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

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

Rewrite Strategy: INNERMOST

Infered types.

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

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

Rules:
active'(U11'(tt', N)) → mark'(N)
active'(U21'(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'(and'(isNat'(V1), isNat'(V2)))
active'(isNat'(s'(V1))) → mark'(isNat'(V1))
active'(plus'(N, 0')) → mark'(U11'(isNat'(N), N))
active'(plus'(N, s'(M))) → mark'(U21'(and'(isNat'(M), isNat'(N)), M, N))
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(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) → mark'(U11'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(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)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U21'(X1, X2, X3)) → U21'(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'(isNat'(X)) → isNat'(proper'(X))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(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))
isNat'(ok'(X)) → ok'(isNat'(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' :: tt':mark':0':ok'
mark' :: 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'
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'
isNat' :: 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'

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', and', isNat', U11', U21', proper', top'

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

Proved the following rewrite lemma:
s'(_gen_tt':mark':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)

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

Induction Step:
s'(_gen_tt':mark':0':ok'3(+(1, +(_$n6, 1)))) →RΩ(1)
mark'(s'(_gen_tt':mark':0':ok'3(+(1, _$n6)))) →IH
mark'(_*4)

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

Rules:
active'(U11'(tt', N)) → mark'(N)
active'(U21'(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'(and'(isNat'(V1), isNat'(V2)))
active'(isNat'(s'(V1))) → mark'(isNat'(V1))
active'(plus'(N, 0')) → mark'(U11'(isNat'(N), N))
active'(plus'(N, s'(M))) → mark'(U21'(and'(isNat'(M), isNat'(N)), M, N))
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(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) → mark'(U11'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(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)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U21'(X1, X2, X3)) → U21'(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'(isNat'(X)) → isNat'(proper'(X))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(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))
isNat'(ok'(X)) → ok'(isNat'(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' :: tt':mark':0':ok'
mark' :: 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'
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'
isNat' :: 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'

Lemmas:
s'(_gen_tt':mark':0':ok'3(+(1, _n5))) → _*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:
plus', active', and', isNat', U11', U21', proper', top'

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

Proved the following rewrite lemma:
plus'(_gen_tt':mark':0':ok'3(+(1, _n1399)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n1399)

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, +(_$n1400, 1))), _gen_tt':mark':0':ok'3(_b2436)) →RΩ(1)
mark'(plus'(_gen_tt':mark':0':ok'3(+(1, _$n1400)), _gen_tt':mark':0':ok'3(_b2436))) →IH
mark'(_*4)

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

Rules:
active'(U11'(tt', N)) → mark'(N)
active'(U21'(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'(and'(isNat'(V1), isNat'(V2)))
active'(isNat'(s'(V1))) → mark'(isNat'(V1))
active'(plus'(N, 0')) → mark'(U11'(isNat'(N), N))
active'(plus'(N, s'(M))) → mark'(U21'(and'(isNat'(M), isNat'(N)), M, N))
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(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) → mark'(U11'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(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)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U21'(X1, X2, X3)) → U21'(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'(isNat'(X)) → isNat'(proper'(X))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(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))
isNat'(ok'(X)) → ok'(isNat'(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' :: tt':mark':0':ok'
mark' :: 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'
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'
isNat' :: 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'

Lemmas:
s'(_gen_tt':mark':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
plus'(_gen_tt':mark':0':ok'3(+(1, _n1399)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n1399)

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', isNat', U11', U21', proper', top'

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

Proved the following rewrite lemma:
and'(_gen_tt':mark':0':ok'3(+(1, _n4121)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n4121)

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, +(_$n4122, 1))), _gen_tt':mark':0':ok'3(_b5266)) →RΩ(1)
mark'(and'(_gen_tt':mark':0':ok'3(+(1, _$n4122)), _gen_tt':mark':0':ok'3(_b5266))) →IH
mark'(_*4)

