```* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
bits(0()) -> 0()
bits(s(x)) -> s(bits(half(s(x))))
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{bits/1,half/1} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {bits,half} and constructors {0,s}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:

Strict DPs
bits#(0()) -> c_1()
bits#(s(x)) -> c_2(bits#(half(s(x))))
half#(0()) -> c_3()
half#(s(0())) -> c_4()
half#(s(s(x))) -> c_5(half#(x))
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
bits#(0()) -> c_1()
bits#(s(x)) -> c_2(bits#(half(s(x))))
half#(0()) -> c_3()
half#(s(0())) -> c_4()
half#(s(s(x))) -> c_5(half#(x))
- Strict TRS:
bits(0()) -> 0()
bits(s(x)) -> s(bits(half(s(x))))
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
bits#(0()) -> c_1()
bits#(s(x)) -> c_2(bits#(half(s(x))))
half#(0()) -> c_3()
half#(s(0())) -> c_4()
half#(s(s(x))) -> c_5(half#(x))
* Step 3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
bits#(0()) -> c_1()
bits#(s(x)) -> c_2(bits#(half(s(x))))
half#(0()) -> c_3()
half#(s(0())) -> c_4()
half#(s(s(x))) -> c_5(half#(x))
- Strict TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(s) = {1},
uargs(bits#) = {1},
uargs(c_2) = {1},
uargs(c_5) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(bits) = 
p(half) =  x1 + 
p(s) =  x1 + 
p(bits#) =  x1 + 
p(half#) = 
p(c_1) = 
p(c_2) =  x1 + 
p(c_3) = 
p(c_4) = 
p(c_5) =  x1 + 

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

half(s(0())) = 
> 
= 0()

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

Following rules are (at-least) weakly oriented:
bits#(0()) =  
>= 
=  c_1()

bits#(s(x)) =   x + 
>=  x + 
=  c_2(bits#(half(s(x))))

half#(0()) =  
>= 
=  c_3()

half#(s(0())) =  
>= 
=  c_4()

half#(s(s(x))) =  
>= 
=  c_5(half#(x))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
bits#(0()) -> c_1()
bits#(s(x)) -> c_2(bits#(half(s(x))))
half#(0()) -> c_3()
half#(s(0())) -> c_4()
half#(s(s(x))) -> c_5(half#(x))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,3,4}
by application of
Pre({1,3,4}) = {2,5}.
Here rules are labelled as follows:
1: bits#(0()) -> c_1()
2: bits#(s(x)) -> c_2(bits#(half(s(x))))
3: half#(0()) -> c_3()
4: half#(s(0())) -> c_4()
5: half#(s(s(x))) -> c_5(half#(x))
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
bits#(s(x)) -> c_2(bits#(half(s(x))))
half#(s(s(x))) -> c_5(half#(x))
- Weak DPs:
bits#(0()) -> c_1()
half#(0()) -> c_3()
half#(s(0())) -> c_4()
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:bits#(s(x)) -> c_2(bits#(half(s(x))))
-->_1 bits#(0()) -> c_1():3
-->_1 bits#(s(x)) -> c_2(bits#(half(s(x)))):1

2:S:half#(s(s(x))) -> c_5(half#(x))
-->_1 half#(s(0())) -> c_4():5
-->_1 half#(0()) -> c_3():4
-->_1 half#(s(s(x))) -> c_5(half#(x)):2

3:W:bits#(0()) -> c_1()

4:W:half#(0()) -> c_3()

5:W:half#(s(0())) -> c_4()

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: half#(0()) -> c_3()
5: half#(s(0())) -> c_4()
3: bits#(0()) -> c_1()
* Step 6: Decompose WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
bits#(s(x)) -> c_2(bits#(half(s(x))))
half#(s(s(x))) -> c_5(half#(x))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.

