```* Step 1: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
f(0()) -> 1()
f(s(x)) -> g(x,s(x))
g(0(),y) -> y
g(s(x),y) -> g(x,+(y,s(x)))
g(s(x),y) -> g(x,s(+(y,x)))
- Signature:
{+/2,f/1,g/2} / {0/0,1/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {+,f,g} and constructors {0,1,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(+) = {1},
uargs(g) = {2},
uargs(s) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(+) = [1] x1 + [10]
p(0) = [0]
p(1) = [2]
p(f) = [8] x1 + [0]
p(g) = [7] x1 + [1] x2 + [0]
p(s) = [1] x1 + [3]

Following rules are strictly oriented:
+(x,0()) = [1] x + [10]
> [1] x + [0]
= x

f(s(x)) = [8] x + [24]
> [8] x + [3]
= g(x,s(x))

g(s(x),y) = [7] x + [1] y + [21]
> [7] x + [1] y + [10]
= g(x,+(y,s(x)))

g(s(x),y) = [7] x + [1] y + [21]
> [7] x + [1] y + [13]
= g(x,s(+(y,x)))

Following rules are (at-least) weakly oriented:
+(x,s(y)) =  [1] x + [10]
>= [1] x + [13]
=  s(+(x,y))

f(0()) =  [0]
>= [2]
=  1()

g(0(),y) =  [1] y + [0]
>= [1] y + [0]
=  y

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
+(x,s(y)) -> s(+(x,y))
f(0()) -> 1()
g(0(),y) -> y
- Weak TRS:
+(x,0()) -> x
f(s(x)) -> g(x,s(x))
g(s(x),y) -> g(x,+(y,s(x)))
g(s(x),y) -> g(x,s(+(y,x)))
- Signature:
{+/2,f/1,g/2} / {0/0,1/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {+,f,g} and constructors {0,1,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(+) = {1},
uargs(g) = {2},
uargs(s) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(+) = [1] x1 + [0]
p(0) = [0]
p(1) = [0]
p(f) = [1] x1 + [1]
p(g) = [1] x2 + [1]
p(s) = [1] x1 + [0]

Following rules are strictly oriented:
f(0()) = [1]
> [0]
= 1()

g(0(),y) = [1] y + [1]
> [1] y + [0]
= y

Following rules are (at-least) weakly oriented:
+(x,0()) =  [1] x + [0]
>= [1] x + [0]
=  x

+(x,s(y)) =  [1] x + [0]
>= [1] x + [0]
=  s(+(x,y))

f(s(x)) =  [1] x + [1]
>= [1] x + [1]
=  g(x,s(x))

g(s(x),y) =  [1] y + [1]
>= [1] y + [1]
=  g(x,+(y,s(x)))

g(s(x),y) =  [1] y + [1]
>= [1] y + [1]
=  g(x,s(+(y,x)))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
+(x,s(y)) -> s(+(x,y))
- Weak TRS:
+(x,0()) -> x
f(0()) -> 1()
f(s(x)) -> g(x,s(x))
g(0(),y) -> y
g(s(x),y) -> g(x,+(y,s(x)))
g(s(x),y) -> g(x,s(+(y,x)))
- Signature:
{+/2,f/1,g/2} / {0/0,1/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {+,f,g} and constructors {0,1,s}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(+) = {1},
uargs(g) = {2},
uargs(s) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(+) = x1 + 2*x2
p(0) = 0
p(1) = 0
p(f) = 4 + x1 + 5*x1^2
p(g) = 1 + 5*x1 + 5*x1^2 + 5*x2
p(s) = 1 + x1

Following rules are strictly oriented:
+(x,s(y)) = 2 + x + 2*y
> 1 + x + 2*y
= s(+(x,y))

Following rules are (at-least) weakly oriented:
+(x,0()) =  x
>= x
=  x

f(0()) =  4
>= 0
=  1()

f(s(x)) =  10 + 11*x + 5*x^2
>= 6 + 10*x + 5*x^2
=  g(x,s(x))

g(0(),y) =  1 + 5*y
>= y
=  y

g(s(x),y) =  11 + 15*x + 5*x^2 + 5*y
>= 11 + 15*x + 5*x^2 + 5*y
=  g(x,+(y,s(x)))

g(s(x),y) =  11 + 15*x + 5*x^2 + 5*y
>= 6 + 15*x + 5*x^2 + 5*y
=  g(x,s(+(y,x)))

* Step 4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
f(0()) -> 1()
f(s(x)) -> g(x,s(x))
g(0(),y) -> y
g(s(x),y) -> g(x,+(y,s(x)))
g(s(x),y) -> g(x,s(+(y,x)))
- Signature:
{+/2,f/1,g/2} / {0/0,1/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {+,f,g} and constructors {0,1,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))
```