### (0) Obligation:

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

a__f(X, X) → a__f(a, b)
a__ba
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → a__b
mark(a) → a
a__f(X1, X2) → f(X1, X2)
a__bb

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(X1, X2)) →+ a__f(mark(X1), X2)
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / f(X1, X2)].
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(X, X) → a__f(a, b)
a__ba
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → a__b
mark(a) → a
a__f(X1, X2) → f(X1, X2)
a__bb

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
a__f(X, X) → a__f(a, b)
a__ba
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → a__b
mark(a) → a
a__f(X1, X2) → f(X1, X2)
a__bb

Types:
a__f :: a:b:f → a:b:f → a:b:f
a :: a:b:f
b :: a:b:f
a__b :: a:b:f
mark :: a:b:f → a:b:f
f :: a:b:f → a:b:f → a:b:f
hole_a:b:f1_0 :: a:b:f
gen_a:b:f2_0 :: Nat → a:b:f

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

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

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

### (8) Obligation:

TRS:
Rules:
a__f(X, X) → a__f(a, b)
a__ba
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → a__b
mark(a) → a
a__f(X1, X2) → f(X1, X2)
a__bb

Types:
a__f :: a:b:f → a:b:f → a:b:f
a :: a:b:f
b :: a:b:f
a__b :: a:b:f
mark :: a:b:f → a:b:f
f :: a:b:f → a:b:f → a:b:f
hole_a:b:f1_0 :: a:b:f
gen_a:b:f2_0 :: Nat → a:b:f

Generator Equations:
gen_a:b:f2_0(0) ⇔ a
gen_a:b:f2_0(+(x, 1)) ⇔ f(gen_a:b:f2_0(x), a)

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

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

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

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

### (10) Obligation:

TRS:
Rules:
a__f(X, X) → a__f(a, b)
a__ba
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → a__b
mark(a) → a
a__f(X1, X2) → f(X1, X2)
a__bb

Types:
a__f :: a:b:f → a:b:f → a:b:f
a :: a:b:f
b :: a:b:f
a__b :: a:b:f
mark :: a:b:f → a:b:f
f :: a:b:f → a:b:f → a:b:f
hole_a:b:f1_0 :: a:b:f
gen_a:b:f2_0 :: Nat → a:b:f

Generator Equations:
gen_a:b:f2_0(0) ⇔ a
gen_a:b:f2_0(+(x, 1)) ⇔ f(gen_a:b:f2_0(x), a)

The following defined symbols remain to be analysed:
mark

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

Proved the following rewrite lemma:
mark(gen_a:b:f2_0(+(1, n25_0))) → *3_0, rt ∈ Ω(n250)

Induction Base:
mark(gen_a:b:f2_0(+(1, 0)))

Induction Step:
mark(gen_a:b:f2_0(+(1, +(n25_0, 1)))) →RΩ(1)
a__f(mark(gen_a:b:f2_0(+(1, n25_0))), a) →IH
a__f(*3_0, a)

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

### (13) Obligation:

TRS:
Rules:
a__f(X, X) → a__f(a, b)
a__ba
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → a__b
mark(a) → a
a__f(X1, X2) → f(X1, X2)
a__bb

Types:
a__f :: a:b:f → a:b:f → a:b:f
a :: a:b:f
b :: a:b:f
a__b :: a:b:f
mark :: a:b:f → a:b:f
f :: a:b:f → a:b:f → a:b:f
hole_a:b:f1_0 :: a:b:f
gen_a:b:f2_0 :: Nat → a:b:f

Lemmas:
mark(gen_a:b:f2_0(+(1, n25_0))) → *3_0, rt ∈ Ω(n250)

Generator Equations:
gen_a:b:f2_0(0) ⇔ a
gen_a:b:f2_0(+(x, 1)) ⇔ f(gen_a:b:f2_0(x), a)

No more defined symbols left to analyse.

### (14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_a:b:f2_0(+(1, n25_0))) → *3_0, rt ∈ Ω(n250)

### (16) Obligation:

TRS:
Rules:
a__f(X, X) → a__f(a, b)
a__ba
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(b) → a__b
mark(a) → a
a__f(X1, X2) → f(X1, X2)
a__bb

Types:
a__f :: a:b:f → a:b:f → a:b:f
a :: a:b:f
b :: a:b:f
a__b :: a:b:f
mark :: a:b:f → a:b:f
f :: a:b:f → a:b:f → a:b:f
hole_a:b:f1_0 :: a:b:f
gen_a:b:f2_0 :: Nat → a:b:f

Lemmas:
mark(gen_a:b:f2_0(+(1, n25_0))) → *3_0, rt ∈ Ω(n250)

Generator Equations:
gen_a:b:f2_0(0) ⇔ a
gen_a:b:f2_0(+(x, 1)) ⇔ f(gen_a:b:f2_0(x), a)

No more defined symbols left to analyse.

### (17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_a:b:f2_0(+(1, n25_0))) → *3_0, rt ∈ Ω(n250)