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

active(U11(tt, V1, V2)) → mark(U12(isNat(V1), V2))
active(U12(tt, V2)) → mark(U13(isNat(V2)))
active(U13(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNat(V1)))
active(U22(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(and(isNatKind(V1), isNatKind(V2)))
active(isNatKind(s(V1))) → mark(isNatKind(V1))
active(plus(N, 0)) → mark(U31(and(isNat(N), isNatKind(N)), N))
active(plus(N, s(M))) → mark(U41(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2)) → U12(active(X1), X2)
active(U13(X)) → U13(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X)) → U22(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(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(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2) → mark(U12(X1, X2))
U13(mark(X)) → mark(U13(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X)) → mark(U22(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(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))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U13(X)) → U13(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X)) → U22(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(0) → ok(0)
proper(isNatKind(X)) → isNatKind(proper(X))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U13(ok(X)) → ok(U13(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X)) → ok(U22(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNatKind(ok(X)) → ok(isNatKind(X))
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', V1, V2)) → mark'(U12'(isNat'(V1), V2))
active'(U12'(tt', V2)) → mark'(U13'(isNat'(V2)))
active'(U13'(tt')) → mark'(tt')
active'(U21'(tt', V1)) → mark'(U22'(isNat'(V1)))
active'(U22'(tt')) → mark'(tt')
active'(U31'(tt', N)) → mark'(N)
active'(U41'(tt', M, N)) → mark'(s'(plus'(N, M)))
active'(and'(tt', X)) → mark'(X)
active'(isNat'(0')) → mark'(tt')
active'(isNat'(plus'(V1, V2))) → mark'(U11'(and'(isNatKind'(V1), isNatKind'(V2)), V1, V2))
active'(isNat'(s'(V1))) → mark'(U21'(isNatKind'(V1), V1))
active'(isNatKind'(0')) → mark'(tt')
active'(isNatKind'(plus'(V1, V2))) → mark'(and'(isNatKind'(V1), isNatKind'(V2)))
active'(isNatKind'(s'(V1))) → mark'(isNatKind'(V1))
active'(plus'(N, 0')) → mark'(U31'(and'(isNat'(N), isNatKind'(N)), N))
active'(plus'(N, s'(M))) → mark'(U41'(and'(and'(isNat'(M), isNatKind'(M)), and'(isNat'(N), isNatKind'(N))), M, N))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2)) → U12'(active'(X1), X2)
active'(U13'(X)) → U13'(active'(X))
active'(U21'(X1, X2)) → U21'(active'(X1), X2)
active'(U22'(X)) → U22'(active'(X))
active'(U31'(X1, X2)) → U31'(active'(X1), X2)
active'(U41'(X1, X2, X3)) → U41'(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'(and'(X1, X2)) → and'(active'(X1), X2)
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2) → mark'(U12'(X1, X2))
U13'(mark'(X)) → mark'(U13'(X))
U21'(mark'(X1), X2) → mark'(U21'(X1, X2))
U22'(mark'(X)) → mark'(U22'(X))
U31'(mark'(X1), X2) → mark'(U31'(X1, X2))
U41'(mark'(X1), X2, X3) → mark'(U41'(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))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2)) → U12'(proper'(X1), proper'(X2))
proper'(isNat'(X)) → isNat'(proper'(X))
proper'(U13'(X)) → U13'(proper'(X))
proper'(U21'(X1, X2)) → U21'(proper'(X1), proper'(X2))
proper'(U22'(X)) → U22'(proper'(X))
proper'(U31'(X1, X2)) → U31'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2, X3)) → U41'(proper'(X1), proper'(X2), proper'(X3))
proper'(s'(X)) → s'(proper'(X))
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(isNatKind'(X)) → isNatKind'(proper'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2)) → ok'(U12'(X1, X2))
isNat'(ok'(X)) → ok'(isNat'(X))
U13'(ok'(X)) → ok'(U13'(X))
U21'(ok'(X1), ok'(X2)) → ok'(U21'(X1, X2))
U22'(ok'(X)) → ok'(U22'(X))
