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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNat(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt, N)) → mark(N)
active(U51(tt, M, N)) → mark(U52(isNat(N), M, N))
active(U52(tt, M, N)) → mark(s(plus(N, M)))
active(U61(tt)) → mark(0)
active(U71(tt, M, N)) → mark(U72(isNat(N), M, N))
active(U72(tt, M, N)) → mark(plus(x(N, M), N))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(isNat(x(V1, V2))) → mark(U31(isNat(V1), V2))
active(plus(N, 0)) → mark(U41(isNat(N), N))
active(plus(N, s(M))) → mark(U51(isNat(M), M, N))
active(x(N, 0)) → mark(U61(isNat(N)))
active(x(N, s(M))) → mark(U71(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X1, X2)) → U41(active(X1), X2)
active(U51(X1, X2, X3)) → U51(active(X1), X2, X3)
active(U52(X1, X2, X3)) → U52(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(U61(X)) → U61(active(X))
active(U71(X1, X2, X3)) → U71(active(X1), X2, X3)
active(U72(X1, X2, X3)) → U72(active(X1), X2, X3)
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X1), X2) → mark(U41(X1, X2))
U51(mark(X1), X2, X3) → mark(U51(X1, X2, X3))
U52(mark(X1), X2, X3) → mark(U52(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))
U61(mark(X)) → mark(U61(X))
U71(mark(X1), X2, X3) → mark(U71(X1, X2, X3))
U72(mark(X1), X2, X3) → mark(U72(X1, X2, X3))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X1, X2)) → U41(proper(X1), proper(X2))
proper(U51(X1, X2, X3)) → U51(proper(X1), proper(X2), proper(X3))
proper(U52(X1, X2, X3)) → U52(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(U61(X)) → U61(proper(X))
proper(0) → ok(0)
proper(U71(X1, X2, X3)) → U71(proper(X1), proper(X2), proper(X3))
proper(U72(X1, X2, X3)) → U72(proper(X1), proper(X2), proper(X3))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X1), ok(X2)) → ok(U41(X1, X2))
U51(ok(X1), ok(X2), ok(X3)) → ok(U51(X1, X2, X3))
U52(ok(X1), ok(X2), ok(X3)) → ok(U52(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
U61(ok(X)) → ok(U61(X))
U71(ok(X1), ok(X2), ok(X3)) → ok(U71(X1, X2, X3))
U72(ok(X1), ok(X2), ok(X3)) → ok(U72(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', V2)) → mark'(U12'(isNat'(V2)))
active'(U12'(tt')) → mark'(tt')
active'(U21'(tt')) → mark'(tt')
active'(U31'(tt', V2)) → mark'(U32'(isNat'(V2)))
active'(U32'(tt')) → mark'(tt')
active'(U41'(tt', N)) → mark'(N)
active'(U51'(tt', M, N)) → mark'(U52'(isNat'(N), M, N))
active'(U52'(tt', M, N)) → mark'(s'(plus'(N, M)))
active'(U61'(tt')) → mark'(0')
active'(U71'(tt', M, N)) → mark'(U72'(isNat'(N), M, N))
active'(U72'(tt', M, N)) → mark'(plus'(x'(N, M), N))
active'(isNat'(0')) → mark'(tt')
active'(isNat'(plus'(V1, V2))) → mark'(U11'(isNat'(V1), V2))
active'(isNat'(s'(V1))) → mark'(U21'(isNat'(V1)))
active'(isNat'(x'(V1, V2))) → mark'(U31'(isNat'(V1), V2))
active'(plus'(N, 0')) → mark'(U41'(isNat'(N), N))
active'(plus'(N, s'(M))) → mark'(U51'(isNat'(M), M, N))
active'(x'(N, 0')) → mark'(U61'(isNat'(N)))
active'(x'(N, s'(M))) → mark'(U71'(isNat'(M), M, N))
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U12'(X)) → U12'(active'(X))
active'(U21'(X)) → U21'(active'(X))
active'(U31'(X1, X2)) → U31'(active'(X1), X2)
active'(U32'(X)) → U32'(active'(X))
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U51'(X1, X2, X3)) → U51'(active'(X1), X2, X3)
