WORST_CASE(?,O(n^2))
* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
and(tt(),X) -> activate(X)
plus(N,0()) -> N
plus(N,s(M)) -> s(plus(N,M))
x(N,0()) -> 0()
x(N,s(M)) -> plus(x(N,M),N)
- Signature:
{activate/1,and/2,plus/2,x/2} / {0/0,s/1,tt/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,and,plus,x} and constructors {0,s,tt}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
activate#(X) -> c_1()
and#(tt(),X) -> c_2(activate#(X))
plus#(N,0()) -> c_3()
plus#(N,s(M)) -> c_4(plus#(N,M))
x#(N,0()) -> c_5()
x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
Weak DPs
and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1()
and#(tt(),X) -> c_2(activate#(X))
plus#(N,0()) -> c_3()
plus#(N,s(M)) -> c_4(plus#(N,M))
x#(N,0()) -> c_5()
x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
- Weak TRS:
activate(X) -> X
and(tt(),X) -> activate(X)
plus(N,0()) -> N
plus(N,s(M)) -> s(plus(N,M))
x(N,0()) -> 0()
x(N,s(M)) -> plus(x(N,M),N)
- Signature:
{activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
plus(N,0()) -> N
plus(N,s(M)) -> s(plus(N,M))
x(N,0()) -> 0()
x(N,s(M)) -> plus(x(N,M),N)
activate#(X) -> c_1()
and#(tt(),X) -> c_2(activate#(X))
plus#(N,0()) -> c_3()
plus#(N,s(M)) -> c_4(plus#(N,M))
x#(N,0()) -> c_5()
x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1()
and#(tt(),X) -> c_2(activate#(X))
plus#(N,0()) -> c_3()
plus#(N,s(M)) -> c_4(plus#(N,M))
x#(N,0()) -> c_5()
x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
- Weak TRS:
plus(N,0()) -> N
plus(N,s(M)) -> s(plus(N,M))
x(N,0()) -> 0()
x(N,s(M)) -> plus(x(N,M),N)
- Signature:
{activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,3,5}
by application of
Pre({1,3,5}) = {2,4,6}.
Here rules are labelled as follows:
1: activate#(X) -> c_1()
2: and#(tt(),X) -> c_2(activate#(X))
3: plus#(N,0()) -> c_3()
4: plus#(N,s(M)) -> c_4(plus#(N,M))
5: x#(N,0()) -> c_5()
6: x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
* Step 4: PredecessorEstimation WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
and#(tt(),X) -> c_2(activate#(X))
plus#(N,s(M)) -> c_4(plus#(N,M))
x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
- Weak DPs:
activate#(X) -> c_1()
plus#(N,0()) -> c_3()
x#(N,0()) -> c_5()
- Weak TRS:
plus(N,0()) -> N
plus(N,s(M)) -> s(plus(N,M))
x(N,0()) -> 0()
x(N,s(M)) -> plus(x(N,M),N)
- Signature:
{activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1}
by application of
Pre({1}) = {}.
Here rules are labelled as follows:
1: and#(tt(),X) -> c_2(activate#(X))
2: plus#(N,s(M)) -> c_4(plus#(N,M))
3: x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
4: activate#(X) -> c_1()
5: plus#(N,0()) -> c_3()
6: x#(N,0()) -> c_5()
* Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
plus#(N,s(M)) -> c_4(plus#(N,M))
x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
- Weak DPs:
activate#(X) -> c_1()
and#(tt(),X) -> c_2(activate#(X))
plus#(N,0()) -> c_3()
x#(N,0()) -> c_5()
- Weak TRS:
plus(N,0()) -> N
plus(N,s(M)) -> s(plus(N,M))
x(N,0()) -> 0()
x(N,s(M)) -> plus(x(N,M),N)
- Signature:
{activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:plus#(N,s(M)) -> c_4(plus#(N,M))
-->_1 plus#(N,0()) -> c_3():5
-->_1 plus#(N,s(M)) -> c_4(plus#(N,M)):1
2:S:x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
-->_2 x#(N,0()) -> c_5():6
-->_1 plus#(N,0()) -> c_3():5
-->_2 x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M)):2
-->_1 plus#(N,s(M)) -> c_4(plus#(N,M)):1
3:W:activate#(X) -> c_1()
4:W:and#(tt(),X) -> c_2(activate#(X))
-->_1 activate#(X) -> c_1():3
5:W:plus#(N,0()) -> c_3()
6:W:x#(N,0()) -> c_5()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: and#(tt(),X) -> c_2(activate#(X))
3: activate#(X) -> c_1()
6: x#(N,0()) -> c_5()
5: plus#(N,0()) -> c_3()
* Step 6: Decompose WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
plus#(N,s(M)) -> c_4(plus#(N,M))
x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
- Weak TRS:
plus(N,0()) -> N
plus(N,s(M)) -> s(plus(N,M))
x(N,0()) -> 0()
x(N,s(M)) -> plus(x(N,M),N)
- Signature:
{activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
+ 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#(N,s(M)) -> c_4(plus#(N,M))
- Weak DPs:
x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
- Weak TRS:
plus(N,0()) -> N
plus(N,s(M)) -> s(plus(N,M))
x(N,0()) -> 0()
x(N,s(M)) -> plus(x(N,M),N)
- Signature:
{activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1
,c_5/0,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
Problem (S)
- Strict DPs:
x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
- Weak DPs:
plus#(N,s(M)) -> c_4(plus#(N,M))
- Weak TRS:
plus(N,0()) -> N
plus(N,s(M)) -> s(plus(N,M))
x(N,0()) -> 0()
x(N,s(M)) -> plus(x(N,M),N)
- Signature:
{activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1
,c_5/0,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
** Step 6.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
plus#(N,s(M)) -> c_4(plus#(N,M))
- Weak DPs:
x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
- Weak TRS:
plus(N,0()) -> N
plus(N,s(M)) -> s(plus(N,M))
x(N,0()) -> 0()
x(N,s(M)) -> plus(x(N,M),N)
- Signature:
{activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
+ 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#(N,s(M)) -> c_4(plus#(N,M))
The strictly oriented rules are moved into the weak component.
