```* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
app(l,nil()) -> l
app(cons(x,l),k) -> cons(x,app(l,k))
app(nil(),k) -> k
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
pred(cons(s(x),nil())) -> cons(x,nil())
sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
sum(cons(x,nil())) -> cons(x,nil())
sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
- Signature:
{app/2,plus/2,pred/1,sum/1} / {0/0,cons/2,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {app,plus,pred,sum} and constructors {0,cons,nil,s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:

Strict DPs
app#(l,nil()) -> c_1()
app#(cons(x,l),k) -> c_2(app#(l,k))
app#(nil(),k) -> c_3()
plus#(0(),y) -> c_4()
plus#(s(x),y) -> c_5(plus#(x,y))
pred#(cons(s(x),nil())) -> c_6()
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
sum#(cons(x,nil())) -> c_8()
sum#(plus(cons(0(),x),cons(y,l))) -> c_9(pred#(sum(cons(s(x),cons(y,l)))),sum#(cons(s(x),cons(y,l))))
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
app#(l,nil()) -> c_1()
app#(cons(x,l),k) -> c_2(app#(l,k))
app#(nil(),k) -> c_3()
plus#(0(),y) -> c_4()
plus#(s(x),y) -> c_5(plus#(x,y))
pred#(cons(s(x),nil())) -> c_6()
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
sum#(cons(x,nil())) -> c_8()
sum#(plus(cons(0(),x),cons(y,l))) -> c_9(pred#(sum(cons(s(x),cons(y,l)))),sum#(cons(s(x),cons(y,l))))
- Weak TRS:
app(l,nil()) -> l
app(cons(x,l),k) -> cons(x,app(l,k))
app(nil(),k) -> k
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
pred(cons(s(x),nil())) -> cons(x,nil())
sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
sum(cons(x,nil())) -> cons(x,nil())
sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
app#(l,nil()) -> c_1()
app#(cons(x,l),k) -> c_2(app#(l,k))
app#(nil(),k) -> c_3()
plus#(0(),y) -> c_4()
plus#(s(x),y) -> c_5(plus#(x,y))
pred#(cons(s(x),nil())) -> c_6()
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
sum#(cons(x,nil())) -> c_8()
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
app#(l,nil()) -> c_1()
app#(cons(x,l),k) -> c_2(app#(l,k))
app#(nil(),k) -> c_3()
plus#(0(),y) -> c_4()
plus#(s(x),y) -> c_5(plus#(x,y))
pred#(cons(s(x),nil())) -> c_6()
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
sum#(cons(x,nil())) -> c_8()
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,3,4,6,8}
by application of
Pre({1,3,4,6,8}) = {2,5,7}.
Here rules are labelled as follows:
1: app#(l,nil()) -> c_1()
2: app#(cons(x,l),k) -> c_2(app#(l,k))
3: app#(nil(),k) -> c_3()
4: plus#(0(),y) -> c_4()
5: plus#(s(x),y) -> c_5(plus#(x,y))
6: pred#(cons(s(x),nil())) -> c_6()
7: sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
8: sum#(cons(x,nil())) -> c_8()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
app#(cons(x,l),k) -> c_2(app#(l,k))
plus#(s(x),y) -> c_5(plus#(x,y))
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
- Weak DPs:
app#(l,nil()) -> c_1()
app#(nil(),k) -> c_3()
plus#(0(),y) -> c_4()
pred#(cons(s(x),nil())) -> c_6()
sum#(cons(x,nil())) -> c_8()
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:app#(cons(x,l),k) -> c_2(app#(l,k))
-->_1 app#(nil(),k) -> c_3():5
-->_1 app#(l,nil()) -> c_1():4
-->_1 app#(cons(x,l),k) -> c_2(app#(l,k)):1

2:S:plus#(s(x),y) -> c_5(plus#(x,y))
-->_1 plus#(0(),y) -> c_4():6
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):2

3:S:sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
-->_1 sum#(cons(x,nil())) -> c_8():8
-->_2 plus#(0(),y) -> c_4():6
-->_1 sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y)):3
-->_2 plus#(s(x),y) -> c_5(plus#(x,y)):2

4:W:app#(l,nil()) -> c_1()

5:W:app#(nil(),k) -> c_3()

6:W:plus#(0(),y) -> c_4()

7:W:pred#(cons(s(x),nil())) -> c_6()

8:W:sum#(cons(x,nil())) -> c_8()

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
7: pred#(cons(s(x),nil())) -> c_6()
8: sum#(cons(x,nil())) -> c_8()
6: plus#(0(),y) -> c_4()
4: app#(l,nil()) -> c_1()
5: app#(nil(),k) -> c_3()
* Step 5: Decompose WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
app#(cons(x,l),k) -> c_2(app#(l,k))
plus#(s(x),y) -> c_5(plus#(x,y))