active'(U52'(X1, X2, X3)) → U52'(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'(U61'(X)) → U61'(active'(X))
active'(U71'(X1, X2, X3)) → U71'(active'(X1), X2, X3)
active'(U72'(X1, X2, X3)) → U72'(active'(X1), X2, X3)
active'(x'(X1, X2)) → x'(active'(X1), X2)
active'(x'(X1, X2)) → x'(X1, active'(X2))
U11'(mark'(X1), X2) → mark'(U11'(X1, X2))
U12'(mark'(X)) → mark'(U12'(X))
U21'(mark'(X)) → mark'(U21'(X))
U31'(mark'(X1), X2) → mark'(U31'(X1, X2))
U32'(mark'(X)) → mark'(U32'(X))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U51'(mark'(X1), X2, X3) → mark'(U51'(X1, X2, X3))
U52'(mark'(X1), X2, X3) → mark'(U52'(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))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X1), X2, X3) → mark'(U71'(X1, X2, X3))
U72'(mark'(X1), X2, X3) → mark'(U72'(X1, X2, X3))
x'(mark'(X1), X2) → mark'(x'(X1, X2))
x'(X1, mark'(X2)) → mark'(x'(X1, X2))
proper'(U11'(X1, X2)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U12'(X)) → U12'(proper'(X))
proper'(isNat'(X)) → isNat'(proper'(X))
proper'(U21'(X)) → U21'(proper'(X))
proper'(U31'(X1, X2)) → U31'(proper'(X1), proper'(X2))
proper'(U32'(X)) → U32'(proper'(X))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U51'(X1, X2, X3)) → U51'(proper'(X1), proper'(X2), proper'(X3))
proper'(U52'(X1, X2, X3)) → U52'(proper'(X1), proper'(X2), proper'(X3))
proper'(s'(X)) → s'(proper'(X))
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(U61'(X)) → U61'(proper'(X))
proper'(0') → ok'(0')
proper'(U71'(X1, X2, X3)) → U71'(proper'(X1), proper'(X2), proper'(X3))
proper'(U72'(X1, X2, X3)) → U72'(proper'(X1), proper'(X2), proper'(X3))
proper'(x'(X1, X2)) → x'(proper'(X1), proper'(X2))
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U12'(ok'(X)) → ok'(U12'(X))
isNat'(ok'(X)) → ok'(isNat'(X))
U21'(ok'(X)) → ok'(U21'(X))
U31'(ok'(X1), ok'(X2)) → ok'(U31'(X1, X2))
U32'(ok'(X)) → ok'(U32'(X))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U51'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U51'(X1, X2, X3))
U52'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U52'(X1, X2, X3))
s'(ok'(X)) → ok'(s'(X))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U71'(X1, X2, X3))
U72'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U72'(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', V2)) → mark'(U12'(isNat'(V2)))
active'(U12'(tt')) → mark'(tt')
active'(U21'(tt')) → mark'(tt')
active'(U31'(tt', V2)) → mark'(U32'(isNat'(V2)))
active'(U32'(tt')) → mark'(tt')
active'(U41'(tt', N)) → mark'(N)
active'(U51'(tt', M, N)) → mark'(U52'(isNat'(N), M, N))
active'(U52'(tt', M, N)) → mark'(s'(plus'(N, M)))
active'(U61'(tt')) → mark'(0')
active'(U71'(tt', M, N)) → mark'(U72'(isNat'(N), M, N))
active'(U72'(tt', M, N)) → mark'(plus'(x'(N, M), N))
active'(isNat'(0')) → mark'(tt')
active'(isNat'(plus'(V1, V2))) → mark'(U11'(isNat'(V1), V2))
active'(isNat'(s'(V1))) → mark'(U21'(isNat'(V1)))
active'(isNat'(x'(V1, V2))) → mark'(U31'(isNat'(V1), V2))
active'(plus'(N, 0')) → mark'(U41'(isNat'(N), N))
active'(plus'(N, s'(M))) → mark'(U51'(isNat'(M), M, N))
active'(x'(N, 0')) → mark'(U61'(isNat'(N)))
active'(x'(N, s'(M))) → mark'(U71'(isNat'(M), M, N))
active'(U11'(X1, X2)) → U11'(active'(X1), X2)
active'(U12'(X)) → U12'(active'(X))
active'(U21'(X)) → U21'(active'(X))
active'(U31'(X1, X2)) → U31'(active'(X1), X2)
