_\$n448, _a707)))

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

Rules:
and'(tt', X) → activate'(X)
plus'(N, 0') → N
plus'(N, s'(M)) → s'(plus'(N, M))
x'(N, 0') → 0'
x'(N, s'(M)) → plus'(x'(N, M), N)
activate'(X) → X

Types:
and' :: tt' → and':activate' → and':activate'
tt' :: tt'
activate' :: and':activate' → and':activate'
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
x' :: 0':s' → 0':s' → 0':s'
_hole_and':activate'1 :: and':activate'
_hole_tt'2 :: tt'
_hole_0':s'3 :: 0':s'
_gen_0':s'4 :: Nat → 0':s'

Lemmas:
plus'(_gen_0':s'4(a), _gen_0':s'4(_n6)) → _gen_0':s'4(+(_n6, a)), rt ∈ Ω(1 + n6)
x'(_gen_0':s'4(a), _gen_0':s'4(_n447)) → _gen_0':s'4(*(_n447, a)), rt ∈ Ω(1 + a707·n447 + n447)

Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))

No more defined symbols left to analyse.

The lowerbound Ω(n2) was proven with the following lemma:
x'(_gen_0':s'4(a), _gen_0':s'4(_n447)) → _gen_0':s'4(*(_n447, a)), rt ∈ Ω(1 + a707·n447 + n447)