* Step 1: Sum WORST_CASE(Omega(n^1),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(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
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:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ 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(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
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:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
app(y,z){y -> cons(x,y)} =
app(cons(x,y),z) ->^+ cons(x,app(y,z))
= C[app(y,z) = app(y,z){}]
** Step 1.b: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(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
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#(app(l,cons(x,cons(y,k)))) -> c_7(sum#(app(l,sum(cons(x,cons(y,k)))))
,app#(l,sum(cons(x,cons(y,k))))
,sum#(cons(x,cons(y,k))))
sum#(cons(x,cons(y,l))) -> c_8(sum#(cons(plus(x,y),l)),plus#(x,y))
sum#(cons(x,nil())) -> c_9()
sum#(plus(cons(0(),x),cons(y,l))) -> c_10(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 1.b: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#(app(l,cons(x,cons(y,k)))) -> c_7(sum#(app(l,sum(cons(x,cons(y,k)))))
,app#(l,sum(cons(x,cons(y,k))))
,sum#(cons(x,cons(y,k))))
sum#(cons(x,cons(y,l))) -> c_8(sum#(cons(plus(x,y),l)),plus#(x,y))
sum#(cons(x,nil())) -> c_9()
sum#(plus(cons(0(),x),cons(y,l))) -> c_10(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(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
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_7/3,c_8/2,c_9/0,c_10/2}
- 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_8(sum#(cons(plus(x,y),l)),plus#(x,y))
sum#(cons(x,nil())) -> c_9()
** Step 1.b: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_8(sum#(cons(plus(x,y),l)),plus#(x,y))
sum#(cons(x,nil())) -> c_9()
- 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_7/3,c_8/2,c_9/0,c_10/2}
- 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_8(sum#(cons(plus(x,y),l)),plus#(x,y))
8: sum#(cons(x,nil())) -> c_9()
** Step 1.b: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_8(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_9()
- 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_7/3,c_8/2,c_9/0,c_10/2}
- 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_8(sum#(cons(plus(x,y),l)),plus#(x,y))
-->_1 sum#(cons(x,nil())) -> c_9():8
-->_2 plus#(0(),y) -> c_4():6
-->_1 sum#(cons(x,cons(y,l))) -> c_8(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_9()
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_9()
6: plus#(0(),y) -> c_4()
4: app#(l,nil()) -> c_1()
5: app#(nil(),k) -> c_3()
** Step 1.b: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_8(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_7/3,c_8/2,c_9/0,c_10/2}
- 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_8(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_7/3,c_8/2,c_9/0,c_10/2}
- 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_8(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_7/3,c_8/2,c_9/0,c_10/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
*** Step 1.b: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_8(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_7/3,c_8/2,c_9/0,c_10/2}
- 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_8(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_8(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_8(sum#(cons(plus(x,y),l)),plus#(x,y))
2: plus#(s(x),y) -> c_5(plus#(x,y))
*** Step 1.b: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_7/3,c_8/2,c_9/0,c_10/2}
- 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 1.b: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_7/3,c_8/2,c_9/0,c_10/2}
- 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 1.b: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_7/3,c_8/2,c_9/0,c_10/2}
- 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) = [4]
p(app) = [1]
p(cons) = [1] x2 + [10]
p(nil) = [1]
p(plus) = [1] x1 + [1] x2 + [0]
p(pred) = [8] x1 + [1]
p(s) = [1]
p(sum) = [1]
p(app#) = [1] x1 + [8] x2 + [8]
p(plus#) = [1] x1 + [2]
p(pred#) = [1]
p(sum#) = [1] x1 + [1]
p(c_1) = [2]
p(c_2) = [1] x1 + [8]
p(c_3) = [1]
p(c_4) = [0]
p(c_5) = [1]
p(c_6) = [0]
p(c_7) = [1] x1 + [1] x2 + [2] x3 + [1]
p(c_8) = [2] x1 + [8] x2 + [1]
p(c_9) = [1]
p(c_10) = [4] x1 + [2] x2 + [0]
Following rules are strictly oriented:
app#(cons(x,l),k) = [8] k + [1] l + [18]
> [8] k + [1] l + [16]
= c_2(app#(l,k))
Following rules are (at-least) weakly oriented:
**** Step 1.b: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_7/3,c_8/2,c_9/0,c_10/2}
- 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 1.b: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_7/3,c_8/2,c_9/0,c_10/2}
- 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 1.b: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_7/3,c_8/2,c_9/0,c_10/2}
- 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 1.b: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_8(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_7/3,c_8/2,c_9/0,c_10/2}
- 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_8(sum#(cons(plus(x,y),l)),plus#(x,y))
-->_1 sum#(cons(x,cons(y,l))) -> c_8(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 1.b: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_8(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_7/3,c_8/2,c_9/0,c_10/2}