active'(U32'(X)) → U32'(active'(X))
active'(U41'(X1, X2)) → U41'(active'(X1), X2)
active'(U51'(X1, X2, X3)) → U51'(active'(X1), X2, X3)
active'(U52'(X1, X2, X3)) → U52'(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'(U61'(X)) → U61'(active'(X))
active'(U71'(X1, X2, X3)) → U71'(active'(X1), X2, X3)
active'(U72'(X1, X2, X3)) → U72'(active'(X1), X2, X3)
active'(x'(X1, X2)) → x'(active'(X1), X2)
active'(x'(X1, X2)) → x'(X1, active'(X2))
U11'(mark'(X1), X2) → mark'(U11'(X1, X2))
U12'(mark'(X)) → mark'(U12'(X))
U21'(mark'(X)) → mark'(U21'(X))
U31'(mark'(X1), X2) → mark'(U31'(X1, X2))
U32'(mark'(X)) → mark'(U32'(X))
U41'(mark'(X1), X2) → mark'(U41'(X1, X2))
U51'(mark'(X1), X2, X3) → mark'(U51'(X1, X2, X3))
U52'(mark'(X1), X2, X3) → mark'(U52'(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))
U61'(mark'(X)) → mark'(U61'(X))
U71'(mark'(X1), X2, X3) → mark'(U71'(X1, X2, X3))
U72'(mark'(X1), X2, X3) → mark'(U72'(X1, X2, X3))
x'(mark'(X1), X2) → mark'(x'(X1, X2))
x'(X1, mark'(X2)) → mark'(x'(X1, X2))
proper'(U11'(X1, X2)) → U11'(proper'(X1), proper'(X2))
proper'(tt') → ok'(tt')
proper'(U12'(X)) → U12'(proper'(X))
proper'(isNat'(X)) → isNat'(proper'(X))
proper'(U21'(X)) → U21'(proper'(X))
proper'(U31'(X1, X2)) → U31'(proper'(X1), proper'(X2))
proper'(U32'(X)) → U32'(proper'(X))
proper'(U41'(X1, X2)) → U41'(proper'(X1), proper'(X2))
proper'(U51'(X1, X2, X3)) → U51'(proper'(X1), proper'(X2), proper'(X3))
proper'(U52'(X1, X2, X3)) → U52'(proper'(X1), proper'(X2), proper'(X3))
proper'(s'(X)) → s'(proper'(X))
proper'(plus'(X1, X2)) → plus'(proper'(X1), proper'(X2))
proper'(U61'(X)) → U61'(proper'(X))
proper'(0') → ok'(0')
proper'(U71'(X1, X2, X3)) → U71'(proper'(X1), proper'(X2), proper'(X3))
proper'(U72'(X1, X2, X3)) → U72'(proper'(X1), proper'(X2), proper'(X3))
proper'(x'(X1, X2)) → x'(proper'(X1), proper'(X2))
U11'(ok'(X1), ok'(X2)) → ok'(U11'(X1, X2))
U12'(ok'(X)) → ok'(U12'(X))
isNat'(ok'(X)) → ok'(isNat'(X))
U21'(ok'(X)) → ok'(U21'(X))
U31'(ok'(X1), ok'(X2)) → ok'(U31'(X1, X2))
U32'(ok'(X)) → ok'(U32'(X))
U41'(ok'(X1), ok'(X2)) → ok'(U41'(X1, X2))
U51'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U51'(X1, X2, X3))
U52'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U52'(X1, X2, X3))
s'(ok'(X)) → ok'(s'(X))
plus'(ok'(X1), ok'(X2)) → ok'(plus'(X1, X2))
U61'(ok'(X)) → ok'(U61'(X))
U71'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U71'(X1, X2, X3))
U72'(ok'(X1), ok'(X2), ok'(X3)) → ok'(U72'(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' :: tt':mark':0':ok'
mark' :: tt':mark':0':ok' → tt':mark':0':ok'
U12' :: tt':mark':0':ok' → tt':mark':0':ok'
isNat' :: tt':mark':0':ok' → tt':mark':0':ok'
U21' :: tt':mark':0':ok' → tt':mark':0':ok'
U31' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U32' :: tt':mark':0':ok' → tt':mark':0':ok'
U41' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U51' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U52' :: 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'
U61' :: tt':mark':0':ok' → tt':mark':0':ok'
0' :: tt':mark':0':ok'
U71' :: tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok' → tt':mark':0':ok'
U72' :: 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'
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', U32', U52', s', plus', U72', x', U11', U21', U31', U41', U51', U61', U71', proper', top'

