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

active(U11(tt, M, N)) → mark(U12(tt, M, N))
active(U12(tt, M, N)) → mark(s(plus(N, M)))
active(U21(tt, M, N)) → mark(U22(tt, M, N))
active(U22(tt, M, N)) → mark(plus(x(N, M), N))
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(U11(tt, M, N))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(U21(tt, M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(U22(X1, X2, X3)) → U22(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
U22(mark(X1), X2, X3) → mark(U22(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(U22(X1, X2, X3)) → U22(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
proper(0) → ok(0)
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
U22(ok(X1), ok(X2), ok(X3)) → ok(U22(X1, X2, X3))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

active'(U11'(tt', M, N)) → mark'(U12'(tt', M, N))
active'(U12'(tt', M, N)) → mark'(s'(plus'(N, M)))
active'(U21'(tt', M, N)) → mark'(U22'(tt', M, N))
active'(U22'(tt', M, N)) → mark'(plus'(x'(N, M), N))
active'(plus'(N, 0')) → mark'(N)
active'(plus'(N, s'(M))) → mark'(U11'(tt', M, N))
active'(x'(N, 0')) → mark'(0')
active'(x'(N, s'(M))) → mark'(U21'(tt', M, N))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(s'(X)) → s'(active'(X))
active'(plus'(X1, X2)) → plus'(active'(X1), X2)
active'(plus'(X1, X2)) → plus'(X1, active'(X2))
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2, X3)) → U22'(active'(X1), X2, X3)
active'(x'(X1, X2)) → x'(active'(X1), X2)
active'(x'(X1, X2)) → x'(X1, active'(X2))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
s'(mark'(X)) → mark'(s'(X))
plus'(mark'(X1), X2) → mark'(plus'(X1, X2))
plus'(X1, mark'(X2)) → mark'(plus'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2, X3) → mark'(U22'(X1, X2, X3))
x'(mark'(X1), X2) → mark'(x'(X1, X2))
x'(X1, mark'(X2)) → mark'(x'(X1, X2))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(s'(X)) → s'(proper'(X))
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2, X3)) → U22'(proper'(X1), proper'(X2), proper'(X3))
proper'(x'(X1, X2)) → x'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
s'(ok'(X)) → ok'(s'(X))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U22'(X1, X2, X3))
x'(ok'(X1), ok'(X2)) → ok'(x'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(U11'(tt', M, N)) → mark'(U12'(tt', M, N))
active'(U12'(tt', M, N)) → mark'(s'(plus'(N, M)))
active'(U21'(tt', M, N)) → mark'(U22'(tt', M, N))
active'(U22'(tt', M, N)) → mark'(plus'(x'(N, M), N))
active'(plus'(N, 0')) → mark'(N)
active'(plus'(N, s'(M))) → mark'(U11'(tt', M, N))
active'(x'(N, 0')) → mark'(0')
active'(x'(N, s'(M))) → mark'(U21'(tt', M, N))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2, X3)) → U12'(active'(X1), X2, X3)
active'(s'(X)) → s'(active'(X))
active'(plus'(X1, X2)) → plus'(active'(X1), X2)
active'(plus'(X1, X2)) → plus'(X1, active'(X2))
active'(U21'(X1, X2, X3)) → U21'(active'(X1), X2, X3)
active'(U22'(X1, X2, X3)) → U22'(active'(X1), X2, X3)
active'(x'(X1, X2)) → x'(active'(X1), X2)
active'(x'(X1, X2)) → x'(X1, active'(X2))
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2, X3) → mark'(U12'(X1, X2, X3))
s'(mark'(X)) → mark'(s'(X))
plus'(mark'(X1), X2) → mark'(plus'(X1, X2))
plus'(X1, mark'(X2)) → mark'(plus'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(X1, X2, X3))
U22'(mark'(X1), X2, X3) → mark'(U22'(X1, X2, X3))
x'(mark'(X1), X2) → mark'(x'(X1, X2))
x'(X1, mark'(X2)) → mark'(x'(X1, X2))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2, X3)) → U12'(proper'(X1), proper'(X2), proper'(X3))
proper'(s'(X)) → s'(proper'(X))
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(U21'(X1, X2, X3)) → U21'(proper'(X1), proper'(X2), proper'(X3))
proper'(U22'(X1, X2, X3)) → U22'(proper'(X1), proper'(X2), proper'(X3))
proper'(x'(X1, X2)) → x'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U12'(X1, X2, X3))
s'(ok'(X)) → ok'(s'(X))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(X1, X2, X3))
U22'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U22'(X1, X2, X3))
x'(ok'(X1), ok'(X2)) → ok'(x'(X1, X2))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Types:
active' :: tt':mark':0':ok' → tt':mark':0':ok'
U11' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
tt' :: tt':mark':0':ok'
mark' :: tt':mark':0':ok' → tt':mark':0':ok'
U12' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
s' :: tt':mark':0':ok' → tt':mark':0':ok'
plus' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U21' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U22' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
x' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
0' :: tt':mark':0':ok'
proper' :: tt':mark':0':ok' → tt':mark':0':ok'
ok' :: tt':mark':0':ok' → tt':mark':0':ok'
top' :: tt':mark':0':ok' → top'
_hole_tt':mark':0':ok'1 :: tt':mark':0':ok'
_hole_top'2 :: top'
_gen_tt':mark':0':ok'3 :: Nat → tt':mark':0':ok'

