```* Step 1: ToInnermost WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
U11(tt(),M,N) -> U12(tt(),activate(M),activate(N))
U12(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
plus(N,0()) -> N
plus(N,s(M)) -> U11(tt(),M,N)
- Signature:
{U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0}
- Obligation:
runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt}
+ Applied Processor:
ToInnermost
+ Details:
switch to innermost, as the system is overlay and right linear and does not contain weak rules
* Step 2: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
U11(tt(),M,N) -> U12(tt(),activate(M),activate(N))
U12(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
plus(N,0()) -> N
plus(N,s(M)) -> U11(tt(),M,N)
- Signature:
{U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:

Strict DPs
U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
activate#(X) -> c_3()
plus#(N,0()) -> c_4()
plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
Weak DPs

and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
activate#(X) -> c_3()
plus#(N,0()) -> c_4()
plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
- Strict TRS:
U11(tt(),M,N) -> U12(tt(),activate(M),activate(N))
U12(tt(),M,N) -> s(plus(activate(N),activate(M)))
activate(X) -> X
plus(N,0()) -> N
plus(N,s(M)) -> U11(tt(),M,N)
- Signature:
{U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0
,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
activate(X) -> X
U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
activate#(X) -> c_3()
plus#(N,0()) -> c_4()
plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
* Step 4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
activate#(X) -> c_3()
plus#(N,0()) -> c_4()
plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
- Strict TRS:
activate(X) -> X
- Signature:
{U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0
,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt}
+ 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(U12#) = {2,3},
uargs(plus#) = {1,2},
uargs(c_1) = {1},
uargs(c_2) = {1},
uargs(c_5) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 
p(U11) = 
p(U12) = 
p(activate) =  x1 + 
p(plus) = 
p(s) =  x1 + 
p(tt) = 
p(U11#) =  x1 +  x2 +  x3 + 
p(U12#) =  x2 +  x3 + 
p(activate#) =  x1 + 
p(plus#) =  x1 +  x2 + 
p(c_1) =  x1 + 
p(c_2) =  x1 + 
p(c_3) = 
p(c_4) = 
p(c_5) =  x1 + 

Following rules are strictly oriented:
U11#(tt(),M,N) =  M +  N + 
>  M +  N + 
= c_1(U12#(tt(),activate(M),activate(N)))

activate#(X) =  X + 
> 
= c_3()

activate(X) =  X + 
>  X + 
= X

Following rules are (at-least) weakly oriented:
U12#(tt(),M,N) =   M +  N + 
>=  M +  N + 
=  c_2(plus#(activate(N),activate(M)))

plus#(N,0()) =   N + 
>= 
=  c_4()

plus#(N,s(M)) =   M +  N + 
>=  M +  N + 
=  c_5(U11#(tt(),M,N))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
plus#(N,0()) -> c_4()
plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
- Weak DPs:
U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
activate#(X) -> c_3()
- Weak TRS:
activate(X) -> X
- Signature:
{U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0
,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{2,3}
by application of
Pre({2,3}) = {1}.
Here rules are labelled as follows:
1: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
2: plus#(N,0()) -> c_4()
3: plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
4: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
5: activate#(X) -> c_3()
* Step 6: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
- Weak DPs:
U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
activate#(X) -> c_3()
plus#(N,0()) -> c_4()
plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
- Weak TRS:
activate(X) -> X
- Signature:
{U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0
,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
-->_1 plus#(N,s(M)) -> c_5(U11#(tt(),M,N)):5
-->_1 plus#(N,0()) -> c_4():4

2:W:U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
-->_1 U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))):1

3:W:activate#(X) -> c_3()

4:W:plus#(N,0()) -> c_4()

5:W:plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
-->_1 U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))):2

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: activate#(X) -> c_3()
4: plus#(N,0()) -> c_4()
* Step 7: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
- Weak DPs:
U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
- Weak TRS:
activate(X) -> X
- Signature:
{U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0
,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt}
+ 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: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))

Consider the set of all dependency pairs
1: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
2: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
5: plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
SPACE(?,?)on application of the dependency pairs
{1}
These cover all (indirect) predecessors of dependency pairs
{1,2,5}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
** Step 7.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
- Weak DPs:
U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
- Weak TRS:
activate(X) -> X
- Signature:
{U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0
,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt}
+ 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_1) = {1},
uargs(c_2) = {1},
uargs(c_5) = {1}

Following symbols are considered usable:
{activate,U11#,U12#,activate#,plus#}
TcT has computed the following interpretation:
p(0) = 
p(U11) =  x1 + 
p(U12) =  x1 +  x3 + 
p(activate) =  x1 + 
p(plus) =  x1 +  x2 + 
p(s) =  x1 + 
p(tt) = 
p(U11#) =  x1 +  x2 +  x3 + 
p(U12#) =  x1 +  x2 +  x3 + 
p(activate#) = 
p(plus#) =  x1 +  x2 + 
p(c_1) =  x1 + 
p(c_2) =  x1 + 
p(c_3) = 
p(c_4) = 
p(c_5) =  x1 + 

Following rules are strictly oriented:
U12#(tt(),M,N) =  M +  N + 
>  M +  N + 
= c_2(plus#(activate(N),activate(M)))

Following rules are (at-least) weakly oriented:
U11#(tt(),M,N) =   M +  N + 
>=  M +  N + 
=  c_1(U12#(tt(),activate(M),activate(N)))

plus#(N,s(M)) =   M +  N + 
>=  M +  N + 
=  c_5(U11#(tt(),M,N))

activate(X) =   X + 
>=  X + 
=  X

** Step 7.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
- Weak TRS:
activate(X) -> X
- Signature:
{U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0
,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

** Step 7.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
- Weak TRS:
activate(X) -> X
- Signature:
{U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0
,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
-->_1 U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))):2

2:W:U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
-->_1 plus#(N,s(M)) -> c_5(U11#(tt(),M,N)):3

3:W:plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
-->_1 U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
3: plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
2: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
** Step 7.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
activate(X) -> X
- Signature:
{U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0
,c_5/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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