```* Step 1: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
f(0()) -> true()
f(1()) -> false()
f(s(x)) -> f(x)
g(x,c(y)) -> g(x,g(s(c(y)),y))
g(s(x),s(y)) -> if(f(x),s(x),s(y))
if(false(),x,y) -> y
if(true(),x,y) -> x
- Signature:
{f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):

The following argument positions are considered usable:
uargs(g) = {2},
uargs(if) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(1) = 
p(c) =  x_1 + 
p(f) = 
p(false) = 
p(g) =  x_2 + 
p(if) =  x_1 +  x_2 +  x_3 + 
p(s) = 
p(true) = 

Following rules are strictly oriented:
if(false(),x,y) =  x +  y + 
>  y + 
= y

if(true(),x,y) =  x +  y + 
>  x + 
= x

Following rules are (at-least) weakly oriented:
f(0()) =  
>= 
=  true()

f(1()) =  
>= 
=  false()

f(s(x)) =  
>= 
=  f(x)

g(x,c(y)) =   y + 
>=  y + 
=  g(x,g(s(c(y)),y))

g(s(x),s(y)) =  
>= 
=  if(f(x),s(x),s(y))

* Step 2: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
f(0()) -> true()
f(1()) -> false()
f(s(x)) -> f(x)
g(x,c(y)) -> g(x,g(s(c(y)),y))
g(s(x),s(y)) -> if(f(x),s(x),s(y))
- Weak TRS:
if(false(),x,y) -> y
if(true(),x,y) -> x
- Signature:
{f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true}
+ 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(g) = {2},
uargs(if) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(1) = 
p(c) =  x1 + 
p(f) = 
p(false) = 
p(g) =  x2 + 
p(if) =  x1 +  x2 +  x3 + 
p(s) = 
p(true) = 

Following rules are strictly oriented:
f(0()) = 
> 
= true()

Following rules are (at-least) weakly oriented:
f(1()) =  
>= 
=  false()

f(s(x)) =  
>= 
=  f(x)

g(x,c(y)) =   y + 
>=  y + 
=  g(x,g(s(c(y)),y))

g(s(x),s(y)) =  
>= 
=  if(f(x),s(x),s(y))

if(false(),x,y) =   x +  y + 
>=  y + 
=  y

if(true(),x,y) =   x +  y + 
>=  x + 
=  x

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
f(1()) -> false()
f(s(x)) -> f(x)
g(x,c(y)) -> g(x,g(s(c(y)),y))
g(s(x),s(y)) -> if(f(x),s(x),s(y))
- Weak TRS:
f(0()) -> true()
if(false(),x,y) -> y
if(true(),x,y) -> x
- Signature:
{f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true}
+ 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(g) = {2},
uargs(if) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(1) = 
p(c) =  x1 + 
p(f) = 
p(false) = 
p(g) =  x1 +  x2 + 
p(if) =  x1 +  x2 +  x3 + 
p(s) = 
p(true) = 

Following rules are strictly oriented:
f(1()) = 
> 
= false()

Following rules are (at-least) weakly oriented:
f(0()) =  
>= 
=  true()

f(s(x)) =  
>= 
=  f(x)

g(x,c(y)) =   x +  y + 
>=  x +  y + 
=  g(x,g(s(c(y)),y))

g(s(x),s(y)) =  
>= 
=  if(f(x),s(x),s(y))

if(false(),x,y) =   x +  y + 
>=  y + 
=  y

if(true(),x,y) =   x +  y + 
>=  x + 
=  x

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
f(s(x)) -> f(x)
g(x,c(y)) -> g(x,g(s(c(y)),y))
g(s(x),s(y)) -> if(f(x),s(x),s(y))
- Weak TRS:
f(0()) -> true()
f(1()) -> false()
if(false(),x,y) -> y
if(true(),x,y) -> x
- Signature:
{f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):

The following argument positions are considered usable:
uargs(g) = {2},
uargs(if) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(1) = 
p(c) =  x_1 + 
p(f) = 
