### (0) Obligation:

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

a__f(0) → cons(0, f(s(0)))
a__f(s(0)) → a__f(a__p(s(0)))
a__p(s(X)) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(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(f(X)) →+ a__f(mark(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / f(X)].
The result substitution is [ ].

### (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:

a__f(0') → cons(0', f(s(0')))
a__f(s(0')) → a__f(a__p(s(0')))
a__p(s(X)) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

S is empty.
Rewrite Strategy: FULL

### (5) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
cons/1

### (6) Obligation:

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

a__f(0') → cons(0')
a__f(s(0')) → a__f(a__p(s(0')))
a__p(s(X)) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0') → 0'
mark(cons(X1)) → cons(mark(X1))
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

S is empty.
Rewrite Strategy: FULL

Infered types.

### (8) Obligation:

TRS:
Rules:
a__f(0') → cons(0')
a__f(s(0')) → a__f(a__p(s(0')))
a__p(s(X)) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0') → 0'
mark(cons(X1)) → cons(mark(X1))
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

Types:
a__f :: 0':cons:s:f:p → 0':cons:s:f:p
0' :: 0':cons:s:f:p
cons :: 0':cons:s:f:p → 0':cons:s:f:p
s :: 0':cons:s:f:p → 0':cons:s:f:p
a__p :: 0':cons:s:f:p → 0':cons:s:f:p
mark :: 0':cons:s:f:p → 0':cons:s:f:p
f :: 0':cons:s:f:p → 0':cons:s:f:p
p :: 0':cons:s:f:p → 0':cons:s:f:p
hole_0':cons:s:f:p1_0 :: 0':cons:s:f:p
gen_0':cons:s:f:p2_0 :: Nat → 0':cons:s:f:p

### (9) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
a__f, a__p, mark

They will be analysed ascendingly in the following order:
a__f = a__p
a__f = mark
a__p = mark

### (10) Obligation:

TRS:
Rules:
a__f(0') → cons(0')
a__f(s(0')) → a__f(a__p(s(0')))
a__p(s(X)) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0') → 0'
mark(cons(X1)) → cons(mark(X1))
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

Types:
a__f :: 0':cons:s:f:p → 0':cons:s:f:p
0' :: 0':cons:s:f:p
cons :: 0':cons:s:f:p → 0':cons:s:f:p
s :: 0':cons:s:f:p → 0':cons:s:f:p
a__p :: 0':cons:s:f:p → 0':cons:s:f:p
mark :: 0':cons:s:f:p → 0':cons:s:f:p
f :: 0':cons:s:f:p → 0':cons:s:f:p
p :: 0':cons:s:f:p → 0':cons:s:f:p
hole_0':cons:s:f:p1_0 :: 0':cons:s:f:p
gen_0':cons:s:f:p2_0 :: Nat → 0':cons:s:f:p

Generator Equations:
gen_0':cons:s:f:p2_0(0) ⇔ 0'
gen_0':cons:s:f:p2_0(+(x, 1)) ⇔ cons(gen_0':cons:s:f:p2_0(x))

The following defined symbols remain to be analysed:
a__p, a__f, mark

They will be analysed ascendingly in the following order:
a__f = a__p
a__f = mark
a__p = mark

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

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

### (12) Obligation:

TRS:
Rules:
a__f(0') → cons(0')
a__f(s(0')) → a__f(a__p(s(0')))
a__p(s(X)) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0') → 0'
mark(cons(X1)) → cons(mark(X1))
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

Types:
a__f :: 0':cons:s:f:p → 0':cons:s:f:p
0' :: 0':cons:s:f:p
cons :: 0':cons:s:f:p → 0':cons:s:f:p
s :: 0':cons:s:f:p → 0':cons:s:f:p
a__p :: 0':cons:s:f:p → 0':cons:s:f:p
mark :: 0':cons:s:f:p → 0':cons:s:f:p
f :: 0':cons:s:f:p → 0':cons:s:f:p
p :: 0':cons:s:f:p → 0':cons:s:f:p
hole_0':cons:s:f:p1_0 :: 0':cons:s:f:p
gen_0':cons:s:f:p2_0 :: Nat → 0':cons:s:f:p

Generator Equations:
gen_0':cons:s:f:p2_0(0) ⇔ 0'
gen_0':cons:s:f:p2_0(+(x, 1)) ⇔ cons(gen_0':cons:s:f:p2_0(x))

The following defined symbols remain to be analysed:
mark, a__f

They will be analysed ascendingly in the following order:
a__f = a__p
a__f = mark
a__p = mark

### (13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mark(gen_0':cons:s:f:p2_0(n10_0)) → gen_0':cons:s:f:p2_0(n10_0), rt ∈ Ω(1 + n100)

Induction Base:
mark(gen_0':cons:s:f:p2_0(0)) →RΩ(1)
0'

Induction Step:
mark(gen_0':cons:s:f:p2_0(+(n10_0, 1))) →RΩ(1)
cons(mark(gen_0':cons:s:f:p2_0(n10_0))) →IH
cons(gen_0':cons:s:f:p2_0(c11_0))

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

### (15) Obligation:

TRS:
Rules:
a__f(0') → cons(0')
a__f(s(0')) → a__f(a__p(s(0')))
a__p(s(X)) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0') → 0'
mark(cons(X1)) → cons(mark(X1))
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

Types:
a__f :: 0':cons:s:f:p → 0':cons:s:f:p
0' :: 0':cons:s:f:p
cons :: 0':cons:s:f:p → 0':cons:s:f:p
s :: 0':cons:s:f:p → 0':cons:s:f:p
a__p :: 0':cons:s:f:p → 0':cons:s:f:p
mark :: 0':cons:s:f:p → 0':cons:s:f:p
f :: 0':cons:s:f:p → 0':cons:s:f:p
p :: 0':cons:s:f:p → 0':cons:s:f:p
hole_0':cons:s:f:p1_0 :: 0':cons:s:f:p
gen_0':cons:s:f:p2_0 :: Nat → 0':cons:s:f:p