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

Rules:
active'(U11'(tt', N)) → mark'(N)
active'(U21'(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'(and'(isNat'(V1), isNat'(V2)))
active'(isNat'(s'(V1))) → mark'(isNat'(V1))
active'(plus'(N, 0')) → mark'(U11'(isNat'(N), N))
active'(plus'(N, s'(M))) → mark'(U21'(and'(isNat'(M), isNat'(N)), M, N))
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(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) → mark'(U11'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(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)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U21'(X1, X2, X3)) → U21'(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'(isNat'(X)) → isNat'(proper'(X))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(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))
isNat'(ok'(X)) → ok'(isNat'(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' :: tt':mark':0':ok'
mark' :: 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'
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'
isNat' :: 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'

Lemmas:
s'(_gen_tt':mark':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
plus'(_gen_tt':mark':0':ok'3(+(1, _n1399)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n1399)
and'(_gen_tt':mark':0':ok'3(+(1, _n4121)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n4121)

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', U11', U21', proper', top'

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

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

Rules:
active'(U11'(tt', N)) → mark'(N)
active'(U21'(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'(and'(isNat'(V1), isNat'(V2)))
active'(isNat'(s'(V1))) → mark'(isNat'(V1))
active'(plus'(N, 0')) → mark'(U11'(isNat'(N), N))
active'(plus'(N, s'(M))) → mark'(U21'(and'(isNat'(M), isNat'(N)), M, N))
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(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) → mark'(U11'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(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)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U21'(X1, X2, X3)) → U21'(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'(isNat'(X)) → isNat'(proper'(X))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(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))
isNat'(ok'(X)) → ok'(isNat'(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' :: tt':mark':0':ok'
mark' :: 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'
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'
isNat' :: 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'

Lemmas:
s'(_gen_tt':mark':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
plus'(_gen_tt':mark':0':ok'3(+(1, _n1399)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n1399)
and'(_gen_tt':mark':0':ok'3(+(1, _n4121)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n4121)

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, _n7024)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n7024)

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

Induction Step:
U11'(_gen_tt':mark':0':ok'3(+(1, +(_$n7025, 1))), _gen_tt':mark':0':ok'3(_b8493)) →RΩ(1)
mark'(U11'(_gen_tt':mark':0':ok'3(+(1, _$n7025)), _gen_tt':mark':0':ok'3(_b8493))) →IH
mark'(_*4)

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

Rules:
active'(U11'(tt', N)) → mark'(N)
active'(U21'(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'(and'(isNat'(V1), isNat'(V2)))
active'(isNat'(s'(V1))) → mark'(isNat'(V1))
active'(plus'(N, 0')) → mark'(U11'(isNat'(N), N))
active'(plus'(N, s'(M))) → mark'(U21'(and'(isNat'(M), isNat'(N)), M, N))
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(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) → mark'(U11'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(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)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U21'(X1, X2, X3)) → U21'(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'(isNat'(X)) → isNat'(proper'(X))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(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))
isNat'(ok'(X)) → ok'(isNat'(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' :: tt':mark':0':ok'
mark' :: 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'
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'
isNat' :: 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'

Lemmas:
s'(_gen_tt':mark':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
plus'(_gen_tt':mark':0':ok'3(+(1, _n1399)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n1399)
and'(_gen_tt':mark':0':ok'3(+(1, _n4121)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n4121)
U11'(_gen_tt':mark':0':ok'3(+(1, _n7024)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n7024)

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, _n10263)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(n10263)

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, +(_$n10264, 1))), _gen_tt':mark':0':ok'3(_b13058), _gen_tt':mark':0':ok'3(_c13059)) →RΩ(1)
mark'(U21'(_gen_tt':mark':0':ok'3(+(1, _$n10264)), _gen_tt':mark':0':ok'3(_b13058), _gen_tt':mark':0':ok'3(_c13059))) →IH
mark'(_*4)

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

Rules:
active'(U11'(tt', N)) → mark'(N)
active'(U21'(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'(and'(isNat'(V1), isNat'(V2)))
active'(isNat'(s'(V1))) → mark'(isNat'(V1))
active'(plus'(N, 0')) → mark'(U11'(isNat'(N), N))
active'(plus'(N, s'(M))) → mark'(U21'(and'(isNat'(M), isNat'(N)), M, N))
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(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) → mark'(U11'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(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)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U21'(X1, X2, X3)) → U21'(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'(isNat'(X)) → isNat'(proper'(X))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(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))
isNat'(ok'(X)) → ok'(isNat'(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' :: tt':mark':0':ok'
mark' :: 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'
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'
isNat' :: 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'

Lemmas:
s'(_gen_tt':mark':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
plus'(_gen_tt':mark':0':ok'3(+(1, _n1399)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n1399)
and'(_gen_tt':mark':0':ok'3(+(1, _n4121)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n4121)
U11'(_gen_tt':mark':0':ok'3(+(1, _n7024)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n7024)
U21'(_gen_tt':mark':0':ok'3(+(1, _n10263)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(n10263)

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'.

