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

a__U11(tt, V2) → a__U12(a__isNat(V2))
a__U12(tt) → tt
a__U21(tt) → tt
a__U31(tt, N) → mark(N)
a__U41(tt, M, N) → a__U42(a__isNat(N), M, N)
a__U42(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__isNat(0) → tt
a__isNat(plus(V1, V2)) → a__U11(a__isNat(V1), V2)
a__isNat(s(V1)) → a__U21(a__isNat(V1))
a__plus(N, 0) → a__U31(a__isNat(N), N)
a__plus(N, s(M)) → a__U41(a__isNat(M), M, N)
mark(U11(X1, X2)) → a__U11(mark(X1), X2)
mark(U12(X)) → a__U12(mark(X))
mark(isNat(X)) → a__isNat(X)
mark(U21(X)) → a__U21(mark(X))
mark(U31(X1, X2)) → a__U31(mark(X1), X2)
mark(U41(X1, X2, X3)) → a__U41(mark(X1), X2, X3)
mark(U42(X1, X2, X3)) → a__U42(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2) → U11(X1, X2)
a__U12(X) → U12(X)
a__isNat(X) → isNat(X)
a__U21(X) → U21(X)
a__U31(X1, X2) → U31(X1, X2)
a__U41(X1, X2, X3) → U41(X1, X2, X3)
a__U42(X1, X2, X3) → U42(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

a__U11'(tt', V2) → a__U12'(a__isNat'(V2))
a__U12'(tt') → tt'
a__U21'(tt') → tt'
a__U31'(tt', N) → mark'(N)
a__U41'(tt', M, N) → a__U42'(a__isNat'(N), M, N)
a__U42'(tt', M, N) → s'(a__plus'(mark'(N), mark'(M)))
a__isNat'(0') → tt'
a__isNat'(plus'(V1, V2)) → a__U11'(a__isNat'(V1), V2)
a__isNat'(s'(V1)) → a__U21'(a__isNat'(V1))
a__plus'(N, 0') → a__U31'(a__isNat'(N), N)
a__plus'(N, s'(M)) → a__U41'(a__isNat'(M), M, N)
mark'(U11'(X1, X2)) → a__U11'(mark'(X1), X2)
mark'(U12'(X)) → a__U12'(mark'(X))
mark'(isNat'(X)) → a__isNat'(X)
mark'(U21'(X)) → a__U21'(mark'(X))
mark'(U31'(X1, X2)) → a__U31'(mark'(X1), X2)
mark'(U41'(X1, X2, X3)) → a__U41'(mark'(X1), X2, X3)
mark'(U42'(X1, X2, X3)) → a__U42'(mark'(X1), X2, X3)
mark'(plus'(X1, X2)) → a__plus'(mark'(X1), mark'(X2))
mark'(tt') → tt'
mark'(s'(X)) → s'(mark'(X))
mark'(0') → 0'
a__U11'(X1, X2) → U11'(X1, X2)
a__U12'(X) → U12'(X)
a__isNat'(X) → isNat'(X)
a__U21'(X) → U21'(X)
a__U31'(X1, X2) → U31'(X1, X2)
a__U41'(X1, X2, X3) → U41'(X1, X2, X3)
a__U42'(X1, X2, X3) → U42'(X1, X2, X3)
a__plus'(X1, X2) → plus'(X1, X2)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
a__U11'(tt', V2) → a__U12'(a__isNat'(V2))
a__U12'(tt') → tt'
a__U21'(tt') → tt'
a__U31'(tt', N) → mark'(N)
a__U41'(tt', M, N) → a__U42'(a__isNat'(N), M, N)
a__U42'(tt', M, N) → s'(a__plus'(mark'(N), mark'(M)))
a__isNat'(0') → tt'
a__isNat'(plus'(V1, V2)) → a__U11'(a__isNat'(V1), V2)
a__isNat'(s'(V1)) → a__U21'(a__isNat'(V1))
a__plus'(N, 0') → a__U31'(a__isNat'(N), N)
a__plus'(N, s'(M)) → a__U41'(a__isNat'(M), M, N)
mark'(U11'(X1, X2)) → a__U11'(mark'(X1), X2)
mark'(U12'(X)) → a__U12'(mark'(X))
mark'(isNat'(X)) → a__isNat'(X)
mark'(U21'(X)) → a__U21'(mark'(X))
mark'(U31'(X1, X2)) → a__U31'(mark'(X1), X2)
mark'(U41'(X1, X2, X3)) → a__U41'(mark'(X1), X2, X3)
mark'(U42'(X1, X2, X3)) → a__U42'(mark'(X1), X2, X3)
mark'(plus'(X1, X2)) → a__plus'(mark'(X1), mark'(X2))
mark'(tt') → tt'
mark'(s'(X)) → s'(mark'(X))
mark'(0') → 0'
a__U11'(X1, X2) → U11'(X1, X2)
a__U12'(X) → U12'(X)
a__isNat'(X) → isNat'(X)
a__U21'(X) → U21'(X)
a__U31'(X1, X2) → U31'(X1, X2)
a__U41'(X1, X2, X3) → U41'(X1, X2, X3)
a__U42'(X1, X2, X3) → U42'(X1, X2, X3)
a__plus'(X1, X2) → plus'(X1, X2)