- 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_8(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_7/3,c_8/2,c_9/0,c_10/2}
- 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_8(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_7/3,c_8/2,c_9/0,c_10/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,plus#,pred#,sum#} and constructors {0,cons,nil,s}
**** Step 1.b: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_8(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_7/3,c_8/2,c_9/0,c_10/2}
- 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 1.b: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_8(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_7/3,c_8/2,c_9/0,c_10/2}
- 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_8) = {1,2}
Following symbols are considered usable:
{plus,app#,plus#,pred#,sum#}
TcT has computed the following interpretation:
p(0) = 0
p(app) = 2*x1 + x1*x2 + x1^2 + 2*x2
p(cons) = 1 + x1 + x2
p(nil) = 1
p(plus) = x1 + x2
p(pred) = 4 + 2*x1 + 4*x1^2
p(s) = 1 + x1
p(sum) = 1 + x1^2
p(app#) = 0
p(plus#) = 5 + x1 + 2*x2
p(pred#) = x1
p(sum#) = 7 + 2*x1^2
p(c_1) = 0
p(c_2) = 0
p(c_3) = 1
p(c_4) = 0
p(c_5) = x1
p(c_6) = 0
p(c_7) = x1
p(c_8) = 1 + x1 + x2
p(c_9) = 0
p(c_10) = 0
Following rules are strictly oriented:
plus#(s(x),y) = 6 + x + 2*y
> 5 + x + 2*y
= c_5(plus#(x,y))
Following rules are (at-least) weakly oriented:
sum#(cons(x,cons(y,l))) = 15 + 8*l + 4*l*x + 4*l*y + 2*l^2 + 8*x + 4*x*y + 2*x^2 + 8*y + 2*y^2
>= 15 + 4*l + 4*l*x + 4*l*y + 2*l^2 + 5*x + 4*x*y + 2*x^2 + 6*y + 2*y^2
= c_8(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 1.b: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_8(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_7/3,c_8/2,c_9/0,c_10/2}
- 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 1.b: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_8(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_7/3,c_8/2,c_9/0,c_10/2}
- 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_8(sum#(cons(plus(x,y),l)),plus#(x,y))
-->_1 sum#(cons(x,cons(y,l))) -> c_8(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_8(sum#(cons(plus(x,y),l)),plus#(x,y))
1: plus#(s(x),y) -> c_5(plus#(x,y))
***** Step 1.b: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_7/3,c_8/2,c_9/0,c_10/2}
- 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 1.b:5.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sum#(cons(x,cons(y,l))) -> c_8(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_7/3,c_8/2,c_9/0,c_10/2}
- 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_8(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_8(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 1.b:5.b:2.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sum#(cons(x,cons(y,l))) -> c_8(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_7/3,c_8/2,c_9/0,c_10/2}
- 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_8(sum#(cons(plus(x,y),l)),plus#(x,y))
-->_1 sum#(cons(x,cons(y,l))) -> c_8(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_8(sum#(cons(plus(x,y),l)))
**** Step 1.b:5.b:2.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sum#(cons(x,cons(y,l))) -> c_8(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_7/3,c_8/1,c_9/0,c_10/2}
- 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_8(sum#(cons(plus(x,y),l)))
The strictly oriented rules are moved into the weak component.
***** Step 1.b:5.b:2.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sum#(cons(x,cons(y,l))) -> c_8(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_7/3,c_8/1,c_9/0,c_10/2}
- 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_8) = {1}
Following symbols are considered usable:
{plus,app#,plus#,pred#,sum#}
TcT has computed the following interpretation:
p(0) = [1]
p(app) = [1]
p(cons) = [1] x1 + [1] x2 + [3]
p(nil) = [2]
p(plus) = [1] x1 + [1] x2 + [1]
p(pred) = [1] x1 + [4]
p(s) = [1] x1 + [8]
p(sum) = [2]
p(app#) = [1] x1 + [4] x2 + [0]
p(plus#) = [2] x2 + [0]
p(pred#) = [1]
p(sum#) = [4] x1 + [0]
p(c_1) = [1]
p(c_2) = [1] x1 + [2]
p(c_3) = [1]
p(c_4) = [1]
p(c_5) = [1] x1 + [1]
p(c_6) = [0]
p(c_7) = [1] x1 + [8] x2 + [1]
p(c_8) = [1] x1 + [4]
p(c_9) = [0]
p(c_10) = [1] x1 + [1] x2 + [4]
Following rules are strictly oriented:
sum#(cons(x,cons(y,l))) = [4] l + [4] x + [4] y + [24]
> [4] l + [4] x + [4] y + [20]
= c_8(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] x + [1] y + [9]
>= [1] x + [1] y + [9]
= s(plus(x,y))
***** Step 1.b:5.b:2.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
sum#(cons(x,cons(y,l))) -> c_8(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_7/3,c_8/1,c_9/0,c_10/2}
- 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 1.b:5.b:2.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
sum#(cons(x,cons(y,l))) -> c_8(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_7/3,c_8/1,c_9/0,c_10/2}
- 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_8(sum#(cons(plus(x,y),l)))
-->_1 sum#(cons(x,cons(y,l))) -> c_8(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_8(sum#(cons(plus(x,y),l)))
***** Step 1.b: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_7/3,c_8/1,c_9/0,c_10/2}
- 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(Omega(n^1),O(n^2))