U31'(ok'(X1), ok'(X2)) → ok'(U31'(X1, X2))
U41'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U41'(X1, X2, X3))
s'(ok'(X)) → ok'(s'(X))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
isNatKind'(ok'(X)) → ok'(isNatKind'(X))
top'(mark'(X)) → top'(proper'(X))
top'(ok'(X)) → top'(active'(X))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
active'(U11'(tt', V1, V2)) → mark'(U12'(isNat'(V1), V2))
active'(U12'(tt', V2)) → mark'(U13'(isNat'(V2)))
active'(U13'(tt')) → mark'(tt')
active'(U21'(tt', V1)) → mark'(U22'(isNat'(V1)))
active'(U22'(tt')) → mark'(tt')
active'(U31'(tt', N)) → mark'(N)
active'(U41'(tt', M, N)) → mark'(s'(plus'(N, M)))
active'(and'(tt', X)) → mark'(X)
active'(isNat'(0')) → mark'(tt')
active'(isNat'(plus'(V1, V2))) → mark'(U11'(and'(isNatKind'(V1), isNatKind'(V2)), V1, V2))
active'(isNat'(s'(V1))) → mark'(U21'(isNatKind'(V1), V1))
active'(isNatKind'(0')) → mark'(tt')
active'(isNatKind'(plus'(V1, V2))) → mark'(and'(isNatKind'(V1), isNatKind'(V2)))
active'(isNatKind'(s'(V1))) → mark'(isNatKind'(V1))
active'(plus'(N, 0')) → mark'(U31'(and'(isNat'(N), isNatKind'(N)), N))
active'(plus'(N, s'(M))) → mark'(U41'(and'(and'(isNat'(M), isNatKind'(M)), and'(isNat'(N), isNatKind'(N))), M, N))
active'(U11'(X1, X2, X3)) → U11'(active'(X1), X2, X3)
active'(U12'(X1, X2)) → U12'(active'(X1), X2)
active'(U13'(X)) → U13'(active'(X))
active'(U21'(X1, X2)) → U21'(active'(X1), X2)
active'(U22'(X)) → U22'(active'(X))
active'(U31'(X1, X2)) → U31'(active'(X1), X2)
active'(U41'(X1, X2, X3)) → U41'(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'(and'(X1, X2)) → and'(active'(X1), X2)
U11'(mark'(X1), X2, X3) → mark'(U11'(X1, X2, X3))
U12'(mark'(X1), X2) → mark'(U12'(X1, X2))
U13'(mark'(X)) → mark'(U13'(X))
U21'(mark'(X1), X2) → mark'(U21'(X1, X2))
U22'(mark'(X)) → mark'(U22'(X))
U31'(mark'(X1), X2) → mark'(U31'(X1, X2))
U41'(mark'(X1), X2, X3) → mark'(U41'(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))
and'(mark'(X1), X2) → mark'(and'(X1, X2))
proper'(U11'(X1, X2, X3)) → U11'(proper'(X1), proper'(X2), proper'(X3))
proper'(tt') → ok'(tt')
proper'(U12'(X1, X2)) → U12'(proper'(X1), proper'(X2))
proper'(isNat'(X)) → isNat'(proper'(X))
proper'(U13'(X)) → U13'(proper'(X))
proper'(U21'(X1, X2)) → U21'(proper'(X1), proper'(X2))
proper'(U22'(X)) → U22'(proper'(X))
proper'(U31'(X1, X2)) → U31'(proper'(X1), proper'(X2))
proper'(U41'(X1, X2, X3)) → U41'(proper'(X1), proper'(X2), proper'(X3))
proper'(s'(X)) → s'(proper'(X))
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(and'(X1, X2)) → and'(proper'(X1), proper'(X2))
proper'(0') → ok'(0')
proper'(isNatKind'(X)) → isNatKind'(proper'(X))
U11'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U11'(X1, X2, X3))
U12'(ok'(X1), ok'(X2)) → ok'(U12'(X1, X2))
isNat'(ok'(X)) → ok'(isNat'(X))
U13'(ok'(X)) → ok'(U13'(X))
U21'(ok'(X1), ok'(X2)) → ok'(U21'(X1, X2))
U22'(ok'(X)) → ok'(U22'(X))
U31'(ok'(X1), ok'(X2)) → ok'(U31'(X1, X2))
U41'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U41'(X1, X2, X3))
s'(ok'(X)) → ok'(s'(X))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
and'(ok'(X1), ok'(X2)) → ok'(and'(X1, X2))
isNatKind'(ok'(X)) → ok'(isNatKind'(X))
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'
isNat' :: tt':mark':0':ok' → tt':mark':0':ok'
U13' :: tt':mark':0':ok' → tt':mark':0':ok'
U21' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U22' :: tt':mark':0':ok' → tt':mark':0':ok'
U31' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U41' :: 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'
and' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
0' :: tt':mark':0':ok'
isNatKind' :: tt':mark':0':ok' → 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', isNat', U13', U22', s', plus', U11', and', isNatKind', U21', U31', U41', proper', top'

