```* Step 1: WeightGap WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
log(s(0())) -> 0()
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
min(X,0()) -> X
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
- Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,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(log) = {1},
uargs(min) = {1},
uargs(quot) = {1},
uargs(s) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(log) =  x1 + 
p(min) =  x1 + 
p(quot) =  x1 + 
p(s) =  x1 + 

Following rules are strictly oriented:
min(X,0()) =  X + 
>  X + 
= X

quot(0(),s(Y)) = 
> 
= 0()

Following rules are (at-least) weakly oriented:
log(s(0())) =  
>= 
=  0()

log(s(s(X))) =   X + 
>=  X + 
=  s(log(s(quot(X,s(s(0()))))))

min(s(X),s(Y)) =   X + 
>=  X + 
=  min(X,Y)

quot(s(X),s(Y)) =   X + 
>=  X + 
=  s(quot(min(X,Y),s(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:
log(s(0())) -> 0()
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
min(s(X),s(Y)) -> min(X,Y)
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
- Weak TRS:
min(X,0()) -> X
quot(0(),s(Y)) -> 0()
- Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,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(log) = {1},
uargs(min) = {1},
uargs(quot) = {1},
uargs(s) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(log) =  x1 + 
p(min) =  x1 + 
p(quot) =  x1 + 
p(s) =  x1 + 

Following rules are strictly oriented:
log(s(0())) = 
> 
= 0()

min(s(X),s(Y)) =  X + 
>  X + 
= min(X,Y)

Following rules are (at-least) weakly oriented:
log(s(s(X))) =   X + 
>=  X + 
=  s(log(s(quot(X,s(s(0()))))))

min(X,0()) =   X + 
>=  X + 
=  X

quot(0(),s(Y)) =  
>= 
=  0()

quot(s(X),s(Y)) =   X + 
>=  X + 
=  s(quot(min(X,Y),s(Y)))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
- Weak TRS:
log(s(0())) -> 0()
min(X,0()) -> X
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
- Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s}
+ 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(log) = {1},
uargs(min) = {1},
uargs(quot) = {1},
uargs(s) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(log) =  x_1 + 
p(min) =  x_1 + 
p(quot) =  x_1 + 
p(s) =  x_1 + 

Following rules are strictly oriented:
log(s(s(X))) =  X + 
>  X + 
= s(log(s(quot(X,s(s(0()))))))

Following rules are (at-least) weakly oriented:
log(s(0())) =  
>= 
=  0()

min(X,0()) =   X + 
>=  X + 
=  X

min(s(X),s(Y)) =   X + 
>=  X + 
=  min(X,Y)

quot(0(),s(Y)) =  
>= 
=  0()

quot(s(X),s(Y)) =   X + 
>=  X + 
=  s(quot(min(X,Y),s(Y)))

* Step 4: MI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
- Weak TRS:
log(s(0())) -> 0()
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
min(X,0()) -> X
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
- Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s}
+ 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(log) = {1},
uargs(min) = {1},
uargs(quot) = {1},
uargs(s) = {1}

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

p(log) = [2 0] x_1 + 
[0 1]       
p(min) = [1 0] x_1 + 
[0 1]       
p(quot) = [1 1] x_1 + 
[0 1]       
p(s) = [1 2] x_1 + 
[0 1]       

Following rules are strictly oriented:
quot(s(X),s(Y)) = [1 3] X + 
[0 1]     
> [1 3] X + 
[0 1]     
= s(quot(min(X,Y),s(Y)))

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

>= 

=  0()

log(s(s(X))) =  [2 8] X + 
[0 1]     
>= [2 8] X + 
[0 1]     
=  s(log(s(quot(X,s(s(0()))))))

min(X,0()) =  [1 0] X + 
[0 1]     
>= [1 0] X + 
[0 1]     
=  X

min(s(X),s(Y)) =  [1 2] X + 
[0 1]     
>= [1 0] X + 
[0 1]     
=  min(X,Y)

quot(0(),s(Y)) =  

>= 

=  0()

* Step 5: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
log(s(0())) -> 0()
log(s(s(X))) -> s(log(s(quot(X,s(s(0()))))))
min(X,0()) -> X
min(s(X),s(Y)) -> min(X,Y)
quot(0(),s(Y)) -> 0()
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
- Signature:
{log/1,min/2,quot/2} / {0/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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