### (0) Obligation:

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

a__f(X, g(X), Y) → a__f(Y, Y, Y)
a__g(b) → c
a__bc
mark(f(X1, X2, X3)) → a__f(X1, X2, X3)
mark(g(X)) → a__g(mark(X))
mark(b) → a__b
mark(c) → c
a__f(X1, X2, X3) → f(X1, X2, X3)
a__g(X) → g(X)
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(g(X)) →+ a__g(mark(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / g(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, g(X), Y) → a__f(Y, Y, Y)
a__g(b) → c
a__bc
mark(f(X1, X2, X3)) → a__f(X1, X2, X3)
mark(g(X)) → a__g(mark(X))
mark(b) → a__b
mark(c) → c
a__f(X1, X2, X3) → f(X1, X2, X3)
a__g(X) → g(X)
a__bb

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
a__f(X, g(X), Y) → a__f(Y, Y, Y)
a__g(b) → c
a__bc
mark(f(X1, X2, X3)) → a__f(X1, X2, X3)
mark(g(X)) → a__g(mark(X))
mark(b) → a__b
mark(c) → c
a__f(X1, X2, X3) → f(X1, X2, X3)
a__g(X) → g(X)
a__bb

Types:
a__f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f
g :: g:b:c:f → g:b:c:f
a__g :: g:b:c:f → g:b:c:f
b :: g:b:c:f
c :: g:b:c:f
a__b :: g:b:c:f
mark :: g:b:c:f → g:b:c:f
f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f
hole_g:b:c:f1_0 :: g:b:c:f
gen_g:b:c:f2_0 :: Nat → g:b:c: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, g(X), Y) → a__f(Y, Y, Y)
a__g(b) → c
a__bc
mark(f(X1, X2, X3)) → a__f(X1, X2, X3)
mark(g(X)) → a__g(mark(X))
mark(b) → a__b
mark(c) → c
a__f(X1, X2, X3) → f(X1, X2, X3)
a__g(X) → g(X)
a__bb

Types:
a__f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f
g :: g:b:c:f → g:b:c:f
a__g :: g:b:c:f → g:b:c:f
b :: g:b:c:f
c :: g:b:c:f
a__b :: g:b:c:f
mark :: g:b:c:f → g:b:c:f
f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f
hole_g:b:c:f1_0 :: g:b:c:f
gen_g:b:c:f2_0 :: Nat → g:b:c:f

Generator Equations:
gen_g:b:c:f2_0(0) ⇔ b
gen_g:b:c:f2_0(+(x, 1)) ⇔ g(gen_g:b:c:f2_0(x))

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, g(X), Y) → a__f(Y, Y, Y)
a__g(b) → c
a__bc
mark(f(X1, X2, X3)) → a__f(X1, X2, X3)
mark(g(X)) → a__g(mark(X))
mark(b) → a__b
mark(c) → c
a__f(X1, X2, X3) → f(X1, X2, X3)
a__g(X) → g(X)
a__bb

Types:
a__f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f
g :: g:b:c:f → g:b:c:f
a__g :: g:b:c:f → g:b:c:f
b :: g:b:c:f
c :: g:b:c:f
a__b :: g:b:c:f
mark :: g:b:c:f → g:b:c:f
f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f
hole_g:b:c:f1_0 :: g:b:c:f
gen_g:b:c:f2_0 :: Nat → g:b:c:f

Generator Equations:
gen_g:b:c:f2_0(0) ⇔ b
gen_g:b:c:f2_0(+(x, 1)) ⇔ g(gen_g:b:c:f2_0(x))

The following defined symbols remain to be analysed:
mark

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

Proved the following rewrite lemma:
mark(gen_g:b:c:f2_0(+(1, n101_0))) → *3_0, rt ∈ Ω(n1010)

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

Induction Step:
mark(gen_g:b:c:f2_0(+(1, +(n101_0, 1)))) →RΩ(1)
a__g(mark(gen_g:b:c:f2_0(+(1, n101_0)))) →IH
a__g(*3_0)

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

### (13) Obligation:

TRS:
Rules:
a__f(X, g(X), Y) → a__f(Y, Y, Y)
a__g(b) → c
a__bc
mark(f(X1, X2, X3)) → a__f(X1, X2, X3)
mark(g(X)) → a__g(mark(X))
mark(b) → a__b
mark(c) → c
a__f(X1, X2, X3) → f(X1, X2, X3)
a__g(X) → g(X)
a__bb

Types:
a__f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f
g :: g:b:c:f → g:b:c:f
a__g :: g:b:c:f → g:b:c:f
b :: g:b:c:f
c :: g:b:c:f
a__b :: g:b:c:f
mark :: g:b:c:f → g:b:c:f
f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f
hole_g:b:c:f1_0 :: g:b:c:f
gen_g:b:c:f2_0 :: Nat → g:b:c:f

Lemmas:
mark(gen_g:b:c:f2_0(+(1, n101_0))) → *3_0, rt ∈ Ω(n1010)

Generator Equations:
gen_g:b:c:f2_0(0) ⇔ b
gen_g:b:c:f2_0(+(x, 1)) ⇔ g(gen_g:b:c:f2_0(x))

No more defined symbols left to analyse.

### (14) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_g:b:c:f2_0(+(1, n101_0))) → *3_0, rt ∈ Ω(n1010)

### (16) Obligation:

TRS:
Rules:
a__f(X, g(X), Y) → a__f(Y, Y, Y)
a__g(b) → c
a__bc
mark(f(X1, X2, X3)) → a__f(X1, X2, X3)
mark(g(X)) → a__g(mark(X))
mark(b) → a__b
mark(c) → c
a__f(X1, X2, X3) → f(X1, X2, X3)
a__g(X) → g(X)
a__bb

Types:
a__f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f
g :: g:b:c:f → g:b:c:f
a__g :: g:b:c:f → g:b:c:f
b :: g:b:c:f
c :: g:b:c:f
a__b :: g:b:c:f
mark :: g:b:c:f → g:b:c:f
f :: g:b:c:f → g:b:c:f → g:b:c:f → g:b:c:f
hole_g:b:c:f1_0 :: g:b:c:f
gen_g:b:c:f2_0 :: Nat → g:b:c:f

Lemmas:
mark(gen_g:b:c:f2_0(+(1, n101_0))) → *3_0, rt ∈ Ω(n1010)

Generator Equations:
gen_g:b:c:f2_0(0) ⇔ b
gen_g:b:c:f2_0(+(x, 1)) ⇔ g(gen_g:b:c:f2_0(x))

No more defined symbols left to analyse.

### (17) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
mark(gen_g:b:c:f2_0(+(1, n101_0))) → *3_0, rt ∈ Ω(n1010)