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,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:
app#(cons(x,l),k) -> c_2(app#(l,k))
- Weak DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}

Problem (S)
- Strict DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
- Weak DPs:
app#(cons(x,l),k) -> c_2(app#(l,k))
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
** Step 5.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
app#(cons(x,l),k) -> c_2(app#(l,k))
- Weak DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:app#(cons(x,l),k) -> c_2(app#(l,k))
-->_1 app#(cons(x,l),k) -> c_2(app#(l,k)):1

2:W:plus#(s(x),y) -> c_5(plus#(x,y))
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):2

3:W:sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
-->_2 plus#(s(x),y) -> c_5(plus#(x,y)):2
-->_1 sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y)):3

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
2: plus#(s(x),y) -> c_5(plus#(x,y))
** Step 5.a:2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
app#(cons(x,l),k) -> c_2(app#(l,k))
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
app#(cons(x,l),k) -> c_2(app#(l,k))
** Step 5.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
app#(cons(x,l),k) -> c_2(app#(l,k))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,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: app#(cons(x,l),k) -> c_2(app#(l,k))

The strictly oriented rules are moved into the weak component.
*** Step 5.a:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
app#(cons(x,l),k) -> c_2(app#(l,k))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,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_2) = {1}

Following symbols are considered usable:
{app#,plus#,pred#,sum#}
TcT has computed the following interpretation:
p(0) = [0]
p(app) = [0]
p(cons) = [1] x2 + [8]
p(nil) = [4]
p(plus) = [2] x2 + [1]
p(pred) = [2] x1 + [1]
p(s) = [1]
p(sum) = [1] x1 + [0]
p(app#) = [2] x1 + [6] x2 + [0]
p(plus#) = [4] x1 + [1] x2 + [0]
p(pred#) = [2] x1 + [2]
p(sum#) = [0]
p(c_1) = [4]
p(c_2) = [1] x1 + [8]
p(c_3) = [2]
p(c_4) = [4]
p(c_5) = [0]
p(c_6) = [2]
p(c_7) = [1] x1 + [1]
p(c_8) = [2]
p(c_9) = [0]

Following rules are strictly oriented:
app#(cons(x,l),k) = [6] k + [2] l + [16]
> [6] k + [2] l + [8]
= c_2(app#(l,k))

Following rules are (at-least) weakly oriented:

*** Step 5.a:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
app#(cons(x,l),k) -> c_2(app#(l,k))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

*** Step 5.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
app#(cons(x,l),k) -> c_2(app#(l,k))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:app#(cons(x,l),k) -> c_2(app#(l,k))
-->_1 app#(cons(x,l),k) -> c_2(app#(l,k)):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: app#(cons(x,l),k) -> c_2(app#(l,k))
*** Step 5.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
- Weak DPs:
app#(cons(x,l),k) -> c_2(app#(l,k))
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:plus#(s(x),y) -> c_5(plus#(x,y))
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):1

2:S:sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
-->_1 sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y)):2
-->_2 plus#(s(x),y) -> c_5(plus#(x,y)):1

3:W:app#(cons(x,l),k) -> c_2(app#(l,k))
-->_1 app#(cons(x,l),k) -> c_2(app#(l,k)):3

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: app#(cons(x,l),k) -> c_2(app#(l,k))
** Step 5.b:2: Decompose WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,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:
plus#(s(x),y) -> c_5(plus#(x,y))
- Weak DPs:
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}

Problem (S)
- Strict DPs:
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
- Weak DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
*** Step 5.b:2.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
- Weak DPs:
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: plus#(s(x),y) -> c_5(plus#(x,y))

The strictly oriented rules are moved into the weak component.