Rules:
active'(U11'(tt', N)) → mark'(N)
active'(U21'(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'(and'(isNat'(V1), isNat'(V2)))
active'(isNat'(s'(V1))) → mark'(isNat'(V1))
active'(plus'(N, 0')) → mark'(U11'(isNat'(N), N))
active'(plus'(N, s'(M))) → mark'(U21'(and'(isNat'(M), isNat'(N)), M, N))
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(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) → mark'(U11'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(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)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U21'(X1, X2, X3)) → U21'(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'(isNat'(X)) → isNat'(proper'(X))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(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))
isNat'(ok'(X)) → ok'(isNat'(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' :: tt':mark':0':ok'
mark' :: 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'
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'
isNat' :: 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'

Lemmas:
s'(_gen_tt':mark':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
plus'(_gen_tt':mark':0':ok'3(+(1, _n1399)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n1399)
and'(_gen_tt':mark':0':ok'3(+(1, _n4121)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n4121)
U11'(_gen_tt':mark':0':ok'3(+(1, _n7024)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n7024)
U21'(_gen_tt':mark':0':ok'3(+(1, _n10263)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(n10263)

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'.

Rules:
active'(U11'(tt', N)) → mark'(N)
active'(U21'(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'(and'(isNat'(V1), isNat'(V2)))
active'(isNat'(s'(V1))) → mark'(isNat'(V1))
active'(plus'(N, 0')) → mark'(U11'(isNat'(N), N))
active'(plus'(N, s'(M))) → mark'(U21'(and'(isNat'(M), isNat'(N)), M, N))
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(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) → mark'(U11'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(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)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U21'(X1, X2, X3)) → U21'(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'(isNat'(X)) → isNat'(proper'(X))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(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))
isNat'(ok'(X)) → ok'(isNat'(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' :: tt':mark':0':ok'
mark' :: 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'
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'
isNat' :: 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'

Lemmas:
s'(_gen_tt':mark':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
plus'(_gen_tt':mark':0':ok'3(+(1, _n1399)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n1399)
and'(_gen_tt':mark':0':ok'3(+(1, _n4121)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n4121)
U11'(_gen_tt':mark':0':ok'3(+(1, _n7024)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n7024)
U21'(_gen_tt':mark':0':ok'3(+(1, _n10263)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(n10263)

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'.

Rules:
active'(U11'(tt', N)) → mark'(N)
active'(U21'(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'(and'(isNat'(V1), isNat'(V2)))
active'(isNat'(s'(V1))) → mark'(isNat'(V1))
active'(plus'(N, 0')) → mark'(U11'(isNat'(N), N))
active'(plus'(N, s'(M))) → mark'(U21'(and'(isNat'(M), isNat'(N)), M, N))
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U21'(X1, X2, X3)) → U21'(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) → mark'(U11'(X1, X2))
U21'(mark'(X1), X2, X3) → mark'(U21'(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)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U21'(X1, X2, X3)) → U21'(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'(isNat'(X)) → isNat'(proper'(X))
proper'(0') → ok'(0')
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U21'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U21'(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))
isNat'(ok'(X)) → ok'(isNat'(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' :: tt':mark':0':ok'
mark' :: 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'
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'
isNat' :: 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'

Lemmas:
s'(_gen_tt':mark':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
plus'(_gen_tt':mark':0':ok'3(+(1, _n1399)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n1399)
and'(_gen_tt':mark':0':ok'3(+(1, _n4121)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n4121)
U11'(_gen_tt':mark':0':ok'3(+(1, _n7024)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(n7024)
U21'(_gen_tt':mark':0':ok'3(+(1, _n10263)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(n10263)

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:
s'(_gen_tt':mark':0':ok'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)