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(eq(0, 0)) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0, X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0)
active(length(cons(X, L))) → mark(s(length(L)))
active(inf(X)) → inf(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(length(X)) → length(active(X))
inf(mark(X)) → mark(inf(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
length(mark(X)) → mark(length(X))
proper(eq(X1, X2)) → eq(proper(X1), proper(X2))
proper(0) → ok(0)
proper(true) → ok(true)
proper(s(X)) → s(proper(X))
proper(false) → ok(false)
proper(inf(X)) → inf(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(length(X)) → length(proper(X))
eq(ok(X1), ok(X2)) → ok(eq(X1, X2))
s(ok(X)) → ok(s(X))
inf(ok(X)) → ok(inf(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
length(ok(X)) → ok(length(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
inf(mark(X)) →⁺ mark(inf(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / mark(X)].
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(eq(0', 0')) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0', X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0')
active(length(cons(X, L))) → mark(s(length(L)))
active(inf(X)) → inf(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(length(X)) → length(active(X))
inf(mark(X)) → mark(inf(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
length(mark(X)) → mark(length(X))
proper(eq(X1, X2)) → eq(proper(X1), proper(X2))
proper(0') → ok(0')
proper(true) → ok(true)
proper(s(X)) → s(proper(X))
proper(false) → ok(false)
proper(inf(X)) → inf(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(length(X)) → length(proper(X))
eq(ok(X1), ok(X2)) → ok(eq(X1, X2))
s(ok(X)) → ok(s(X))
inf(ok(X)) → ok(inf(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
length(ok(X)) → ok(length(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(eq(0', 0')) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0', X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0')
active(length(cons(X, L))) → mark(s(length(L)))
active(inf(X)) → inf(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(length(X)) → length(active(X))
inf(mark(X)) → mark(inf(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
length(mark(X)) → mark(length(X))
proper(eq(X1, X2)) → eq(proper(X1), proper(X2))
proper(0') → ok(0')
proper(true) → ok(true)
proper(s(X)) → s(proper(X))
proper(false) → ok(false)
proper(inf(X)) → inf(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(length(X)) → length(proper(X))
eq(ok(X1), ok(X2)) → ok(eq(X1, X2))
s(ok(X)) → ok(s(X))
inf(ok(X)) → ok(inf(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
eq :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
0' :: 0':true:mark:false:nil:ok
mark :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
true :: 0':true:mark:false:nil:ok
s :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
false :: 0':true:mark:false:nil:ok
inf :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
cons :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
take :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
nil :: 0':true:mark:false:nil:ok
length :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
proper :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
ok :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
top :: 0':true:mark:false:nil:ok → top
hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok
hole_top2_0 :: top
gen_0':true:mark:false:nil:ok3_0 :: Nat → 0':true:mark:false:nil:ok

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

Heuristically decided to analyse the following defined symbols:
active, eq, cons, inf, s, take, length, proper, top

They will be analysed ascendingly in the following order:
eq < active
cons < active
inf < active
s < active
take < active
length < active
active < top
eq < proper
cons < proper
inf < proper
s < proper
take < proper
length < proper
proper < top

(8) Obligation:

TRS:
Rules:
active(eq(0', 0')) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0', X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0')
active(length(cons(X, L))) → mark(s(length(L)))
active(inf(X)) → inf(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(length(X)) → length(active(X))
inf(mark(X)) → mark(inf(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
length(mark(X)) → mark(length(X))
proper(eq(X1, X2)) → eq(proper(X1), proper(X2))
proper(0') → ok(0')
proper(true) → ok(true)
proper(s(X)) → s(proper(X))
proper(false) → ok(false)
proper(inf(X)) → inf(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(length(X)) → length(proper(X))
eq(ok(X1), ok(X2)) → ok(eq(X1, X2))
s(ok(X)) → ok(s(X))
inf(ok(X)) → ok(inf(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
eq :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
0' :: 0':true:mark:false:nil:ok
mark :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
true :: 0':true:mark:false:nil:ok
s :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
false :: 0':true:mark:false:nil:ok
inf :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
cons :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
take :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
nil :: 0':true:mark:false:nil:ok
length :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
proper :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
ok :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
top :: 0':true:mark:false:nil:ok → top
hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok
hole_top2_0 :: top
gen_0':true:mark:false:nil:ok3_0 :: Nat → 0':true:mark:false:nil:ok