They will be analysed ascendingly in the following order:
U12' < active'
isNat' < active'
U13' < active'
U22' < active'
s' < active'
plus' < active'
U11' < active'
and' < active'
isNatKind' < active'
U21' < active'
U31' < active'
U41' < active'
active' < top'
U12' < proper'
isNat' < proper'
U13' < proper'
U22' < proper'
s' < proper'
plus' < proper'
U11' < proper'
and' < proper'
isNatKind' < proper'
U21' < proper'
U31' < proper'
U41' < 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', isNat', U13', U22', s', plus', U11', and', isNatKind', U21', U31', U41', proper', top'

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

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

Induction Step:
U12'(_gen_tt':mark':0':ok'3(+(1, +(_$n6, 1))), _gen_tt':mark':0':ok'3(_b610)) →_R^Ω(1)
mark'(U12'(_gen_tt':mark':0':ok'3(+(1, _$n6)), _gen_tt':mark':0':ok'3(_b610))) →_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)) → _*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:
isNat', active', U13', U22', s', plus', U11', and', isNatKind', U21', U31', U41', proper', top'

They will be analysed ascendingly in the following order:
isNat' < active'
U13' < active'
U22' < active'
s' < active'
plus' < active'
U11' < active'
and' < active'
isNatKind' < active'
U21' < active'
U31' < active'
U41' < active'
active' < top'
isNat' < proper'
U13' < proper'
U22' < proper'
s' < proper'
plus' < proper'
U11' < proper'
and' < proper'
isNatKind' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
proper' < top'

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

Lemmas:
U12'(_gen_tt':mark':0':ok'3(+(1, _n5)), _gen_tt':mark':0':ok'3(b)) → _*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:
U13', active', U22', s', plus', U11', and', isNatKind', U21', U31', U41', proper', top'

They will be analysed ascendingly in the following order:
U13' < active'
U22' < active'
s' < active'
plus' < active'
U11' < active'
and' < active'
isNatKind' < active'
U21' < active'
U31' < active'
U41' < active'
active' < top'
U13' < proper'
U22' < proper'
s' < proper'
plus' < proper'
U11' < proper'
and' < proper'
isNatKind' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
proper' < top'

Proved the following rewrite lemma:
U13'(_gen_tt':mark':0':ok'3(+(1, _n3588))) → _*4, rt ∈ Ω(_n3588)

Induction Base:
U13'(_gen_tt':mark':0':ok'3(+(1, 0)))

Induction Step:
U13'(_gen_tt':mark':0':ok'3(+(1, +(_$n3589, 1)))) →_R^Ω(1)
mark'(U13'(_gen_tt':mark':0':ok'3(+(1, _$n3589)))) →_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)) → _*4, rt ∈ Ω(_n5)
U13'(_gen_tt':mark':0':ok'3(+(1, _n3588))) → _*4, rt ∈ Ω(_n3588)

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', s', plus', U11', and', isNatKind', U21', U31', U41', proper', top'

They will be analysed ascendingly in the following order:
U22' < active'
s' < active'
plus' < active'
U11' < active'
and' < active'
isNatKind' < active'
U21' < active'
U31' < active'
U41' < active'
active' < top'
U22' < proper'
s' < proper'
plus' < proper'
U11' < proper'
and' < proper'
isNatKind' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
proper' < top'

Proved the following rewrite lemma:
U22'(_gen_tt':mark':0':ok'3(+(1, _n6028))) → _*4, rt ∈ Ω(_n6028)

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

Induction Step:
U22'(_gen_tt':mark':0':ok'3(+(1, +(_$n6029, 1)))) →_R^Ω(1)
mark'(U22'(_gen_tt':mark':0':ok'3(+(1, _$n6029)))) →_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:
s', active', plus', U11', and', isNatKind', U21', U31', U41', proper', top'

They will be analysed ascendingly in the following order:
s' < active'
plus' < active'
U11' < active'
and' < active'
isNatKind' < active'
U21' < active'
U31' < active'
U41' < active'
active' < top'
s' < proper'
plus' < proper'
U11' < proper'
and' < proper'
isNatKind' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
proper' < top'

Proved the following rewrite lemma:
s'(_gen_tt':mark':0':ok'3(+(1, _n8592))) → _*4, rt ∈ Ω(_n8592)

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

Induction Step:
s'(_gen_tt':mark':0':ok'3(+(1, +(_$n8593, 1)))) →_R^Ω(1)
mark'(s'(_gen_tt':mark':0':ok'3(+(1, _$n8593)))) →_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:
plus', active', U11', and', isNatKind', U21', U31', U41', proper', top'

They will be analysed ascendingly in the following order:
plus' < active'
U11' < active'
and' < active'
isNatKind' < active'
U21' < active'
U31' < active'
U41' < active'
active' < top'
plus' < proper'
U11' < proper'
and' < proper'
isNatKind' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
proper' < top'

Proved the following rewrite lemma:
plus'(_gen_tt':mark':0':ok'3(+(1, _n11280)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n11280)

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, +(_$n11281, 1))), _gen_tt':mark':0':ok'3(_b13073)) →_R^Ω(1)
mark'(plus'(_gen_tt':mark':0':ok'3(+(1, _$n11281)), _gen_tt':mark':0':ok'3(_b13073))) →_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', and', isNatKind', U21', U31', U41', proper', top'