Lemmas:
mark(gen_0':cons:s:f:p2_0(n10_0)) → gen_0':cons:s:f:p2_0(n10_0), rt ∈ Ω(1 + n100)

Generator Equations:
gen_0':cons:s:f:p2_0(0) ⇔ 0'
gen_0':cons:s:f:p2_0(+(x, 1)) ⇔ cons(gen_0':cons:s:f:p2_0(x))

The following defined symbols remain to be analysed:
a__f, a__p

They will be analysed ascendingly in the following order:
a__f = a__p
a__f = mark
a__p = mark

### (16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

### (17) Obligation:

TRS:
Rules:
a__f(0') → cons(0')
a__f(s(0')) → a__f(a__p(s(0')))
a__p(s(X)) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0') → 0'
mark(cons(X1)) → cons(mark(X1))
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

Types:
a__f :: 0':cons:s:f:p → 0':cons:s:f:p
0' :: 0':cons:s:f:p
cons :: 0':cons:s:f:p → 0':cons:s:f:p
s :: 0':cons:s:f:p → 0':cons:s:f:p
a__p :: 0':cons:s:f:p → 0':cons:s:f:p
mark :: 0':cons:s:f:p → 0':cons:s:f:p
f :: 0':cons:s:f:p → 0':cons:s:f:p
p :: 0':cons:s:f:p → 0':cons:s:f:p
hole_0':cons:s:f:p1_0 :: 0':cons:s:f:p
gen_0':cons:s:f:p2_0 :: Nat → 0':cons:s:f:p

Lemmas:
mark(gen_0':cons:s:f:p2_0(n10_0)) → gen_0':cons:s:f:p2_0(n10_0), rt ∈ Ω(1 + n100)

Generator Equations:
gen_0':cons:s:f:p2_0(0) ⇔ 0'
gen_0':cons:s:f:p2_0(+(x, 1)) ⇔ cons(gen_0':cons:s:f:p2_0(x))

The following defined symbols remain to be analysed:
a__p

They will be analysed ascendingly in the following order:
a__f = a__p
a__f = mark
a__p = mark

### (18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

### (19) Obligation:

TRS:
Rules:
a__f(0') → cons(0')
a__f(s(0')) → a__f(a__p(s(0')))
a__p(s(X)) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0') → 0'
mark(cons(X1)) → cons(mark(X1))
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

Types:
a__f :: 0':cons:s:f:p → 0':cons:s:f:p
0' :: 0':cons:s:f:p
cons :: 0':cons:s:f:p → 0':cons:s:f:p
s :: 0':cons:s:f:p → 0':cons:s:f:p
a__p :: 0':cons:s:f:p → 0':cons:s:f:p
mark :: 0':cons:s:f:p → 0':cons:s:f:p
f :: 0':cons:s:f:p → 0':cons:s:f:p
p :: 0':cons:s:f:p → 0':cons:s:f:p
hole_0':cons:s:f:p1_0 :: 0':cons:s:f:p
gen_0':cons:s:f:p2_0 :: Nat → 0':cons:s:f:p

Lemmas:
mark(gen_0':cons:s:f:p2_0(n10_0)) → gen_0':cons:s:f:p2_0(n10_0), rt ∈ Ω(1 + n100)

Generator Equations:
gen_0':cons:s:f:p2_0(0) ⇔ 0'
gen_0':cons:s:f:p2_0(+(x, 1)) ⇔ cons(gen_0':cons:s:f:p2_0(x))

No more defined symbols left to analyse.

### (20) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_0':cons:s:f:p2_0(n10_0)) → gen_0':cons:s:f:p2_0(n10_0), rt ∈ Ω(1 + n100)

### (22) Obligation:

TRS:
Rules:
a__f(0') → cons(0')
a__f(s(0')) → a__f(a__p(s(0')))
a__p(s(X)) → mark(X)
mark(f(X)) → a__f(mark(X))
mark(p(X)) → a__p(mark(X))
mark(0') → 0'
mark(cons(X1)) → cons(mark(X1))
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__p(X) → p(X)

Types:
a__f :: 0':cons:s:f:p → 0':cons:s:f:p
0' :: 0':cons:s:f:p
cons :: 0':cons:s:f:p → 0':cons:s:f:p
s :: 0':cons:s:f:p → 0':cons:s:f:p
a__p :: 0':cons:s:f:p → 0':cons:s:f:p
mark :: 0':cons:s:f:p → 0':cons:s:f:p
f :: 0':cons:s:f:p → 0':cons:s:f:p
p :: 0':cons:s:f:p → 0':cons:s:f:p
hole_0':cons:s:f:p1_0 :: 0':cons:s:f:p
gen_0':cons:s:f:p2_0 :: Nat → 0':cons:s:f:p

Lemmas:
mark(gen_0':cons:s:f:p2_0(n10_0)) → gen_0':cons:s:f:p2_0(n10_0), rt ∈ Ω(1 + n100)

Generator Equations:
gen_0':cons:s:f:p2_0(0) ⇔ 0'
gen_0':cons:s:f:p2_0(+(x, 1)) ⇔ cons(gen_0':cons:s:f:p2_0(x))

No more defined symbols left to analyse.

### (23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_0':cons:s:f:p2_0(n10_0)) → gen_0':cons:s:f:p2_0(n10_0), rt ∈ Ω(1 + n100)