Generator Equations:
gen_0':true:mark:false:nil:ok3_0(0) ⇔ 0'
gen_0':true:mark:false:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':true:mark:false:nil:ok3_0(x))

The following defined symbols remain to be analysed:
eq, active, cons, inf, s, take, length, proper, top

They will be analysed ascendingly in the following order:
eq < active
cons < active
inf < active
s < active
take < active
length < active
active < top
eq < proper
cons < proper
inf < proper
s < proper
take < proper
length < proper
proper < top

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

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

(10) Obligation:

TRS:
Rules:
active(eq(0', 0')) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0', X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0')
active(length(cons(X, L))) → mark(s(length(L)))
active(inf(X)) → inf(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(length(X)) → length(active(X))
inf(mark(X)) → mark(inf(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
length(mark(X)) → mark(length(X))
proper(eq(X1, X2)) → eq(proper(X1), proper(X2))
proper(0') → ok(0')
proper(true) → ok(true)
proper(s(X)) → s(proper(X))
proper(false) → ok(false)
proper(inf(X)) → inf(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(length(X)) → length(proper(X))
eq(ok(X1), ok(X2)) → ok(eq(X1, X2))
s(ok(X)) → ok(s(X))
inf(ok(X)) → ok(inf(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
eq :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
0' :: 0':true:mark:false:nil:ok
mark :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
true :: 0':true:mark:false:nil:ok
s :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
false :: 0':true:mark:false:nil:ok
inf :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
cons :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
take :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
nil :: 0':true:mark:false:nil:ok
length :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
proper :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
ok :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
top :: 0':true:mark:false:nil:ok → top
hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok
hole_top2_0 :: top
gen_0':true:mark:false:nil:ok3_0 :: Nat → 0':true:mark:false:nil:ok

Generator Equations:
gen_0':true:mark:false:nil:ok3_0(0) ⇔ 0'
gen_0':true:mark:false:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':true:mark:false:nil:ok3_0(x))

The following defined symbols remain to be analysed:
cons, active, inf, s, take, length, proper, top

They will be analysed ascendingly in the following order:
cons < active
inf < active
s < active
take < active
length < active
active < top
cons < proper
inf < proper
s < proper
take < proper
length < proper
proper < top

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

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

(12) Obligation:

TRS:
Rules:
active(eq(0', 0')) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0', X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0')
active(length(cons(X, L))) → mark(s(length(L)))
active(inf(X)) → inf(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(length(X)) → length(active(X))
inf(mark(X)) → mark(inf(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
length(mark(X)) → mark(length(X))
proper(eq(X1, X2)) → eq(proper(X1), proper(X2))
proper(0') → ok(0')
proper(true) → ok(true)
proper(s(X)) → s(proper(X))
proper(false) → ok(false)
proper(inf(X)) → inf(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(length(X)) → length(proper(X))
eq(ok(X1), ok(X2)) → ok(eq(X1, X2))
s(ok(X)) → ok(s(X))
inf(ok(X)) → ok(inf(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
eq :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
0' :: 0':true:mark:false:nil:ok
mark :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
true :: 0':true:mark:false:nil:ok
s :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
false :: 0':true:mark:false:nil:ok
inf :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
cons :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
take :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
nil :: 0':true:mark:false:nil:ok
length :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
proper :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
ok :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
top :: 0':true:mark:false:nil:ok → top
hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok
hole_top2_0 :: top
gen_0':true:mark:false:nil:ok3_0 :: Nat → 0':true:mark:false:nil:ok