They will be analysed ascendingly in the following order:
U11' < active'
and' < active'
isNatKind' < active'
U21' < active'
U31' < active'
U41' < active'
active' < top'
U11' < proper'
and' < proper'
isNatKind' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
proper' < top'

Proved the following rewrite lemma:
U11'(_gen_tt':mark':0':ok'3(+(1, _n16157)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n16157)

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, +(_$n16158, 1))), _gen_tt':mark':0':ok'3(_b19114), _gen_tt':mark':0':ok'3(_c19115)) →_R^Ω(1)
mark'(U11'(_gen_tt':mark':0':ok'3(+(1, _$n16158)), _gen_tt':mark':0':ok'3(_b19114), _gen_tt':mark':0':ok'3(_c19115))) →_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:
and', active', isNatKind', U21', U31', U41', proper', top'

They will be analysed ascendingly in the following order:
and' < active'
isNatKind' < active'
U21' < active'
U31' < active'
U41' < active'
active' < top'
and' < proper'
isNatKind' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
proper' < top'

Proved the following rewrite lemma:
and'(_gen_tt':mark':0':ok'3(+(1, _n23320)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n23320)

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

Induction Step:
and'(_gen_tt':mark':0':ok'3(+(1, +(_$n23321, 1))), _gen_tt':mark':0':ok'3(_b25653)) →_R^Ω(1)
mark'(and'(_gen_tt':mark':0':ok'3(+(1, _$n23321)), _gen_tt':mark':0':ok'3(_b25653))) →_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:
isNatKind', active', U21', U31', U41', proper', top'

They will be analysed ascendingly in the following order:
isNatKind' < active'
U21' < active'
U31' < active'
U41' < active'
active' < top'
isNatKind' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
proper' < top'

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

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', U31', U41', proper', top'

They will be analysed ascendingly in the following order:
U21' < active'
U31' < active'
U41' < active'
active' < top'
U21' < proper'
U31' < proper'
U41' < proper'
proper' < top'

Proved the following rewrite lemma:
U21'(_gen_tt':mark':0':ok'3(+(1, _n28909)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n28909)

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

Induction Step:
U21'(_gen_tt':mark':0':ok'3(+(1, +(_$n28910, 1))), _gen_tt':mark':0':ok'3(_b31566)) →_R^Ω(1)
mark'(U21'(_gen_tt':mark':0':ok'3(+(1, _$n28910)), _gen_tt':mark':0':ok'3(_b31566))) →_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)) → _*4, rt ∈ Ω(_n5)
U13'(_gen_tt':mark':0':ok'3(+(1, _n3588))) → _*4, rt ∈ Ω(_n3588)
U22'(_gen_tt':mark':0':ok'3(+(1, _n6028))) → _*4, rt ∈ Ω(_n6028)
s'(_gen_tt':mark':0':ok'3(+(1, _n8592))) → _*4, rt ∈ Ω(_n8592)
plus'(_gen_tt':mark':0':ok'3(+(1, _n11280)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n11280)
U11'(_gen_tt':mark':0':ok'3(+(1, _n16157)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n16157)
and'(_gen_tt':mark':0':ok'3(+(1, _n23320)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n23320)
U21'(_gen_tt':mark':0':ok'3(+(1, _n28909)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n28909)

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:
U31', active', U41', proper', top'

They will be analysed ascendingly in the following order:
U31' < active'
U41' < active'
active' < top'
U31' < proper'
U41' < proper'
proper' < top'

Proved the following rewrite lemma:
U31'(_gen_tt':mark':0':ok'3(+(1, _n34798)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n34798)

Induction Base:
U31'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b))

Induction Step:
U31'(_gen_tt':mark':0':ok'3(+(1, +(_$n34799, 1))), _gen_tt':mark':0':ok'3(_b37779)) →_R^Ω(1)
mark'(U31'(_gen_tt':mark':0':ok'3(+(1, _$n34799)), _gen_tt':mark':0':ok'3(_b37779))) →_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:
U41', active', proper', top'

They will be analysed ascendingly in the following order:
U41' < active'
active' < top'
U41' < proper'
proper' < top'

Proved the following rewrite lemma:
U41'(_gen_tt':mark':0':ok'3(+(1, _n41055)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n41055)

Induction Base:
U41'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))

Induction Step:
U41'(_gen_tt':mark':0':ok'3(+(1, +(_$n41056, 1))), _gen_tt':mark':0':ok'3(_b46118), _gen_tt':mark':0':ok'3(_c46119)) →_R^Ω(1)
mark'(U41'(_gen_tt':mark':0':ok'3(+(1, _$n41056)), _gen_tt':mark':0':ok'3(_b46118), _gen_tt':mark':0':ok'3(_c46119))) →_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'