Heuristically decided to analyse the following defined symbols:
active', U12', s', plus', U22', x', U11', U21', proper', top'

They will be analysed ascendingly in the following order:
U12' < active'
s' < active'
plus' < active'
U22' < active'
x' < active'
U11' < active'
U21' < active'
active' < top'
U12' < proper'
s' < proper'
plus' < proper'
U22' < proper'
x' < proper'
U11' < proper'
U21' < proper'
proper' < top'

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
U12', active', s', plus', U22', x', U11', U21', proper', top'

Proved the following rewrite lemma:
U12'(_gen_tt':mark':0':ok'3(+(1, _n5)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n5)

Induction Base:
U12'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))

Induction Step:
U12'(_gen_tt':mark':0':ok'3(+(1, +(_$n6, 1))), _gen_tt':mark':0':ok'3(_b1018), _gen_tt':mark':0':ok'3(_c1019)) →_R^Ω(1)
mark'(U12'(_gen_tt':mark':0':ok'3(+(1, _$n6)), _gen_tt':mark':0':ok'3(_b1018), _gen_tt':mark':0':ok'3(_c1019))) →_IH
mark'(_*4)

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

Lemmas:
U12'(_gen_tt':mark':0':ok'3(+(1, _n5)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n5)

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
s', active', plus', U22', x', U11', U21', proper', top'

They will be analysed ascendingly in the following order:
s' < active'
plus' < active'
U22' < active'
x' < active'
U11' < active'
U21' < active'
active' < top'
s' < proper'
plus' < proper'
U22' < proper'
x' < proper'
U11' < proper'
U21' < proper'
proper' < top'

Proved the following rewrite lemma:
s'(_gen_tt':mark':0':ok'3(+(1, _n4311))) → _*4, rt ∈ Ω(_n4311)

Induction Base:
s'(_gen_tt':mark':0':ok'3(+(1, 0)))

Induction Step:
s'(_gen_tt':mark':0':ok'3(+(1, +(_$n4312, 1)))) →_R^Ω(1)
mark'(s'(_gen_tt':mark':0':ok'3(+(1, _$n4312)))) →_IH
mark'(_*4)

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

Lemmas:
U12'(_gen_tt':mark':0':ok'3(+(1, _n5)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n5)
s'(_gen_tt':mark':0':ok'3(+(1, _n4311))) → _*4, rt ∈ Ω(_n4311)

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
plus', active', U22', x', U11', U21', proper', top'

They will be analysed ascendingly in the following order:
plus' < active'
U22' < active'
x' < active'
U11' < active'
U21' < active'
active' < top'
plus' < proper'
U22' < proper'
x' < proper'
U11' < proper'
U21' < proper'
proper' < top'

Proved the following rewrite lemma:
plus'(_gen_tt':mark':0':ok'3(+(1, _n6521)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n6521)

Induction Base:
plus'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b))

Induction Step:
plus'(_gen_tt':mark':0':ok'3(+(1, +(_$n6522, 1))), _gen_tt':mark':0':ok'3(_b7990)) →_R^Ω(1)
mark'(plus'(_gen_tt':mark':0':ok'3(+(1, _$n6522)), _gen_tt':mark':0':ok'3(_b7990))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
U22', active', x', U11', U21', proper', top'

They will be analysed ascendingly in the following order:
U22' < active'
x' < active'
U11' < active'
U21' < active'
active' < top'
U22' < proper'
x' < proper'
U11' < proper'
U21' < proper'
proper' < top'

