```* Step 1: ToInnermost WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
half(0()) -> 0()
half(s(s(x))) -> s(half(x))
log(s(0())) -> 0()
log(s(s(x))) -> s(log(s(half(x))))
- Signature:
{half/1,log/1} / {0/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {half,log} and constructors {0,s}
+ Applied Processor:
ToInnermost
+ Details:
switch to innermost, as the system is overlay and right linear and does not contain weak rules
* Step 2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
half(0()) -> 0()
half(s(s(x))) -> s(half(x))
log(s(0())) -> 0()
log(s(s(x))) -> s(log(s(half(x))))
- Signature:
{half/1,log/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {half,log} 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(s) = {1}

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

Following rules are strictly oriented:
half(0()) = 
> 
= 0()

log(s(0())) = 
> 
= 0()

Following rules are (at-least) weakly oriented:
half(s(s(x))) =  
>= 
=  s(half(x))

log(s(s(x))) =   x + 
>= 
=  s(log(s(half(x))))

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

Following symbols are considered usable:
{half,log}
TcT has computed the following interpretation:
p(0) = 
p(half) =  x_1 + 
p(log) =  x_1 + 
p(s) =  x_1 + 

Following rules are strictly oriented:
half(s(s(x))) =  x + 
>  x + 
= s(half(x))

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

log(s(0())) =  
>= 
=  0()

log(s(s(x))) =   x + 
>=  x + 
=  s(log(s(half(x))))

* Step 4: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
log(s(s(x))) -> s(log(s(half(x))))
- Weak TRS:
half(0()) -> 0()
half(s(s(x))) -> s(half(x))
log(s(0())) -> 0()
- Signature:
{half/1,log/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {half,log} 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(s) = {1}

Following symbols are considered usable:
{half,log}
TcT has computed the following interpretation:
p(0) = 
p(half) =  x_1 + 
p(log) =  x_1 + 
p(s) =  x_1 + 

Following rules are strictly oriented:
log(s(s(x))) =  x + 
>  x + 
= s(log(s(half(x))))

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

half(s(s(x))) =   x + 
>=  x + 
=  s(half(x))

log(s(0())) =  
>= 
=  0()

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

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