Types:
a__U11' :: tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'
tt' :: tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'
a__U12' :: tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'
a__isNat' :: tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'
a__U21' :: tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'
a__U31' :: tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'
mark' :: tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'
a__U41' :: tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'
a__U42' :: tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'
s' :: tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'
a__plus' :: tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'
0' :: tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'
plus' :: tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'
U11' :: tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'
U12' :: tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'
isNat' :: tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'
U21' :: tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'
U31' :: tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'
U41' :: tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'
U42' :: tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42' → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'
_hole_tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'1 :: tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'
_gen_tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'2 :: Nat → tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'

Heuristically decided to analyse the following defined symbols:
a__U11', a__isNat', a__U31', mark', a__U41', a__U42', a__plus'

They will be analysed ascendingly in the following order:
a__U11' = a__isNat'
a__U11' < mark'
a__isNat' < mark'
a__isNat' < a__U41'
a__isNat' < a__plus'
a__U31' = mark'
a__U31' = a__U41'
a__U31' = a__U42'
a__U31' = a__plus'
mark' = a__U41'
mark' = a__U42'
mark' = a__plus'
a__U41' = a__U42'
a__U41' = a__plus'
a__U42' = a__plus'

Generator Equations:
_gen_tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'2(0) ⇔ tt'
_gen_tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'2(+(x, 1)) ⇔ s'(_gen_tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'2(x))

The following defined symbols remain to be analysed:
a__isNat', a__U11', a__U31', mark', a__U41', a__U42', a__plus'

Proved the following rewrite lemma:
a__isNat'(_gen_tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'2(+(1, _n4))) → _*3, rt ∈ Ω(_n4)

Induction Base:
a__isNat'(_gen_tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'2(+(1, 0)))

Induction Step:
a__isNat'(_gen_tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'2(+(1, +(_$n5, 1)))) →_R^Ω(1)
a__U21'(a__isNat'(_gen_tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'2(+(1, _$n5)))) →_IH
a__U21'(_*3)

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

Lemmas:
a__isNat'(_gen_tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'2(+(1, _n4))) → _*3, rt ∈ Ω(_n4)

The following defined symbols remain to be analysed:
a__U11', a__U31', mark', a__U41', a__U42', a__plus'

Could not prove a rewrite lemma for the defined symbol a__U11'.

Lemmas:
a__isNat'(_gen_tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'2(+(1, _n4))) → _*3, rt ∈ Ω(_n4)

The following defined symbols remain to be analysed:
mark', a__U31', a__U41', a__U42', a__plus'

They will be analysed ascendingly in the following order:
a__U31' = mark'
a__U31' = a__U41'
a__U31' = a__U42'
a__U31' = a__plus'
mark' = a__U41'
mark' = a__U42'
mark' = a__plus'
a__U41' = a__U42'
a__U41' = a__plus'
a__U42' = a__plus'

Proved the following rewrite lemma:
mark'(_gen_tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'2(_n5684)) → _gen_tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'2(_n5684), rt ∈ Ω(1 + _n5684)

Induction Base:
mark'(_gen_tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'2(0)) →_R^Ω(1)
tt'

Induction Step:
mark'(_gen_tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'2(+(_$n5685, 1))) →_R^Ω(1)
s'(mark'(_gen_tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'2(_$n5685))) →_IH
s'(_gen_tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'2(_$n5685))

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

Lemmas:
a__isNat'(_gen_tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'2(+(1, _n4))) → _*3, rt ∈ Ω(_n4)
mark'(_gen_tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'2(_n5684)) → _gen_tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'2(_n5684), rt ∈ Ω(1 + _n5684)

The following defined symbols remain to be analysed:
a__U31', a__U41', a__U42', a__plus'

Could not prove a rewrite lemma for the defined symbol a__U31'.

The following defined symbols remain to be analysed:
a__U41', a__U42', a__plus'

Could not prove a rewrite lemma for the defined symbol a__U41'.

The following defined symbols remain to be analysed:
a__U42', a__plus'

Could not prove a rewrite lemma for the defined symbol a__U42'.

The following defined symbols remain to be analysed:
a__plus'

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

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
a__isNat'(_gen_tt':s':0':plus':U11':U12':isNat':U21':U31':U41':U42'2(+(1, _n4))) → _*3, rt ∈ Ω(_n4)