### (0) Obligation:

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

a__f(g(X), Y) → a__f(mark(X), f(g(X), Y))
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(g(X)) → g(mark(X))
a__f(X1, X2) → f(X1, X2)

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
mark(f(g(X2938_0), X2)) →+ a__f(mark(mark(X2938_0)), f(g(mark(X2938_0)), X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [X2938_0 / f(g(X2938_0), X2)].
The result substitution is [ ].

The rewrite sequence
mark(f(g(X2938_0), X2)) →+ a__f(mark(mark(X2938_0)), f(g(mark(X2938_0)), X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
The pumping substitution is [X2938_0 / f(g(X2938_0), 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(g(X), Y) → a__f(mark(X), f(g(X), Y))
mark(f(X1, X2)) → a__f(mark(X1), X2)
mark(g(X)) → g(mark(X))
a__f(X1, X2) → f(X1, X2)

S is empty.
Rewrite Strategy: FULL

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

Sliced the following arguments:
a__f/1
f/1

### (6) Obligation:

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

a__f(g(X)) → a__f(mark(X))
mark(f(X1)) → a__f(mark(X1))
mark(g(X)) → g(mark(X))
a__f(X1) → f(X1)

S is empty.
Rewrite Strategy: FULL

Infered types.

### (8) Obligation:

TRS:
Rules:
a__f(g(X)) → a__f(mark(X))
mark(f(X1)) → a__f(mark(X1))
mark(g(X)) → g(mark(X))
a__f(X1) → f(X1)

Types:
a__f :: g:f → g:f
g :: g:f → g:f
mark :: g:f → g:f
f :: g:f → g:f
hole_g:f1_0 :: g:f
gen_g:f2_0 :: Nat → g:f

### (9) 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

### (10) Obligation:

TRS:
Rules:
a__f(g(X)) → a__f(mark(X))
mark(f(X1)) → a__f(mark(X1))
mark(g(X)) → g(mark(X))
a__f(X1) → f(X1)

Types:
a__f :: g:f → g:f
g :: g:f → g:f
mark :: g:f → g:f
f :: g:f → g:f
hole_g:f1_0 :: g:f
gen_g:f2_0 :: Nat → g:f

Generator Equations:
gen_g:f2_0(0) ⇔ hole_g:f1_0
gen_g:f2_0(+(x, 1)) ⇔ g(gen_g:f2_0(x))

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

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

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

Proved the following rewrite lemma:
mark(gen_g:f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

Induction Base:
mark(gen_g:f2_0(+(1, 0)))

Induction Step:
mark(gen_g:f2_0(+(1, +(n4_0, 1)))) →RΩ(1)
g(mark(gen_g:f2_0(+(1, n4_0)))) →IH
g(*3_0)

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

### (13) Obligation:

TRS:
Rules:
a__f(g(X)) → a__f(mark(X))
mark(f(X1)) → a__f(mark(X1))
mark(g(X)) → g(mark(X))
a__f(X1) → f(X1)

Types:
a__f :: g:f → g:f
g :: g:f → g:f
mark :: g:f → g:f
f :: g:f → g:f
hole_g:f1_0 :: g:f
gen_g:f2_0 :: Nat → g:f

Lemmas:
mark(gen_g:f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_g:f2_0(0) ⇔ hole_g:f1_0
gen_g:f2_0(+(x, 1)) ⇔ g(gen_g:f2_0(x))

The following defined symbols remain to be analysed:
a__f

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

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

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

### (15) Obligation:

TRS:
Rules:
a__f(g(X)) → a__f(mark(X))
mark(f(X1)) → a__f(mark(X1))
mark(g(X)) → g(mark(X))
a__f(X1) → f(X1)

Types:
a__f :: g:f → g:f
g :: g:f → g:f
mark :: g:f → g:f
f :: g:f → g:f
hole_g:f1_0 :: g:f
gen_g:f2_0 :: Nat → g:f

Lemmas:
mark(gen_g:f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_g:f2_0(0) ⇔ hole_g:f1_0
gen_g:f2_0(+(x, 1)) ⇔ g(gen_g:f2_0(x))

No more defined symbols left to analyse.

### (16) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_g:f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

### (18) Obligation:

TRS:
Rules:
a__f(g(X)) → a__f(mark(X))
mark(f(X1)) → a__f(mark(X1))
mark(g(X)) → g(mark(X))
a__f(X1) → f(X1)

Types:
a__f :: g:f → g:f
g :: g:f → g:f
mark :: g:f → g:f
f :: g:f → g:f
hole_g:f1_0 :: g:f
gen_g:f2_0 :: Nat → g:f

Lemmas:
mark(gen_g:f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)

Generator Equations:
gen_g:f2_0(0) ⇔ hole_g:f1_0
gen_g:f2_0(+(x, 1)) ⇔ g(gen_g:f2_0(x))

No more defined symbols left to analyse.

### (19) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_g:f2_0(+(1, n4_0))) → *3_0, rt ∈ Ω(n40)