Proved the following rewrite lemma:
U22'(_gen_tt':mark':0':ok'3(+(1, _n10548)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n10548)

Induction Base:
U22'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))

Induction Step:
U22'(_gen_tt':mark':0':ok'3(+(1, +(_$n10549, 1))), _gen_tt':mark':0':ok'3(_b13019), _gen_tt':mark':0':ok'3(_c13020)) →_R^Ω(1)
mark'(U22'(_gen_tt':mark':0':ok'3(+(1, _$n10549)), _gen_tt':mark':0':ok'3(_b13019), _gen_tt':mark':0':ok'3(_c13020))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
x', active', U11', U21', proper', top'

They will be analysed ascendingly in the following order:
x' < active'
U11' < active'
U21' < active'
active' < top'
x' < proper'
U11' < proper'
U21' < proper'
proper' < top'

Proved the following rewrite lemma:
x'(_gen_tt':mark':0':ok'3(+(1, _n16489)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n16489)

Induction Base:
x'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b))

Induction Step:
x'(_gen_tt':mark':0':ok'3(+(1, +(_$n16490, 1))), _gen_tt':mark':0':ok'3(_b18714)) →_R^Ω(1)
mark'(x'(_gen_tt':mark':0':ok'3(+(1, _$n16490)), _gen_tt':mark':0':ok'3(_b18714))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
U11', active', U21', proper', top'

They will be analysed ascendingly in the following order:
U11' < active'
U21' < active'
active' < top'
U11' < proper'
U21' < proper'
proper' < top'

Proved the following rewrite lemma:
U11'(_gen_tt':mark':0':ok'3(+(1, _n21379)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n21379)

Induction Base:
U11'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))

Induction Step:
U11'(_gen_tt':mark':0':ok'3(+(1, +(_$n21380, 1))), _gen_tt':mark':0':ok'3(_b24984), _gen_tt':mark':0':ok'3(_c24985)) →_R^Ω(1)
mark'(U11'(_gen_tt':mark':0':ok'3(+(1, _$n21380)), _gen_tt':mark':0':ok'3(_b24984), _gen_tt':mark':0':ok'3(_c24985))) →_IH
mark'(_*4)

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

Lemmas:
U12'(_gen_tt':mark':0':ok'3(+(1, _n5)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n5)
s'(_gen_tt':mark':0':ok'3(+(1, _n4311))) → _*4, rt ∈ Ω(_n4311)
plus'(_gen_tt':mark':0':ok'3(+(1, _n6521)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n6521)
U22'(_gen_tt':mark':0':ok'3(+(1, _n10548)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n10548)
x'(_gen_tt':mark':0':ok'3(+(1, _n16489)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n16489)
U11'(_gen_tt':mark':0':ok'3(+(1, _n21379)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n21379)

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
U21', active', proper', top'

They will be analysed ascendingly in the following order:
U21' < active'
active' < top'
U21' < proper'
proper' < top'

Proved the following rewrite lemma:
U21'(_gen_tt':mark':0':ok'3(+(1, _n28597)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n28597)

Induction Base:
U21'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))

Induction Step:
U21'(_gen_tt':mark':0':ok'3(+(1, +(_$n28598, 1))), _gen_tt':mark':0':ok'3(_b32850), _gen_tt':mark':0':ok'3(_c32851)) →_R^Ω(1)
mark'(U21'(_gen_tt':mark':0':ok'3(+(1, _$n28598)), _gen_tt':mark':0':ok'3(_b32850), _gen_tt':mark':0':ok'3(_c32851))) →_IH
mark'(_*4)

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

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
active', proper', top'

They will be analysed ascendingly in the following order:
active' < top'
proper' < top'

Could not prove a rewrite lemma for the defined symbol active'.

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
proper', top'

They will be analysed ascendingly in the following order:
proper' < top'

Could not prove a rewrite lemma for the defined symbol proper'.

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

The following defined symbols remain to be analysed:
top'

Could not prove a rewrite lemma for the defined symbol top'.

Generator Equations:
_gen_tt':mark':0':ok'3(0) ⇔ tt'
_gen_tt':mark':0':ok'3(+(x, 1)) ⇔ mark'(_gen_tt':mark':0':ok'3(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
U12'(_gen_tt':mark':0':ok'3(+(1, _n5)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n5)