The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
U11(mark(X1), X2) →⁺ mark(U11(X1, X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / mark(X1)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative 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, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
active, U12, isNat, U42, s, plus, U11, U21, U31, U41, proper, top

They will be analysed ascendingly in the following order:
U12 < active
isNat < active
U42 < active
s < active
plus < active
U11 < active
U21 < active
U31 < active
U41 < active
active < top
U12 < proper
isNat < proper
U42 < proper
s < proper
plus < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
proper < top

(8) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))

The following defined symbols remain to be analysed:
U12, active, isNat, U42, s, plus, U11, U21, U31, U41, proper, top

They will be analysed ascendingly in the following order:
U12 < active
isNat < active
U42 < active
s < active
plus < active
U11 < active
U21 < active
U31 < active
U41 < active
active < top
U12 < proper
isNat < proper
U42 < proper
s < proper
plus < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
proper < top

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Induction Base:
U12(gen_tt:mark:0':ok3_0(+(1, 0)))

Induction Step:
U12(gen_tt:mark:0':ok3_0(+(1, +(n5_0, 1)))) →_R^Ω(1)
mark(U12(gen_tt:mark:0':ok3_0(+(1, n5_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))

The following defined symbols remain to be analysed:
isNat, active, U42, s, plus, U11, U21, U31, U41, proper, top

They will be analysed ascendingly in the following order:
isNat < active
U42 < active
s < active
plus < active
U11 < active
U21 < active
U31 < active
U41 < active
active < top
isNat < proper
U42 < proper
s < proper
plus < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
proper < top

(12) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(13) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))

The following defined symbols remain to be analysed:
U42, active, s, plus, U11, U21, U31, U41, proper, top

They will be analysed ascendingly in the following order:
U42 < active
s < active
plus < active
U11 < active
U21 < active
U31 < active
U41 < active
active < top
U42 < proper
s < proper
plus < proper
U11 < proper
U21 < proper
U31 < proper
U41 < proper
proper < top

(14) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U42(gen_tt:mark:0':ok3_0(+(1, n432_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n432₀)

Induction Base:
U42(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))

Induction Step:
U42(gen_tt:mark:0':ok3_0(+(1, +(n432_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) →_R^Ω(1)
mark(U42(gen_tt:mark:0':ok3_0(+(1, n432_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(15) Complex Obligation (BEST)

(16) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
U42(gen_tt:mark:0':ok3_0(+(1, n432_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n432₀)

Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))

The following defined symbols remain to be analysed:
s, active, plus, U11, U21, U31, U41, proper, top

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

(17) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
s(gen_tt:mark:0':ok3_0(+(1, n2725_0))) → *4_0, rt ∈ Ω(n2725₀)

Induction Base:
s(gen_tt:mark:0':ok3_0(+(1, 0)))

Induction Step:
s(gen_tt:mark:0':ok3_0(+(1, +(n2725_0, 1)))) →_R^Ω(1)
mark(s(gen_tt:mark:0':ok3_0(+(1, n2725_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(18) Complex Obligation (BEST)

(19) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
U42(gen_tt:mark:0':ok3_0(+(1, n432_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n432₀)
s(gen_tt:mark:0':ok3_0(+(1, n2725_0))) → *4_0, rt ∈ Ω(n2725₀)

Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))

The following defined symbols remain to be analysed:
plus, active, U11, U21, U31, U41, proper, top

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

(20) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
plus(gen_tt:mark:0':ok3_0(+(1, n3445_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3445₀)

Induction Base:
plus(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))

Induction Step:
plus(gen_tt:mark:0':ok3_0(+(1, +(n3445_0, 1))), gen_tt:mark:0':ok3_0(b)) →_R^Ω(1)
mark(plus(gen_tt:mark:0':ok3_0(+(1, n3445_0)), gen_tt:mark:0':ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(21) Complex Obligation (BEST)

(22) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
U42(gen_tt:mark:0':ok3_0(+(1, n432_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n432₀)
s(gen_tt:mark:0':ok3_0(+(1, n2725_0))) → *4_0, rt ∈ Ω(n2725₀)
plus(gen_tt:mark:0':ok3_0(+(1, n3445_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3445₀)

Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))

The following defined symbols remain to be analysed:
U11, active, U21, U31, U41, proper, top

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

(23) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U11(gen_tt:mark:0':ok3_0(+(1, n5607_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5607₀)

Induction Base:
U11(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))

Induction Step:
U11(gen_tt:mark:0':ok3_0(+(1, +(n5607_0, 1))), gen_tt:mark:0':ok3_0(b)) →_R^Ω(1)
mark(U11(gen_tt:mark:0':ok3_0(+(1, n5607_0)), gen_tt:mark:0':ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(24) Complex Obligation (BEST)

(25) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
U42(gen_tt:mark:0':ok3_0(+(1, n432_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n432₀)
s(gen_tt:mark:0':ok3_0(+(1, n2725_0))) → *4_0, rt ∈ Ω(n2725₀)
plus(gen_tt:mark:0':ok3_0(+(1, n3445_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3445₀)
U11(gen_tt:mark:0':ok3_0(+(1, n5607_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5607₀)

Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(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

(26) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U21(gen_tt:mark:0':ok3_0(+(1, n7874_0))) → *4_0, rt ∈ Ω(n7874₀)

Induction Base:
U21(gen_tt:mark:0':ok3_0(+(1, 0)))

Induction Step:
U21(gen_tt:mark:0':ok3_0(+(1, +(n7874_0, 1)))) →_R^Ω(1)
mark(U21(gen_tt:mark:0':ok3_0(+(1, n7874_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(27) Complex Obligation (BEST)

(28) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
U42(gen_tt:mark:0':ok3_0(+(1, n432_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n432₀)
s(gen_tt:mark:0':ok3_0(+(1, n2725_0))) → *4_0, rt ∈ Ω(n2725₀)
plus(gen_tt:mark:0':ok3_0(+(1, n3445_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3445₀)
U11(gen_tt:mark:0':ok3_0(+(1, n5607_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5607₀)
U21(gen_tt:mark:0':ok3_0(+(1, n7874_0))) → *4_0, rt ∈ Ω(n7874₀)

Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(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

(29) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U31(gen_tt:mark:0':ok3_0(+(1, n8995_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n8995₀)

Induction Base:
U31(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b))

Induction Step:
U31(gen_tt:mark:0':ok3_0(+(1, +(n8995_0, 1))), gen_tt:mark:0':ok3_0(b)) →_R^Ω(1)
mark(U31(gen_tt:mark:0':ok3_0(+(1, n8995_0)), gen_tt:mark:0':ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(30) Complex Obligation (BEST)

(31) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
U42(gen_tt:mark:0':ok3_0(+(1, n432_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n432₀)
s(gen_tt:mark:0':ok3_0(+(1, n2725_0))) → *4_0, rt ∈ Ω(n2725₀)
plus(gen_tt:mark:0':ok3_0(+(1, n3445_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3445₀)
U11(gen_tt:mark:0':ok3_0(+(1, n5607_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5607₀)
U21(gen_tt:mark:0':ok3_0(+(1, n7874_0))) → *4_0, rt ∈ Ω(n7874₀)
U31(gen_tt:mark:0':ok3_0(+(1, n8995_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n8995₀)

Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(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

(32) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U41(gen_tt:mark:0':ok3_0(+(1, n11772_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11772₀)

Induction Base:
U41(gen_tt:mark:0':ok3_0(+(1, 0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))

Induction Step:
U41(gen_tt:mark:0':ok3_0(+(1, +(n11772_0, 1))), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) →_R^Ω(1)
mark(U41(gen_tt:mark:0':ok3_0(+(1, n11772_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(33) Complex Obligation (BEST)

(34) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
U42(gen_tt:mark:0':ok3_0(+(1, n432_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n432₀)
s(gen_tt:mark:0':ok3_0(+(1, n2725_0))) → *4_0, rt ∈ Ω(n2725₀)
plus(gen_tt:mark:0':ok3_0(+(1, n3445_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3445₀)
U11(gen_tt:mark:0':ok3_0(+(1, n5607_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5607₀)
U21(gen_tt:mark:0':ok3_0(+(1, n7874_0))) → *4_0, rt ∈ Ω(n7874₀)
U31(gen_tt:mark:0':ok3_0(+(1, n8995_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n8995₀)
U41(gen_tt:mark:0':ok3_0(+(1, n11772_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11772₀)

Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(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

(35) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(36) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
U42(gen_tt:mark:0':ok3_0(+(1, n432_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n432₀)
s(gen_tt:mark:0':ok3_0(+(1, n2725_0))) → *4_0, rt ∈ Ω(n2725₀)
plus(gen_tt:mark:0':ok3_0(+(1, n3445_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3445₀)
U11(gen_tt:mark:0':ok3_0(+(1, n5607_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5607₀)
U21(gen_tt:mark:0':ok3_0(+(1, n7874_0))) → *4_0, rt ∈ Ω(n7874₀)
U31(gen_tt:mark:0':ok3_0(+(1, n8995_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n8995₀)
U41(gen_tt:mark:0':ok3_0(+(1, n11772_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11772₀)

Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))

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

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

(37) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(38) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
U42(gen_tt:mark:0':ok3_0(+(1, n432_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n432₀)
s(gen_tt:mark:0':ok3_0(+(1, n2725_0))) → *4_0, rt ∈ Ω(n2725₀)
plus(gen_tt:mark:0':ok3_0(+(1, n3445_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3445₀)
U11(gen_tt:mark:0':ok3_0(+(1, n5607_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5607₀)
U21(gen_tt:mark:0':ok3_0(+(1, n7874_0))) → *4_0, rt ∈ Ω(n7874₀)
U31(gen_tt:mark:0':ok3_0(+(1, n8995_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n8995₀)
U41(gen_tt:mark:0':ok3_0(+(1, n11772_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11772₀)

Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))

The following defined symbols remain to be analysed:
top

(39) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(40) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
U42(gen_tt:mark:0':ok3_0(+(1, n432_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n432₀)
s(gen_tt:mark:0':ok3_0(+(1, n2725_0))) → *4_0, rt ∈ Ω(n2725₀)
plus(gen_tt:mark:0':ok3_0(+(1, n3445_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3445₀)
U11(gen_tt:mark:0':ok3_0(+(1, n5607_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5607₀)
U21(gen_tt:mark:0':ok3_0(+(1, n7874_0))) → *4_0, rt ∈ Ω(n7874₀)
U31(gen_tt:mark:0':ok3_0(+(1, n8995_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n8995₀)
U41(gen_tt:mark:0':ok3_0(+(1, n11772_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11772₀)

Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))

No more defined symbols left to analyse.

