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

Strict DPs
cons#(x,y) -> c_1(x)
cons#(x,y) -> c_2(y)
f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
g#(f(s(x),s(y),z)) -> c_4(g#(f(x,y,z)))
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
cons#(x,y) -> c_1(x)
cons#(x,y) -> c_2(y)
f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
g#(f(s(x),s(y),z)) -> c_4(g#(f(x,y,z)))
- Strict TRS:
cons(x,y) -> x
cons(x,y) -> y
f(s(a()),s(b()),x) -> f(x,x,x)
g(f(s(x),s(y),z)) -> g(f(x,y,z))
- Signature:
{cons/2,f/3,g/1,cons#/2,f#/3,g#/1} / {a/0,b/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
- Obligation:
runtime complexity wrt. defined symbols {cons#,f#,g#} and constructors {a,b,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
cons#(x,y) -> c_1(x)
cons#(x,y) -> c_2(y)
f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
* Step 3: WeightGap WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
cons#(x,y) -> c_1(x)
cons#(x,y) -> c_2(y)
f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
- Signature:
{cons/2,f/3,g/1,cons#/2,f#/3,g#/1} / {a/0,b/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
- Obligation:
runtime complexity wrt. defined symbols {cons#,f#,g#} and constructors {a,b,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
none

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a) = [0]
p(b) = [0]
p(cons) = [0]
p(f) = [0]
p(g) = [0]
p(s) = [0]
p(cons#) = [0]
p(f#) = [15]
p(g#) = [0]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [0]
p(c_4) = [0]

Following rules are strictly oriented:
f#(s(a()),s(b()),x) = [15]
> [0]
= c_3(f#(x,x,x))

Following rules are (at-least) weakly oriented:
cons#(x,y) =  [0]
>= [0]
=  c_1(x)

cons#(x,y) =  [0]
>= [0]
=  c_2(y)

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: WeightGap WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
cons#(x,y) -> c_1(x)
cons#(x,y) -> c_2(y)
- Weak DPs:
f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
- Signature:
{cons/2,f/3,g/1,cons#/2,f#/3,g#/1} / {a/0,b/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
- Obligation:
runtime complexity wrt. defined symbols {cons#,f#,g#} and constructors {a,b,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
none

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(a) = [0]
p(b) = [0]
p(cons) = [0]
p(f) = [0]
p(g) = [0]
p(s) = [11]
p(cons#) = [1]
p(f#) = [0]
p(g#) = [0]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [0]
p(c_4) = [0]

Following rules are strictly oriented:
cons#(x,y) = [1]
> [0]
= c_1(x)

cons#(x,y) = [1]
> [0]
= c_2(y)

Following rules are (at-least) weakly oriented:
f#(s(a()),s(b()),x) =  [0]
>= [0]
=  c_3(f#(x,x,x))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
cons#(x,y) -> c_1(x)
cons#(x,y) -> c_2(y)
f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
- Signature:
{cons/2,f/3,g/1,cons#/2,f#/3,g#/1} / {a/0,b/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
- Obligation:
runtime complexity wrt. defined symbols {cons#,f#,g#} and constructors {a,b,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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