They will be analysed ascendingly in the following order:
U12' < active'
isNat' < active'
U32' < active'
U52' < active'
s' < active'
plus' < active'
U72' < active'
x' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
active' < top'
U12' < proper'
isNat' < proper'
U32' < proper'
U52' < proper'
s' < proper'
plus' < proper'
U72' < proper'
x' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < 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', U32', U52', s', plus', U72', x', U11', U21', U31', U41', U51', U61', U71', proper', top'

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

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

Induction Step:
U12'(_gen_tt':mark':0':ok'3(+(1, +(_$n6, 1)))) →_R^Ω(1)
mark'(U12'(_gen_tt':mark':0':ok'3(+(1, _$n6)))) →_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))) → _*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', U32', U52', s', plus', U72', x', U11', U21', U31', U41', U51', U61', U71', proper', top'

They will be analysed ascendingly in the following order:
isNat' < active'
U32' < active'
U52' < active'
s' < active'
plus' < active'
U72' < active'
x' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
active' < top'
isNat' < proper'
U32' < proper'
U52' < proper'
s' < proper'
plus' < proper'
U72' < proper'
x' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
proper' < top'

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

Lemmas:
U12'(_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:
U32', active', U52', s', plus', U72', x', U11', U21', U31', U41', U51', U61', U71', proper', top'

They will be analysed ascendingly in the following order:
U32' < active'
U52' < active'
s' < active'
plus' < active'
U72' < active'
x' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
active' < top'
U32' < proper'
U52' < proper'
s' < proper'
plus' < proper'
U72' < proper'
x' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
proper' < top'

Proved the following rewrite lemma:
U32'(_gen_tt':mark':0':ok'3(+(1, _n2965))) → _*4, rt ∈ Ω(_n2965)

Induction Base:
U32'(_gen_tt':mark':0':ok'3(+(1, 0)))

Induction Step:
U32'(_gen_tt':mark':0':ok'3(+(1, +(_$n2966, 1)))) →_R^Ω(1)
mark'(U32'(_gen_tt':mark':0':ok'3(+(1, _$n2966)))) →_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))) → _*4, rt ∈ Ω(_n5)
U32'(_gen_tt':mark':0':ok'3(+(1, _n2965))) → _*4, rt ∈ Ω(_n2965)

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:
U52', active', s', plus', U72', x', U11', U21', U31', U41', U51', U61', U71', proper', top'

They will be analysed ascendingly in the following order:
U52' < active'
s' < active'
plus' < active'
U72' < active'
x' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
active' < top'
U52' < proper'
s' < proper'
plus' < proper'
U72' < proper'
x' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
proper' < top'

Proved the following rewrite lemma:
U52'(_gen_tt':mark':0':ok'3(+(1, _n6037)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n6037)

Induction Base:
U52'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))

Induction Step:
U52'(_gen_tt':mark':0':ok'3(+(1, +(_$n6038, 1))), _gen_tt':mark':0':ok'3(_b7698), _gen_tt':mark':0':ok'3(_c7699)) →_R^Ω(1)
mark'(U52'(_gen_tt':mark':0':ok'3(+(1, _$n6038)), _gen_tt':mark':0':ok'3(_b7698), _gen_tt':mark':0':ok'3(_c7699))) →_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', U72', x', U11', U21', U31', U41', U51', U61', U71', proper', top'

They will be analysed ascendingly in the following order:
s' < active'
plus' < active'
U72' < active'
x' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
active' < top'
s' < proper'
plus' < proper'
U72' < proper'
x' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
proper' < top'

Proved the following rewrite lemma:
s'(_gen_tt':mark':0':ok'3(+(1, _n13147))) → _*4, rt ∈ Ω(_n13147)