p(false) = 
p(g) =  x_2 + 
p(if) =  x_1 +  x_2 +  x_3 + 
p(s) = 
p(true) = 

Following rules are strictly oriented:
g(x,c(y)) =  y + 
>  y + 
= g(x,g(s(c(y)),y))

Following rules are (at-least) weakly oriented:
f(0()) =  
>= 
=  true()

f(1()) =  
>= 
=  false()

f(s(x)) =  
>= 
=  f(x)

g(s(x),s(y)) =  
>= 
=  if(f(x),s(x),s(y))

if(false(),x,y) =   x +  y + 
>=  y + 
=  y

if(true(),x,y) =   x +  y + 
>=  x + 
=  x

* Step 5: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
f(s(x)) -> f(x)
g(s(x),s(y)) -> if(f(x),s(x),s(y))
- Weak TRS:
f(0()) -> true()
f(1()) -> false()
g(x,c(y)) -> g(x,g(s(c(y)),y))
if(false(),x,y) -> y
if(true(),x,y) -> x
- Signature:
{f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):

The following argument positions are considered usable:
uargs(g) = {2},
uargs(if) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(1) = 
p(c) =  x_1 + 
p(f) = 
p(false) = 
p(g) =  x_1 +  x_2 + 
p(if) =  x_1 +  x_2 +  x_3 + 
p(s) = 
p(true) = 

Following rules are strictly oriented:
g(s(x),s(y)) = 
> 
= if(f(x),s(x),s(y))

Following rules are (at-least) weakly oriented:
f(0()) =  
>= 
=  true()

f(1()) =  
>= 
=  false()

f(s(x)) =  
>= 
=  f(x)

g(x,c(y)) =   x +  y + 
>=  x +  y + 
=  g(x,g(s(c(y)),y))

if(false(),x,y) =   x +  y + 
>=  y + 
=  y

if(true(),x,y) =   x +  y + 
>=  x + 
=  x

* Step 6: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
f(s(x)) -> f(x)
- Weak TRS:
f(0()) -> true()
f(1()) -> false()
g(x,c(y)) -> g(x,g(s(c(y)),y))
g(s(x),s(y)) -> if(f(x),s(x),s(y))
if(false(),x,y) -> y
if(true(),x,y) -> x
- Signature:
{f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):

The following argument positions are considered usable:
uargs(g) = {2},
uargs(if) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 

p(1) = 

p(c) = [1 0] x_1 + 
[0 0]       
p(f) = [0 2] x_1 + 
[0 1]       
p(false) = 

p(g) = [3 0] x_1 + [1 0] x_2 + 
[4 3]       [4 0]       
p(if) = [2 0] x_1 + [1 0] x_2 + [1 0] x_3 + 
[4 1]       [0 1]       [0 2]       
p(s) = [0 2] x_1 + 
[0 1]       
p(true) = 


Following rules are strictly oriented:
f(s(x)) = [0 2] x + 
[0 1]     
> [0 2] x + 
[0 1]     
= f(x)

Following rules are (at-least) weakly oriented:
f(0()) =  

>= 

=  true()

f(1()) =  

>= 

=  false()

g(x,c(y)) =  [3 0] x + [1 0] y + 
[4 3]     [4 0]     
>= [3 0] x + [1 0] y + 
[4 3]     [4 0]     
=  g(x,g(s(c(y)),y))

g(s(x),s(y)) =  [0  6] x + [0 2] y + 
[0 11]     [0 8]     
>= [0  6] x + [0 2] y + 
[0 10]     [0 2]     
=  if(f(x),s(x),s(y))

if(false(),x,y) =  [1 0] x + [1 0] y + 
[0 1]     [0 2]     
>= [1 0] y + 
[0 1]     
=  y

if(true(),x,y) =  [1 0] x + [1 0] y + 
[0 1]     [0 2]     
>= [1 0] x + 
[0 1]     
=  x

* Step 7: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
f(0()) -> true()
f(1()) -> false()
f(s(x)) -> f(x)
g(x,c(y)) -> g(x,g(s(c(y)),y))
g(s(x),s(y)) -> if(f(x),s(x),s(y))
if(false(),x,y) -> y
if(true(),x,y) -> x
- Signature:
{f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0}
- Obligation:
runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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