Generator Equations:
gen_0':true:mark:false:nil:ok3_0(0) ⇔ 0'
gen_0':true:mark:false:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':true:mark:false:nil:ok3_0(x))

The following defined symbols remain to be analysed:
inf, active, s, take, length, proper, top

They will be analysed ascendingly in the following order:
inf < active
s < active
take < active
length < active
active < top
inf < proper
s < proper
take < proper
length < proper
proper < top

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

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

(14) Obligation:

TRS:
Rules:
active(eq(0', 0')) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0', X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0')
active(length(cons(X, L))) → mark(s(length(L)))
active(inf(X)) → inf(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(length(X)) → length(active(X))
inf(mark(X)) → mark(inf(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
length(mark(X)) → mark(length(X))
proper(eq(X1, X2)) → eq(proper(X1), proper(X2))
proper(0') → ok(0')
proper(true) → ok(true)
proper(s(X)) → s(proper(X))
proper(false) → ok(false)
proper(inf(X)) → inf(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(length(X)) → length(proper(X))
eq(ok(X1), ok(X2)) → ok(eq(X1, X2))
s(ok(X)) → ok(s(X))
inf(ok(X)) → ok(inf(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
eq :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
0' :: 0':true:mark:false:nil:ok
mark :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
true :: 0':true:mark:false:nil:ok
s :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
false :: 0':true:mark:false:nil:ok
inf :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
cons :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
take :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
nil :: 0':true:mark:false:nil:ok
length :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
proper :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
ok :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
top :: 0':true:mark:false:nil:ok → top
hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok
hole_top2_0 :: top
gen_0':true:mark:false:nil:ok3_0 :: Nat → 0':true:mark:false:nil:ok

Generator Equations:
gen_0':true:mark:false:nil:ok3_0(0) ⇔ 0'
gen_0':true:mark:false:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':true:mark:false:nil:ok3_0(x))

The following defined symbols remain to be analysed:
s, active, take, length, proper, top

They will be analysed ascendingly in the following order:
s < active
take < active
length < active
active < top
s < proper
take < proper
length < proper
proper < top

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

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

(16) Obligation:

TRS:
Rules:
active(eq(0', 0')) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0', X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0')
active(length(cons(X, L))) → mark(s(length(L)))
active(inf(X)) → inf(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(length(X)) → length(active(X))
inf(mark(X)) → mark(inf(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
length(mark(X)) → mark(length(X))
proper(eq(X1, X2)) → eq(proper(X1), proper(X2))
proper(0') → ok(0')
proper(true) → ok(true)
proper(s(X)) → s(proper(X))
proper(false) → ok(false)
proper(inf(X)) → inf(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(length(X)) → length(proper(X))
eq(ok(X1), ok(X2)) → ok(eq(X1, X2))
s(ok(X)) → ok(s(X))
inf(ok(X)) → ok(inf(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
eq :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
0' :: 0':true:mark:false:nil:ok
mark :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
true :: 0':true:mark:false:nil:ok
s :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
false :: 0':true:mark:false:nil:ok
inf :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
cons :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
take :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
nil :: 0':true:mark:false:nil:ok
length :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
proper :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
ok :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
top :: 0':true:mark:false:nil:ok → top
hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok
hole_top2_0 :: top
gen_0':true:mark:false:nil:ok3_0 :: Nat → 0':true:mark:false:nil:ok

Generator Equations:
gen_0':true:mark:false:nil:ok3_0(0) ⇔ 0'
gen_0':true:mark:false:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':true:mark:false:nil:ok3_0(x))

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

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

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

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

(18) Obligation:

TRS:
Rules:
active(eq(0', 0')) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0', X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0')
active(length(cons(X, L))) → mark(s(length(L)))
active(inf(X)) → inf(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(length(X)) → length(active(X))
inf(mark(X)) → mark(inf(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
length(mark(X)) → mark(length(X))
proper(eq(X1, X2)) → eq(proper(X1), proper(X2))
proper(0') → ok(0')
proper(true) → ok(true)
proper(s(X)) → s(proper(X))
proper(false) → ok(false)
proper(inf(X)) → inf(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(length(X)) → length(proper(X))
eq(ok(X1), ok(X2)) → ok(eq(X1, X2))
s(ok(X)) → ok(s(X))
inf(ok(X)) → ok(inf(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
eq :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
0' :: 0':true:mark:false:nil:ok
mark :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
true :: 0':true:mark:false:nil:ok
s :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
false :: 0':true:mark:false:nil:ok
inf :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
cons :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
take :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
nil :: 0':true:mark:false:nil:ok
length :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
proper :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
ok :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
top :: 0':true:mark:false:nil:ok → top
hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok
hole_top2_0 :: top
gen_0':true:mark:false:nil:ok3_0 :: Nat → 0':true:mark:false:nil:ok

Generator Equations:
gen_0':true:mark:false:nil:ok3_0(0) ⇔ 0'
gen_0':true:mark:false:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':true:mark:false:nil:ok3_0(x))

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

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

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

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

(20) Obligation:

TRS:
Rules:
active(eq(0', 0')) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0', X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0')
active(length(cons(X, L))) → mark(s(length(L)))
active(inf(X)) → inf(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(length(X)) → length(active(X))
inf(mark(X)) → mark(inf(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
length(mark(X)) → mark(length(X))
proper(eq(X1, X2)) → eq(proper(X1), proper(X2))
proper(0') → ok(0')
proper(true) → ok(true)
proper(s(X)) → s(proper(X))
proper(false) → ok(false)
proper(inf(X)) → inf(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(length(X)) → length(proper(X))
eq(ok(X1), ok(X2)) → ok(eq(X1, X2))
s(ok(X)) → ok(s(X))
inf(ok(X)) → ok(inf(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
eq :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
0' :: 0':true:mark:false:nil:ok
mark :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
true :: 0':true:mark:false:nil:ok
s :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
false :: 0':true:mark:false:nil:ok
inf :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
cons :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
take :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
nil :: 0':true:mark:false:nil:ok
length :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
proper :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
ok :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
top :: 0':true:mark:false:nil:ok → top
hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok
hole_top2_0 :: top
gen_0':true:mark:false:nil:ok3_0 :: Nat → 0':true:mark:false:nil:ok

Generator Equations:
gen_0':true:mark:false:nil:ok3_0(0) ⇔ 0'
gen_0':true:mark:false:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':true:mark:false: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

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

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

(22) Obligation:

TRS:
Rules:
active(eq(0', 0')) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0', X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0')
active(length(cons(X, L))) → mark(s(length(L)))
active(inf(X)) → inf(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(length(X)) → length(active(X))
inf(mark(X)) → mark(inf(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
length(mark(X)) → mark(length(X))
proper(eq(X1, X2)) → eq(proper(X1), proper(X2))
proper(0') → ok(0')
proper(true) → ok(true)
proper(s(X)) → s(proper(X))
proper(false) → ok(false)
proper(inf(X)) → inf(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(length(X)) → length(proper(X))
eq(ok(X1), ok(X2)) → ok(eq(X1, X2))
s(ok(X)) → ok(s(X))
inf(ok(X)) → ok(inf(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
eq :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
0' :: 0':true:mark:false:nil:ok
mark :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
true :: 0':true:mark:false:nil:ok
s :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
false :: 0':true:mark:false:nil:ok
inf :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
cons :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
take :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
nil :: 0':true:mark:false:nil:ok
length :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
proper :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
ok :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
top :: 0':true:mark:false:nil:ok → top
hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok
hole_top2_0 :: top
gen_0':true:mark:false:nil:ok3_0 :: Nat → 0':true:mark:false:nil:ok