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

Induction Step:
s'(_gen_tt':mark':0':ok'3(+(1, +(_$n13148, 1)))) →_R^Ω(1)
mark'(s'(_gen_tt':mark':0':ok'3(+(1, _$n13148)))) →_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', U72', x', U11', U21', U31', U41', U51', U61', U71', proper', top'

They will be analysed ascendingly in the following order:
plus' < active'
U72' < active'
x' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
active' < top'
plus' < proper'
U72' < proper'
x' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
proper' < top'

Proved the following rewrite lemma:
plus'(_gen_tt':mark':0':ok'3(+(1, _n16601)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n16601)

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, +(_$n16602, 1))), _gen_tt':mark':0':ok'3(_b18502)) →_R^Ω(1)
mark'(plus'(_gen_tt':mark':0':ok'3(+(1, _$n16602)), _gen_tt':mark':0':ok'3(_b18502))) →_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:
U72', active', x', U11', U21', U31', U41', U51', U61', U71', proper', top'

They will be analysed ascendingly in the following order:
U72' < active'
x' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
active' < top'
U72' < proper'
x' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
proper' < top'

Proved the following rewrite lemma:
U72'(_gen_tt':mark':0':ok'3(+(1, _n22652)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n22652)

Induction Base:
U72'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))

Induction Step:
U72'(_gen_tt':mark':0':ok'3(+(1, +(_$n22653, 1))), _gen_tt':mark':0':ok'3(_b25771), _gen_tt':mark':0':ok'3(_c25772)) →_R^Ω(1)
mark'(U72'(_gen_tt':mark':0':ok'3(+(1, _$n22653)), _gen_tt':mark':0':ok'3(_b25771), _gen_tt':mark':0':ok'3(_c25772))) →_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', U31', U41', U51', U61', U71', proper', top'

They will be analysed ascendingly in the following order:
x' < active'
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
active' < top'
x' < proper'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
proper' < top'

Proved the following rewrite lemma:
x'(_gen_tt':mark':0':ok'3(+(1, _n31397)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n31397)

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, +(_$n31398, 1))), _gen_tt':mark':0':ok'3(_b34054)) →_R^Ω(1)
mark'(x'(_gen_tt':mark':0':ok'3(+(1, _$n31398)), _gen_tt':mark':0':ok'3(_b34054))) →_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', U31', U41', U51', U61', U71', proper', top'

They will be analysed ascendingly in the following order:
U11' < active'
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
active' < top'
U11' < proper'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
proper' < top'

Proved the following rewrite lemma:
U11'(_gen_tt':mark':0':ok'3(+(1, _n38311)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n38311)

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, +(_$n38312, 1))), _gen_tt':mark':0':ok'3(_b41076)) →_R^Ω(1)
mark'(U11'(_gen_tt':mark':0':ok'3(+(1, _$n38312)), _gen_tt':mark':0':ok'3(_b41076))) →_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:
U21', active', U31', U41', U51', U61', U71', proper', top'

They will be analysed ascendingly in the following order:
U21' < active'
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
active' < top'
U21' < proper'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
proper' < top'

Proved the following rewrite lemma:
U21'(_gen_tt':mark':0':ok'3(+(1, _n45374))) → _*4, rt ∈ Ω(_n45374)

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

Induction Step:
U21'(_gen_tt':mark':0':ok'3(+(1, +(_$n45375, 1)))) →_R^Ω(1)
mark'(U21'(_gen_tt':mark':0':ok'3(+(1, _$n45375)))) →_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:
U31', active', U41', U51', U61', U71', proper', top'

They will be analysed ascendingly in the following order:
U31' < active'
U41' < active'
U51' < active'
U61' < active'
U71' < active'
active' < top'
U31' < proper'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
proper' < top'

Proved the following rewrite lemma:
U31'(_gen_tt':mark':0':ok'3(+(1, _n49783)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n49783)

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, +(_$n49784, 1))), _gen_tt':mark':0':ok'3(_b53088)) →_R^Ω(1)
mark'(U31'(_gen_tt':mark':0':ok'3(+(1, _$n49784)), _gen_tt':mark':0':ok'3(_b53088))) →_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', U51', U61', U71', proper', top'

They will be analysed ascendingly in the following order:
U41' < active'
U51' < active'
U61' < active'
U71' < active'
active' < top'
U41' < proper'
U51' < proper'
U61' < proper'
U71' < proper'
proper' < top'

