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(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
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
cons(mark(X1), X2) →⁺ mark(cons(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(pairNs) → mark(cons(0', incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0', XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
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(pairNs) → mark(cons(0', incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0', XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok

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

Heuristically decided to analyse the following defined symbols:
active, cons, incr, s, take, pair, zip, repItems, tail, proper, top

They will be analysed ascendingly in the following order:
cons < active
incr < active
s < active
take < active
pair < active
zip < active
repItems < active
tail < active
active < top
cons < proper
incr < proper
s < proper
take < proper
pair < proper
zip < proper
repItems < proper
tail < proper
proper < top

(8) Obligation:

TRS:
Rules:
active(pairNs) → mark(cons(0', incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0', XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok

Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))

The following defined symbols remain to be analysed:
cons, active, incr, s, take, pair, zip, repItems, tail, proper, top

They will be analysed ascendingly in the following order:
cons < active
incr < active
s < active
take < active
pair < active
zip < active
repItems < active
tail < active
active < top
cons < proper
incr < proper
s < proper
take < proper
pair < proper
zip < proper
repItems < proper
tail < proper
proper < top

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(10) Obligation:

TRS:
Rules:
active(pairNs) → mark(cons(0', incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0', XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok

Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))

The following defined symbols remain to be analysed:
incr, active, s, take, pair, zip, repItems, tail, proper, top

They will be analysed ascendingly in the following order:
incr < active
s < active
take < active
pair < active
zip < active
repItems < active
tail < active
active < top
incr < proper
s < proper
take < proper
pair < proper
zip < proper
repItems < proper
tail < proper
proper < top

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(12) Obligation:

TRS:
Rules:
active(pairNs) → mark(cons(0', incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0', XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok

Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))

The following defined symbols remain to be analysed:
s, active, take, pair, zip, repItems, tail, proper, top

They will be analysed ascendingly in the following order:
s < active
take < active
pair < active
zip < active
repItems < active
tail < active
active < top
s < proper
take < proper
pair < proper
zip < proper
repItems < proper
tail < proper
proper < top

(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(14) Obligation:

TRS:
Rules:
active(pairNs) → mark(cons(0', incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0', XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok

Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))

The following defined symbols remain to be analysed:
take, active, pair, zip, repItems, tail, proper, top

They will be analysed ascendingly in the following order:
take < active
pair < active
zip < active
repItems < active
tail < active
active < top
take < proper
pair < proper
zip < proper
repItems < proper
tail < proper
proper < top

(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(16) Obligation:

TRS:
Rules:
active(pairNs) → mark(cons(0', incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0', XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok

Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))

The following defined symbols remain to be analysed:
pair, active, zip, repItems, tail, proper, top

They will be analysed ascendingly in the following order:
pair < active
zip < active
repItems < active
tail < active
active < top
pair < proper
zip < proper
repItems < proper
tail < proper
proper < top

(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(18) Obligation:

TRS:
Rules:
active(pairNs) → mark(cons(0', incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0', XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok

Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))

The following defined symbols remain to be analysed:
zip, active, repItems, tail, proper, top

They will be analysed ascendingly in the following order:
zip < active
repItems < active
tail < active
active < top
zip < proper
repItems < proper
tail < proper
proper < top

(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(20) Obligation:

TRS:
Rules:
active(pairNs) → mark(cons(0', incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0', XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok

Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))

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

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

(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(22) Obligation:

TRS:
Rules:
active(pairNs) → mark(cons(0', incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0', XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok

Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))

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

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

(23) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(24) Obligation:

TRS:
Rules:
active(pairNs) → mark(cons(0', incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0', XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok

Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil: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

(25) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(26) Obligation:

TRS:
Rules:
active(pairNs) → mark(cons(0', incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0', XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok

Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))

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

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

(27) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(28) Obligation:

TRS:
Rules:
active(pairNs) → mark(cons(0', incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0', XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok

Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))

The following defined symbols remain to be analysed:
top

(29) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(30) Obligation:

TRS:
Rules:
active(pairNs) → mark(cons(0', incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0', XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pairNs :: pairNs:0':oddNs:mark:nil:ok
mark :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
cons :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
0' :: pairNs:0':oddNs:mark:nil:ok
incr :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
oddNs :: pairNs:0':oddNs:mark:nil:ok
s :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
take :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
nil :: pairNs:0':oddNs:mark:nil:ok
zip :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
pair :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
tail :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
repItems :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
proper :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
ok :: pairNs:0':oddNs:mark:nil:ok → pairNs:0':oddNs:mark:nil:ok
top :: pairNs:0':oddNs:mark:nil:ok → top
hole_pairNs:0':oddNs:mark:nil:ok1_0 :: pairNs:0':oddNs:mark:nil:ok
hole_top2_0 :: top
gen_pairNs:0':oddNs:mark:nil:ok3_0 :: Nat → pairNs:0':oddNs:mark:nil:ok

Generator Equations:
gen_pairNs:0':oddNs:mark:nil:ok3_0(0) ⇔ pairNs
gen_pairNs:0':oddNs:mark:nil:ok3_0(+(x, 1)) ⇔ mark(gen_pairNs:0':oddNs:mark:nil:ok3_0(x))

No more defined symbols left to analyse.