### (0) Obligation:

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

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

The rewrite sequence
mark(f(X)) →+ cons(mark(mark(X)), f(g(mark(X))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
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(X) → cons(mark(X), f(g(X)))
a__g(0') → s(0')
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0', cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(X)
a__sel(X1, X2) → sel(X1, X2)

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
a__f(X) → cons(mark(X), f(g(X)))
a__g(0') → s(0')
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0', cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(X)
a__sel(X1, X2) → sel(X1, X2)

Types:
a__f :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
cons :: g:f:cons:0':s:sel → g:f:cons:0':s:sel → g:f:cons:0':s:sel
mark :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
f :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
g :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
a__g :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
0' :: g:f:cons:0':s:sel
s :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
a__sel :: g:f:cons:0':s:sel → g:f:cons:0':s:sel → g:f:cons:0':s:sel
sel :: g:f:cons:0':s:sel → g:f:cons:0':s:sel → g:f:cons:0':s:sel
hole_g:f:cons:0':s:sel1_0 :: g:f:cons:0':s:sel
gen_g:f:cons:0':s:sel2_0 :: Nat → g:f:cons:0':s:sel

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

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

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

### (8) Obligation:

TRS:
Rules:
a__f(X) → cons(mark(X), f(g(X)))
a__g(0') → s(0')
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0', cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(X)
a__sel(X1, X2) → sel(X1, X2)

Types:
a__f :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
cons :: g:f:cons:0':s:sel → g:f:cons:0':s:sel → g:f:cons:0':s:sel
mark :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
f :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
g :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
a__g :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
0' :: g:f:cons:0':s:sel
s :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
a__sel :: g:f:cons:0':s:sel → g:f:cons:0':s:sel → g:f:cons:0':s:sel
sel :: g:f:cons:0':s:sel → g:f:cons:0':s:sel → g:f:cons:0':s:sel
hole_g:f:cons:0':s:sel1_0 :: g:f:cons:0':s:sel
gen_g:f:cons:0':s:sel2_0 :: Nat → g:f:cons:0':s:sel

Generator Equations:
gen_g:f:cons:0':s:sel2_0(0) ⇔ 0'
gen_g:f:cons:0':s:sel2_0(+(x, 1)) ⇔ cons(0', gen_g:f:cons:0':s:sel2_0(x))

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

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

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

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

### (10) Obligation:

TRS:
Rules:
a__f(X) → cons(mark(X), f(g(X)))
a__g(0') → s(0')
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0', cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(X)
a__sel(X1, X2) → sel(X1, X2)

Types:
a__f :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
cons :: g:f:cons:0':s:sel → g:f:cons:0':s:sel → g:f:cons:0':s:sel
mark :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
f :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
g :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
a__g :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
0' :: g:f:cons:0':s:sel
s :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
a__sel :: g:f:cons:0':s:sel → g:f:cons:0':s:sel → g:f:cons:0':s:sel
sel :: g:f:cons:0':s:sel → g:f:cons:0':s:sel → g:f:cons:0':s:sel
hole_g:f:cons:0':s:sel1_0 :: g:f:cons:0':s:sel
gen_g:f:cons:0':s:sel2_0 :: Nat → g:f:cons:0':s:sel

Generator Equations:
gen_g:f:cons:0':s:sel2_0(0) ⇔ 0'
gen_g:f:cons:0':s:sel2_0(+(x, 1)) ⇔ cons(0', gen_g:f:cons:0':s:sel2_0(x))

The following defined symbols remain to be analysed:
a__f, a__g, a__sel

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

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

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

### (12) Obligation:

TRS:
Rules:
a__f(X) → cons(mark(X), f(g(X)))
a__g(0') → s(0')
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0', cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(X)
a__sel(X1, X2) → sel(X1, X2)

Types:
a__f :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
cons :: g:f:cons:0':s:sel → g:f:cons:0':s:sel → g:f:cons:0':s:sel
mark :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
f :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
g :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
a__g :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
0' :: g:f:cons:0':s:sel
s :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
a__sel :: g:f:cons:0':s:sel → g:f:cons:0':s:sel → g:f:cons:0':s:sel
sel :: g:f:cons:0':s:sel → g:f:cons:0':s:sel → g:f:cons:0':s:sel
hole_g:f:cons:0':s:sel1_0 :: g:f:cons:0':s:sel
gen_g:f:cons:0':s:sel2_0 :: Nat → g:f:cons:0':s:sel

Generator Equations:
gen_g:f:cons:0':s:sel2_0(0) ⇔ 0'
gen_g:f:cons:0':s:sel2_0(+(x, 1)) ⇔ cons(0', gen_g:f:cons:0':s:sel2_0(x))

The following defined symbols remain to be analysed:
a__g, a__sel

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

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

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

### (14) Obligation:

TRS:
Rules:
a__f(X) → cons(mark(X), f(g(X)))
a__g(0') → s(0')
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0', cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(X)
a__sel(X1, X2) → sel(X1, X2)

Types:
a__f :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
cons :: g:f:cons:0':s:sel → g:f:cons:0':s:sel → g:f:cons:0':s:sel
mark :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
f :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
g :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
a__g :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
0' :: g:f:cons:0':s:sel
s :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
a__sel :: g:f:cons:0':s:sel → g:f:cons:0':s:sel → g:f:cons:0':s:sel
sel :: g:f:cons:0':s:sel → g:f:cons:0':s:sel → g:f:cons:0':s:sel
hole_g:f:cons:0':s:sel1_0 :: g:f:cons:0':s:sel
gen_g:f:cons:0':s:sel2_0 :: Nat → g:f:cons:0':s:sel

Generator Equations:
gen_g:f:cons:0':s:sel2_0(0) ⇔ 0'
gen_g:f:cons:0':s:sel2_0(+(x, 1)) ⇔ cons(0', gen_g:f:cons:0':s:sel2_0(x))

The following defined symbols remain to be analysed:
a__sel

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

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

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

### (16) Obligation:

TRS:
Rules:
a__f(X) → cons(mark(X), f(g(X)))
a__g(0') → s(0')
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0', cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0') → 0'
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(X)
a__sel(X1, X2) → sel(X1, X2)

Types:
a__f :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
cons :: g:f:cons:0':s:sel → g:f:cons:0':s:sel → g:f:cons:0':s:sel
mark :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
f :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
g :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
a__g :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
0' :: g:f:cons:0':s:sel
s :: g:f:cons:0':s:sel → g:f:cons:0':s:sel
a__sel :: g:f:cons:0':s:sel → g:f:cons:0':s:sel → g:f:cons:0':s:sel
sel :: g:f:cons:0':s:sel → g:f:cons:0':s:sel → g:f:cons:0':s:sel
hole_g:f:cons:0':s:sel1_0 :: g:f:cons:0':s:sel
gen_g:f:cons:0':s:sel2_0 :: Nat → g:f:cons:0':s:sel

Generator Equations:
gen_g:f:cons:0':s:sel2_0(0) ⇔ 0'
gen_g:f:cons:0':s:sel2_0(+(x, 1)) ⇔ cons(0', gen_g:f:cons:0':s:sel2_0(x))

No more defined symbols left to analyse.