Proved the following rewrite lemma:
U41'(_gen_tt':mark':0':ok'3(+(1, _n57455)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n57455)

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

Induction Step:
U41'(_gen_tt':mark':0':ok'3(+(1, +(_$n57456, 1))), _gen_tt':mark':0':ok'3(_b61084)) →_R^Ω(1)
mark'(U41'(_gen_tt':mark':0':ok'3(+(1, _$n57456)), _gen_tt':mark':0':ok'3(_b61084))) →_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))) → _*4, rt ∈ Ω(_n5)
U32'(_gen_tt':mark':0':ok'3(+(1, _n2965))) → _*4, rt ∈ Ω(_n2965)
U52'(_gen_tt':mark':0':ok'3(+(1, _n6037)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n6037)
s'(_gen_tt':mark':0':ok'3(+(1, _n13147))) → _*4, rt ∈ Ω(_n13147)
plus'(_gen_tt':mark':0':ok'3(+(1, _n16601)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n16601)
U72'(_gen_tt':mark':0':ok'3(+(1, _n22652)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n22652)
x'(_gen_tt':mark':0':ok'3(+(1, _n31397)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n31397)
U11'(_gen_tt':mark':0':ok'3(+(1, _n38311)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n38311)
U21'(_gen_tt':mark':0':ok'3(+(1, _n45374))) → _*4, rt ∈ Ω(_n45374)
U31'(_gen_tt':mark':0':ok'3(+(1, _n49783)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n49783)
U41'(_gen_tt':mark':0':ok'3(+(1, _n57455)), _gen_tt':mark':0':ok'3(b)) → _*4, rt ∈ Ω(_n57455)

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:
U51', active', U61', U71', proper', top'

They will be analysed ascendingly in the following order:
U51' < active'
U61' < active'
U71' < active'
active' < top'
U51' < proper'
U61' < proper'
U71' < proper'
proper' < top'

Proved the following rewrite lemma:
U51'(_gen_tt':mark':0':ok'3(+(1, _n65495)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n65495)

Induction Base:
U51'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))

Induction Step:
U51'(_gen_tt':mark':0':ok'3(+(1, +(_$n65496, 1))), _gen_tt':mark':0':ok'3(_b71530), _gen_tt':mark':0':ok'3(_c71531)) →_R^Ω(1)
mark'(U51'(_gen_tt':mark':0':ok'3(+(1, _$n65496)), _gen_tt':mark':0':ok'3(_b71530), _gen_tt':mark':0':ok'3(_c71531))) →_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:
U61', active', U71', proper', top'

They will be analysed ascendingly in the following order:
U61' < active'
U71' < active'
active' < top'
U61' < proper'
U71' < proper'
proper' < top'

Proved the following rewrite lemma:
U61'(_gen_tt':mark':0':ok'3(+(1, _n77510))) → _*4, rt ∈ Ω(_n77510)

Induction Base:
U61'(_gen_tt':mark':0':ok'3(+(1, 0)))

Induction Step:
U61'(_gen_tt':mark':0':ok'3(+(1, +(_$n77511, 1)))) →_R^Ω(1)
mark'(U61'(_gen_tt':mark':0':ok'3(+(1, _$n77511)))) →_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:
U71', active', proper', top'

They will be analysed ascendingly in the following order:
U71' < active'
active' < top'
U71' < proper'
proper' < top'

Proved the following rewrite lemma:
U71'(_gen_tt':mark':0':ok'3(+(1, _n82683)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c)) → _*4, rt ∈ Ω(_n82683)

Induction Base:
U71'(_gen_tt':mark':0':ok'3(+(1, 0)), _gen_tt':mark':0':ok'3(b), _gen_tt':mark':0':ok'3(c))

Induction Step:
U71'(_gen_tt':mark':0':ok'3(+(1, +(_$n82684, 1))), _gen_tt':mark':0':ok'3(_b89690), _gen_tt':mark':0':ok'3(_c89691)) →_R^Ω(1)
mark'(U71'(_gen_tt':mark':0':ok'3(+(1, _$n82684)), _gen_tt':mark':0':ok'3(_b89690), _gen_tt':mark':0':ok'3(_c89691))) →_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'