```* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(cons(x,k),l) -> g(k,l,cons(x,k))
f(empty(),l) -> l
g(a,b,c) -> f(a,cons(b,c))
- Signature:
{f/2,g/3} / {cons/2,empty/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {f,g} and constructors {cons,empty}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:

Strict DPs
f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
f#(empty(),l) -> c_2()
g#(a,b,c) -> c_3(f#(a,cons(b,c)))
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
f#(empty(),l) -> c_2()
g#(a,b,c) -> c_3(f#(a,cons(b,c)))
- Strict TRS:
f(cons(x,k),l) -> g(k,l,cons(x,k))
f(empty(),l) -> l
g(a,b,c) -> f(a,cons(b,c))
- Signature:
{f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {cons,empty}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
f#(empty(),l) -> c_2()
g#(a,b,c) -> c_3(f#(a,cons(b,c)))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
f#(empty(),l) -> c_2()
g#(a,b,c) -> c_3(f#(a,cons(b,c)))
- Signature:
{f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {cons,empty}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{2}
by application of
Pre({2}) = {3}.
Here rules are labelled as follows:
1: f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
2: f#(empty(),l) -> c_2()
3: g#(a,b,c) -> c_3(f#(a,cons(b,c)))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
g#(a,b,c) -> c_3(f#(a,cons(b,c)))
- Weak DPs:
f#(empty(),l) -> c_2()
- Signature:
{f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {cons,empty}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
-->_1 g#(a,b,c) -> c_3(f#(a,cons(b,c))):2

2:S:g#(a,b,c) -> c_3(f#(a,cons(b,c)))
-->_1 f#(empty(),l) -> c_2():3
-->_1 f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k))):1

3:W:f#(empty(),l) -> c_2()

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: f#(empty(),l) -> c_2()
* Step 5: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
g#(a,b,c) -> c_3(f#(a,cons(b,c)))
- Signature:
{f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {cons,empty}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))

Consider the set of all dependency pairs
1: f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
2: g#(a,b,c) -> c_3(f#(a,cons(b,c)))
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{1}
These cover all (indirect) predecessors of dependency pairs
{1,2}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
** Step 5.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
g#(a,b,c) -> c_3(f#(a,cons(b,c)))
- Signature:
{f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {cons,empty}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_1) = {1},
uargs(c_3) = {1}

Following symbols are considered usable:
{f#,g#}
TcT has computed the following interpretation:
p(cons) = [1] x1 + [1] x2 + [2]
p(empty) = [1]
p(f) = [1] x1 + [0]
p(g) = [2] x1 + [1] x2 + [1]
p(f#) = [8] x1 + [0]
p(g#) = [8] x1 + [0]
p(c_1) = [1] x1 + [8]
p(c_2) = [1]
p(c_3) = [1] x1 + [0]

Following rules are strictly oriented:
f#(cons(x,k),l) = [8] k + [8] x + [16]
> [8] k + [8]
= c_1(g#(k,l,cons(x,k)))

Following rules are (at-least) weakly oriented:
g#(a,b,c) =  [8] a + [0]
>= [8] a + [0]
=  c_3(f#(a,cons(b,c)))

** Step 5.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
g#(a,b,c) -> c_3(f#(a,cons(b,c)))
- Weak DPs:
f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
- Signature:
{f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {cons,empty}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
g#(a,b,c) -> c_3(f#(a,cons(b,c)))
- Signature:
{f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {cons,empty}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
-->_1 g#(a,b,c) -> c_3(f#(a,cons(b,c))):2

2:W:g#(a,b,c) -> c_3(f#(a,cons(b,c)))
-->_1 f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k))):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
2: g#(a,b,c) -> c_3(f#(a,cons(b,c)))
** Step 5.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {cons,empty}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))
```