Generator Equations:
gen_0':true:mark:false:nil:ok3_0(0) ⇔ 0'
gen_0':true:mark:false:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':true:mark:false: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

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

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

(24) Obligation:

TRS:
Rules:
active(eq(0', 0')) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0', X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0')
active(length(cons(X, L))) → mark(s(length(L)))
active(inf(X)) → inf(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(length(X)) → length(active(X))
inf(mark(X)) → mark(inf(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
length(mark(X)) → mark(length(X))
proper(eq(X1, X2)) → eq(proper(X1), proper(X2))
proper(0') → ok(0')
proper(true) → ok(true)
proper(s(X)) → s(proper(X))
proper(false) → ok(false)
proper(inf(X)) → inf(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(length(X)) → length(proper(X))
eq(ok(X1), ok(X2)) → ok(eq(X1, X2))
s(ok(X)) → ok(s(X))
inf(ok(X)) → ok(inf(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
eq :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
0' :: 0':true:mark:false:nil:ok
mark :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
true :: 0':true:mark:false:nil:ok
s :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
false :: 0':true:mark:false:nil:ok
inf :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
cons :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
take :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
nil :: 0':true:mark:false:nil:ok
length :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
proper :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
ok :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
top :: 0':true:mark:false:nil:ok → top
hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok
hole_top2_0 :: top
gen_0':true:mark:false:nil:ok3_0 :: Nat → 0':true:mark:false:nil:ok

Generator Equations:
gen_0':true:mark:false:nil:ok3_0(0) ⇔ 0'
gen_0':true:mark:false:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':true:mark:false:nil:ok3_0(x))

The following defined symbols remain to be analysed:
top

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

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

(26) Obligation:

TRS:
Rules:
active(eq(0', 0')) → mark(true)
active(eq(s(X), s(Y))) → mark(eq(X, Y))
active(eq(X, Y)) → mark(false)
active(inf(X)) → mark(cons(X, inf(s(X))))
active(take(0', X)) → mark(nil)
active(take(s(X), cons(Y, L))) → mark(cons(Y, take(X, L)))
active(length(nil)) → mark(0')
active(length(cons(X, L))) → mark(s(length(L)))
active(inf(X)) → inf(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(length(X)) → length(active(X))
inf(mark(X)) → mark(inf(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
length(mark(X)) → mark(length(X))
proper(eq(X1, X2)) → eq(proper(X1), proper(X2))
proper(0') → ok(0')
proper(true) → ok(true)
proper(s(X)) → s(proper(X))
proper(false) → ok(false)
proper(inf(X)) → inf(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(length(X)) → length(proper(X))
eq(ok(X1), ok(X2)) → ok(eq(X1, X2))
s(ok(X)) → ok(s(X))
inf(ok(X)) → ok(inf(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
length(ok(X)) → ok(length(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
eq :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
0' :: 0':true:mark:false:nil:ok
mark :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
true :: 0':true:mark:false:nil:ok
s :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
false :: 0':true:mark:false:nil:ok
inf :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
cons :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
take :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
nil :: 0':true:mark:false:nil:ok
length :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
proper :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
ok :: 0':true:mark:false:nil:ok → 0':true:mark:false:nil:ok
top :: 0':true:mark:false:nil:ok → top
hole_0':true:mark:false:nil:ok1_0 :: 0':true:mark:false:nil:ok
hole_top2_0 :: top
gen_0':true:mark:false:nil:ok3_0 :: Nat → 0':true:mark:false:nil:ok

Generator Equations:
gen_0':true:mark:false:nil:ok3_0(0) ⇔ 0'
gen_0':true:mark:false:nil:ok3_0(+(x, 1)) ⇔ mark(gen_0':true:mark:false:nil:ok3_0(x))

No more defined symbols left to analyse.