Problem (R)
- Strict DPs:
bits#(s(x)) -> c_2(bits#(half(s(x))))
- Weak DPs:
half#(s(s(x))) -> c_5(half#(x))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}

Problem (S)
- Strict DPs:
half#(s(s(x))) -> c_5(half#(x))
- Weak DPs:
bits#(s(x)) -> c_2(bits#(half(s(x))))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
** Step 6.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
bits#(s(x)) -> c_2(bits#(half(s(x))))
- Weak DPs:
half#(s(s(x))) -> c_5(half#(x))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:bits#(s(x)) -> c_2(bits#(half(s(x))))
-->_1 bits#(s(x)) -> c_2(bits#(half(s(x)))):1

2:W:half#(s(s(x))) -> c_5(half#(x))
-->_1 half#(s(s(x))) -> c_5(half#(x)):2

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: half#(s(s(x))) -> c_5(half#(x))
** Step 6.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
bits#(s(x)) -> c_2(bits#(half(s(x))))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: bits#(s(x)) -> c_2(bits#(half(s(x))))

The strictly oriented rules are moved into the weak component.
*** Step 6.a:2.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
bits#(s(x)) -> c_2(bits#(half(s(x))))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
+ Applied Processor:
NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_2) = {1}

Following symbols are considered usable:
{half,bits#,half#}
TcT has computed the following interpretation:
p(0) = 

p(bits) = [0 1] x1 + 
[1 1]      
p(half) = [0 1] x1 + 
[0 1]      
p(s) = [0 1] x1 + 
[0 1]      
p(bits#) = [2 0] x1 + 
[0 2]      
p(half#) = 

p(c_1) = 

p(c_2) = [1 0] x1 + 
[0 0]      
p(c_3) = 

p(c_4) = 

p(c_5) = [0 1] x1 + 
[0 1]      

Following rules are strictly oriented:
bits#(s(x)) = [0 2] x + 
[0 2]     
> [0 2] x + 
[0 0]     
= c_2(bits#(half(s(x))))

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

>= 

=  0()

half(s(0())) =  

>= 

=  0()

half(s(s(x))) =  [0 1] x + 
[0 1]     
>= [0 1] x + 
[0 1]     
=  s(half(x))

*** Step 6.a:2.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
bits#(s(x)) -> c_2(bits#(half(s(x))))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

*** Step 6.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
bits#(s(x)) -> c_2(bits#(half(s(x))))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:bits#(s(x)) -> c_2(bits#(half(s(x))))
-->_1 bits#(s(x)) -> c_2(bits#(half(s(x)))):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: bits#(s(x)) -> c_2(bits#(half(s(x))))
*** Step 6.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
half#(s(s(x))) -> c_5(half#(x))
- Weak DPs:
bits#(s(x)) -> c_2(bits#(half(s(x))))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:half#(s(s(x))) -> c_5(half#(x))
-->_1 half#(s(s(x))) -> c_5(half#(x)):1

2:W:bits#(s(x)) -> c_2(bits#(half(s(x))))
-->_1 bits#(s(x)) -> c_2(bits#(half(s(x)))):2

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: bits#(s(x)) -> c_2(bits#(half(s(x))))
** Step 6.b:2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
half#(s(s(x))) -> c_5(half#(x))
- Weak TRS:
half(0()) -> 0()
half(s(0())) -> 0()
half(s(s(x))) -> s(half(x))
- Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
half#(s(s(x))) -> c_5(half#(x))
** Step 6.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
half#(s(s(x))) -> c_5(half#(x))
- Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: half#(s(s(x))) -> c_5(half#(x))

The strictly oriented rules are moved into the weak component.
*** Step 6.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
half#(s(s(x))) -> c_5(half#(x))
- Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_5) = {1}

Following symbols are considered usable:
{bits#,half#}
TcT has computed the following interpretation:
p(0) = 
p(bits) = 
p(half) = 
p(s) =  x1 + 
p(bits#) = 
p(half#) =  x1 + 
p(c_1) = 
p(c_2) =  x1 + 
p(c_3) = 
p(c_4) = 
p(c_5) =  x1 + 

Following rules are strictly oriented:
half#(s(s(x))) =  x + 
>  x + 
= c_5(half#(x))

Following rules are (at-least) weakly oriented:

*** Step 6.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
half#(s(s(x))) -> c_5(half#(x))
- Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

*** Step 6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
half#(s(s(x))) -> c_5(half#(x))
- Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:half#(s(s(x))) -> c_5(half#(x))
-->_1 half#(s(s(x))) -> c_5(half#(x)):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: half#(s(s(x))) -> c_5(half#(x))
*** Step 6.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{bits/1,half/1,bits#/1,half#/1} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {bits#,half#} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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