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(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isNePal(__(I, __(P, I)))) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(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
__(mark(X1), X2) →⁺ mark(__(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(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isNePal(__(I, __(P, I)))) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(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(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isNePal(__(I, __(P, I)))) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:nil:tt:ok → mark:nil:tt:ok
__ :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
mark :: mark:nil:tt:ok → mark:nil:tt:ok
nil :: mark:nil:tt:ok
and :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
tt :: mark:nil:tt:ok
isNePal :: mark:nil:tt:ok → mark:nil:tt:ok
proper :: mark:nil:tt:ok → mark:nil:tt:ok
ok :: mark:nil:tt:ok → mark:nil:tt:ok
top :: mark:nil:tt:ok → top
hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok
hole_top2_0 :: top
gen_mark:nil:tt:ok3_0 :: Nat → mark:nil:tt:ok

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

Heuristically decided to analyse the following defined symbols:
active, __, and, isNePal, proper, top

They will be analysed ascendingly in the following order:
__ < active
and < active
isNePal < active
active < top
__ < proper
and < proper
isNePal < proper
proper < top

(8) Obligation:

TRS:
Rules:
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isNePal(__(I, __(P, I)))) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:nil:tt:ok → mark:nil:tt:ok
__ :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
mark :: mark:nil:tt:ok → mark:nil:tt:ok
nil :: mark:nil:tt:ok
and :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
tt :: mark:nil:tt:ok
isNePal :: mark:nil:tt:ok → mark:nil:tt:ok
proper :: mark:nil:tt:ok → mark:nil:tt:ok
ok :: mark:nil:tt:ok → mark:nil:tt:ok
top :: mark:nil:tt:ok → top
hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok
hole_top2_0 :: top
gen_mark:nil:tt:ok3_0 :: Nat → mark:nil:tt:ok

Generator Equations:
gen_mark:nil:tt:ok3_0(0) ⇔ nil
gen_mark:nil:tt:ok3_0(+(x, 1)) ⇔ mark(gen_mark:nil:tt:ok3_0(x))

The following defined symbols remain to be analysed:
__, active, and, isNePal, proper, top

They will be analysed ascendingly in the following order:
__ < active
and < active
isNePal < active
active < top
__ < proper
and < proper
isNePal < proper
proper < top

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

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

(10) Obligation:

TRS:
Rules:
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isNePal(__(I, __(P, I)))) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:nil:tt:ok → mark:nil:tt:ok
__ :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
mark :: mark:nil:tt:ok → mark:nil:tt:ok
nil :: mark:nil:tt:ok
and :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
tt :: mark:nil:tt:ok
isNePal :: mark:nil:tt:ok → mark:nil:tt:ok
proper :: mark:nil:tt:ok → mark:nil:tt:ok
ok :: mark:nil:tt:ok → mark:nil:tt:ok
top :: mark:nil:tt:ok → top
hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok
hole_top2_0 :: top
gen_mark:nil:tt:ok3_0 :: Nat → mark:nil:tt:ok

Generator Equations:
gen_mark:nil:tt:ok3_0(0) ⇔ nil
gen_mark:nil:tt:ok3_0(+(x, 1)) ⇔ mark(gen_mark:nil:tt:ok3_0(x))

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

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

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

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

(12) Obligation:

TRS:
Rules:
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isNePal(__(I, __(P, I)))) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:nil:tt:ok → mark:nil:tt:ok
__ :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
mark :: mark:nil:tt:ok → mark:nil:tt:ok
nil :: mark:nil:tt:ok
and :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
tt :: mark:nil:tt:ok
isNePal :: mark:nil:tt:ok → mark:nil:tt:ok
proper :: mark:nil:tt:ok → mark:nil:tt:ok
ok :: mark:nil:tt:ok → mark:nil:tt:ok
top :: mark:nil:tt:ok → top
hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok
hole_top2_0 :: top
gen_mark:nil:tt:ok3_0 :: Nat → mark:nil:tt:ok

Generator Equations:
gen_mark:nil:tt:ok3_0(0) ⇔ nil
gen_mark:nil:tt:ok3_0(+(x, 1)) ⇔ mark(gen_mark:nil:tt:ok3_0(x))

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

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

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

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

(14) Obligation:

TRS:
Rules:
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isNePal(__(I, __(P, I)))) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:nil:tt:ok → mark:nil:tt:ok
__ :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
mark :: mark:nil:tt:ok → mark:nil:tt:ok
nil :: mark:nil:tt:ok
and :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
tt :: mark:nil:tt:ok
isNePal :: mark:nil:tt:ok → mark:nil:tt:ok
proper :: mark:nil:tt:ok → mark:nil:tt:ok
ok :: mark:nil:tt:ok → mark:nil:tt:ok
top :: mark:nil:tt:ok → top
hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok
hole_top2_0 :: top
gen_mark:nil:tt:ok3_0 :: Nat → mark:nil:tt:ok

Generator Equations:
gen_mark:nil:tt:ok3_0(0) ⇔ nil
gen_mark:nil:tt:ok3_0(+(x, 1)) ⇔ mark(gen_mark:nil:tt: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

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

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

(16) Obligation:

TRS:
Rules:
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isNePal(__(I, __(P, I)))) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:nil:tt:ok → mark:nil:tt:ok
__ :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
mark :: mark:nil:tt:ok → mark:nil:tt:ok
nil :: mark:nil:tt:ok
and :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
tt :: mark:nil:tt:ok
isNePal :: mark:nil:tt:ok → mark:nil:tt:ok
proper :: mark:nil:tt:ok → mark:nil:tt:ok
ok :: mark:nil:tt:ok → mark:nil:tt:ok
top :: mark:nil:tt:ok → top
hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok
hole_top2_0 :: top
gen_mark:nil:tt:ok3_0 :: Nat → mark:nil:tt:ok

Generator Equations:
gen_mark:nil:tt:ok3_0(0) ⇔ nil
gen_mark:nil:tt:ok3_0(+(x, 1)) ⇔ mark(gen_mark:nil:tt:ok3_0(x))

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

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

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

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

(18) Obligation:

TRS:
Rules:
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isNePal(__(I, __(P, I)))) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:nil:tt:ok → mark:nil:tt:ok
__ :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
mark :: mark:nil:tt:ok → mark:nil:tt:ok
nil :: mark:nil:tt:ok
and :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
tt :: mark:nil:tt:ok
isNePal :: mark:nil:tt:ok → mark:nil:tt:ok
proper :: mark:nil:tt:ok → mark:nil:tt:ok
ok :: mark:nil:tt:ok → mark:nil:tt:ok
top :: mark:nil:tt:ok → top
hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok
hole_top2_0 :: top
gen_mark:nil:tt:ok3_0 :: Nat → mark:nil:tt:ok

Generator Equations:
gen_mark:nil:tt:ok3_0(0) ⇔ nil
gen_mark:nil:tt:ok3_0(+(x, 1)) ⇔ mark(gen_mark:nil:tt:ok3_0(x))

The following defined symbols remain to be analysed:
top

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

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

(20) Obligation:

TRS:
Rules:
active(__(__(X, Y), Z)) → mark(__(X, __(Y, Z)))
active(__(X, nil)) → mark(X)
active(__(nil, X)) → mark(X)
active(and(tt, X)) → mark(X)
active(isNePal(__(I, __(P, I)))) → mark(tt)
active(__(X1, X2)) → __(active(X1), X2)
active(__(X1, X2)) → __(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
active(isNePal(X)) → isNePal(active(X))
__(mark(X1), X2) → mark(__(X1, X2))
__(X1, mark(X2)) → mark(__(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
isNePal(mark(X)) → mark(isNePal(X))
proper(__(X1, X2)) → __(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(isNePal(X)) → isNePal(proper(X))
__(ok(X1), ok(X2)) → ok(__(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNePal(ok(X)) → ok(isNePal(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: mark:nil:tt:ok → mark:nil:tt:ok
__ :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
mark :: mark:nil:tt:ok → mark:nil:tt:ok
nil :: mark:nil:tt:ok
and :: mark:nil:tt:ok → mark:nil:tt:ok → mark:nil:tt:ok
tt :: mark:nil:tt:ok
isNePal :: mark:nil:tt:ok → mark:nil:tt:ok
proper :: mark:nil:tt:ok → mark:nil:tt:ok
ok :: mark:nil:tt:ok → mark:nil:tt:ok
top :: mark:nil:tt:ok → top
hole_mark:nil:tt:ok1_0 :: mark:nil:tt:ok
hole_top2_0 :: top
gen_mark:nil:tt:ok3_0 :: Nat → mark:nil:tt:ok

Generator Equations:
gen_mark:nil:tt:ok3_0(0) ⇔ nil
gen_mark:nil:tt:ok3_0(+(x, 1)) ⇔ mark(gen_mark:nil:tt:ok3_0(x))

No more defined symbols left to analyse.