(41) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(42) BOUNDS(n^1, INF)

(43) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
U42(gen_tt:mark:0':ok3_0(+(1, n432_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n432₀)
s(gen_tt:mark:0':ok3_0(+(1, n2725_0))) → *4_0, rt ∈ Ω(n2725₀)
plus(gen_tt:mark:0':ok3_0(+(1, n3445_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3445₀)
U11(gen_tt:mark:0':ok3_0(+(1, n5607_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5607₀)
U21(gen_tt:mark:0':ok3_0(+(1, n7874_0))) → *4_0, rt ∈ Ω(n7874₀)
U31(gen_tt:mark:0':ok3_0(+(1, n8995_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n8995₀)
U41(gen_tt:mark:0':ok3_0(+(1, n11772_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n11772₀)

Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))

No more defined symbols left to analyse.

(44) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(45) BOUNDS(n^1, INF)

(46) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
U42(gen_tt:mark:0':ok3_0(+(1, n432_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n432₀)
s(gen_tt:mark:0':ok3_0(+(1, n2725_0))) → *4_0, rt ∈ Ω(n2725₀)
plus(gen_tt:mark:0':ok3_0(+(1, n3445_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3445₀)
U11(gen_tt:mark:0':ok3_0(+(1, n5607_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5607₀)
U21(gen_tt:mark:0':ok3_0(+(1, n7874_0))) → *4_0, rt ∈ Ω(n7874₀)
U31(gen_tt:mark:0':ok3_0(+(1, n8995_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n8995₀)

Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))

No more defined symbols left to analyse.

(47) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(48) BOUNDS(n^1, INF)

(49) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
U42(gen_tt:mark:0':ok3_0(+(1, n432_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n432₀)
s(gen_tt:mark:0':ok3_0(+(1, n2725_0))) → *4_0, rt ∈ Ω(n2725₀)
plus(gen_tt:mark:0':ok3_0(+(1, n3445_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3445₀)
U11(gen_tt:mark:0':ok3_0(+(1, n5607_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5607₀)
U21(gen_tt:mark:0':ok3_0(+(1, n7874_0))) → *4_0, rt ∈ Ω(n7874₀)

Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))

No more defined symbols left to analyse.

(50) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(51) BOUNDS(n^1, INF)

(52) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
U42(gen_tt:mark:0':ok3_0(+(1, n432_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n432₀)
s(gen_tt:mark:0':ok3_0(+(1, n2725_0))) → *4_0, rt ∈ Ω(n2725₀)
plus(gen_tt:mark:0':ok3_0(+(1, n3445_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3445₀)
U11(gen_tt:mark:0':ok3_0(+(1, n5607_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n5607₀)

Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))

No more defined symbols left to analyse.

(53) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(54) BOUNDS(n^1, INF)

(55) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
U42(gen_tt:mark:0':ok3_0(+(1, n432_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n432₀)
s(gen_tt:mark:0':ok3_0(+(1, n2725_0))) → *4_0, rt ∈ Ω(n2725₀)
plus(gen_tt:mark:0':ok3_0(+(1, n3445_0)), gen_tt:mark:0':ok3_0(b)) → *4_0, rt ∈ Ω(n3445₀)

Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))

No more defined symbols left to analyse.

(56) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(57) BOUNDS(n^1, INF)

(58) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
U42(gen_tt:mark:0':ok3_0(+(1, n432_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n432₀)
s(gen_tt:mark:0':ok3_0(+(1, n2725_0))) → *4_0, rt ∈ Ω(n2725₀)

Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))

No more defined symbols left to analyse.

(59) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(60) BOUNDS(n^1, INF)

(61) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)
U42(gen_tt:mark:0':ok3_0(+(1, n432_0)), gen_tt:mark:0':ok3_0(b), gen_tt:mark:0':ok3_0(c)) → *4_0, rt ∈ Ω(n432₀)

Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))

No more defined symbols left to analyse.

(62) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(63) BOUNDS(n^1, INF)

(64) Obligation:

TRS:
Rules:
active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0')) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0')) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(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(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
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))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
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(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
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))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(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
U41 :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
U42 :: 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
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':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok

Lemmas:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))

No more defined symbols left to analyse.

(65) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
U12(gen_tt:mark:0':ok3_0(+(1, n5_0))) → *4_0, rt ∈ Ω(n5₀)

(66) BOUNDS(n^1, INF)