**** Step 5.b:2.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
- Weak DPs:
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_5) = {1},
uargs(c_7) = {1,2}

Following symbols are considered usable:
{plus,app#,plus#,pred#,sum#}
TcT has computed the following interpretation:
p(0) = 0
p(app) = 4 + 2*x1 + x1*x2 + 2*x1^2 + x2^2
p(cons) = 1 + x1 + x2
p(nil) = 1
p(plus) = x1 + x2
p(pred) = 0
p(s) = 1 + x1
p(sum) = 2 + x1^2
p(app#) = 1 + x1*x2 + 4*x2
p(plus#) = x1
p(pred#) = 2*x1 + x1^2
p(sum#) = x1^2
p(c_1) = 1
p(c_2) = 0
p(c_3) = 0
p(c_4) = 1
p(c_5) = x1
p(c_6) = 1
p(c_7) = x1 + x2
p(c_8) = 0
p(c_9) = x1 + x2

Following rules are strictly oriented:
plus#(s(x),y) = 1 + x
> x
= c_5(plus#(x,y))

Following rules are (at-least) weakly oriented:
sum#(cons(x,cons(y,l))) =  4 + 4*l + 2*l*x + 2*l*y + l^2 + 4*x + 2*x*y + x^2 + 4*y + y^2
>= 1 + 2*l + 2*l*x + 2*l*y + l^2 + 3*x + 2*x*y + x^2 + 2*y + y^2
=  c_7(sum#(cons(plus(x,y),l)),plus#(x,y))

plus(0(),y) =  y
>= y
=  y

plus(s(x),y) =  1 + x + y
>= 1 + x + y
=  s(plus(x,y))

**** Step 5.b:2.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

**** Step 5.b:2.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:plus#(s(x),y) -> c_5(plus#(x,y))
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):1

2:W:sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
-->_1 sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y)):2
-->_2 plus#(s(x),y) -> c_5(plus#(x,y)):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
1: plus#(s(x),y) -> c_5(plus#(x,y))
**** Step 5.b:2.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

*** Step 5.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
- Weak DPs:
plus#(s(x),y) -> c_5(plus#(x,y))
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
-->_2 plus#(s(x),y) -> c_5(plus#(x,y)):2
-->_1 sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y)):1

2:W:plus#(s(x),y) -> c_5(plus#(x,y))
-->_1 plus#(s(x),y) -> c_5(plus#(x,y)):2

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: plus#(s(x),y) -> c_5(plus#(x,y))
*** Step 5.b:2.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y))
-->_1 sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)),plus#(x,y)):1

Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)))
*** Step 5.b:2.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)))
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,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: sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)))

The strictly oriented rules are moved into the weak component.
**** Step 5.b:2.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)))
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,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_7) = {1}

Following symbols are considered usable:
{plus,app#,plus#,pred#,sum#}
TcT has computed the following interpretation:
p(0) = [1]
p(app) = [0]
p(cons) = [1] x1 + [1] x2 + [3]
p(nil) = [2]
p(plus) = [1] x1 + [1] x2 + [1]
p(pred) = [4] x1 + [4]
p(s) = [0]
p(sum) = [1] x1 + [4]
p(app#) = [2] x1 + [4] x2 + [1]
p(plus#) = [0]
p(pred#) = [2] x1 + [1]
p(sum#) = [1] x1 + [11]
p(c_1) = [0]
p(c_2) = [1] x1 + [0]
p(c_3) = [0]
p(c_4) = [1]
p(c_5) = [1] x1 + [2]
p(c_6) = [1]
p(c_7) = [1] x1 + [0]
p(c_8) = [1]
p(c_9) = [2] x1 + [0]

Following rules are strictly oriented:
sum#(cons(x,cons(y,l))) = [1] l + [1] x + [1] y + [17]
> [1] l + [1] x + [1] y + [15]
= c_7(sum#(cons(plus(x,y),l)))

Following rules are (at-least) weakly oriented:
plus(0(),y) =  [1] y + [2]
>= [1] y + [0]
=  y

plus(s(x),y) =  [1] y + [1]
>= [0]
=  s(plus(x,y))

**** Step 5.b:2.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)))
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

**** Step 5.b:2.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)))
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)))
-->_1 sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l))):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: sum#(cons(x,cons(y,l))) -> c_7(sum#(cons(plus(x,y),l)))
**** Step 5.b:2.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
plus(0(),y) -> y
plus(s(x),y) -> s(plus(x,y))
- Signature:
{app/2,plus/2,pred/1,sum/1,app#/2,plus#/2,pred#/1,sum#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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