*** Step 6.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
plus#(N,s(M)) -> c_4(plus#(N,M))
- Weak DPs:
x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
- Weak TRS:
plus(N,0()) -> N
plus(N,s(M)) -> s(plus(N,M))
x(N,0()) -> 0()
x(N,s(M)) -> plus(x(N,M),N)
- Signature:
{activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
+ 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_4) = {1},
uargs(c_6) = {1,2}
Following symbols are considered usable:
{activate#,and#,plus#,x#}
TcT has computed the following interpretation:
p(0) = 3
p(activate) = 1 + 2*x1 + x1^2
p(and) = x2
p(plus) = 1 + x1^2 + 2*x2
p(s) = 2 + x1
p(tt) = 0
p(x) = x1^2
p(activate#) = 1 + 4*x1 + 4*x1^2
p(and#) = 4*x1^2
p(plus#) = 4*x2
p(x#) = 2*x1 + 4*x1*x2 + x1^2
p(c_1) = 0
p(c_2) = 1
p(c_3) = 0
p(c_4) = 4 + x1
p(c_5) = 1
p(c_6) = x1 + x2
Following rules are strictly oriented:
plus#(N,s(M)) = 8 + 4*M
> 4 + 4*M
= c_4(plus#(N,M))
Following rules are (at-least) weakly oriented:
x#(N,s(M)) = 4*M*N + 10*N + N^2
>= 4*M*N + 6*N + N^2
= c_6(plus#(x(N,M),N),x#(N,M))
*** Step 6.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
plus#(N,s(M)) -> c_4(plus#(N,M))
x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
- Weak TRS:
plus(N,0()) -> N
plus(N,s(M)) -> s(plus(N,M))
x(N,0()) -> 0()
x(N,s(M)) -> plus(x(N,M),N)
- Signature:
{activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
*** Step 6.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
plus#(N,s(M)) -> c_4(plus#(N,M))
x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
- Weak TRS:
plus(N,0()) -> N
plus(N,s(M)) -> s(plus(N,M))
x(N,0()) -> 0()
x(N,s(M)) -> plus(x(N,M),N)
- Signature:
{activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:plus#(N,s(M)) -> c_4(plus#(N,M))
-->_1 plus#(N,s(M)) -> c_4(plus#(N,M)):1
2:W:x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
-->_2 x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M)):2
-->_1 plus#(N,s(M)) -> c_4(plus#(N,M)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
1: plus#(N,s(M)) -> c_4(plus#(N,M))
*** Step 6.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
plus(N,0()) -> N
plus(N,s(M)) -> s(plus(N,M))
x(N,0()) -> 0()
x(N,s(M)) -> plus(x(N,M),N)
- Signature:
{activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
+ 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:
x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
- Weak DPs:
plus#(N,s(M)) -> c_4(plus#(N,M))
- Weak TRS:
plus(N,0()) -> N
plus(N,s(M)) -> s(plus(N,M))
x(N,0()) -> 0()
x(N,s(M)) -> plus(x(N,M),N)
- Signature:
{activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
-->_1 plus#(N,s(M)) -> c_4(plus#(N,M)):2
-->_2 x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M)):1
2:W:plus#(N,s(M)) -> c_4(plus#(N,M))
-->_1 plus#(N,s(M)) -> c_4(plus#(N,M)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: plus#(N,s(M)) -> c_4(plus#(N,M))
** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
- Weak TRS:
plus(N,0()) -> N
plus(N,s(M)) -> s(plus(N,M))
x(N,0()) -> 0()
x(N,s(M)) -> plus(x(N,M),N)
- Signature:
{activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
,c_6/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M))
-->_2 x#(N,s(M)) -> c_6(plus#(x(N,M),N),x#(N,M)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
x#(N,s(M)) -> c_6(x#(N,M))
** Step 6.b:3: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
x#(N,s(M)) -> c_6(x#(N,M))
- Weak TRS:
plus(N,0()) -> N
plus(N,s(M)) -> s(plus(N,M))
x(N,0()) -> 0()
x(N,s(M)) -> plus(x(N,M),N)
- Signature:
{activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
x#(N,s(M)) -> c_6(x#(N,M))
** Step 6.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
x#(N,s(M)) -> c_6(x#(N,M))
- Signature:
{activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} 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: x#(N,s(M)) -> c_6(x#(N,M))
The strictly oriented rules are moved into the weak component.
*** Step 6.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
x#(N,s(M)) -> c_6(x#(N,M))
- Signature:
{activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} 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_6) = {1}
Following symbols are considered usable:
{activate#,and#,plus#,x#}
TcT has computed the following interpretation:
p(0) = [2]
p(activate) = [1] x1 + [1]
p(and) = [1] x1 + [0]
p(plus) = [1] x1 + [1]
p(s) = [1] x1 + [4]
p(tt) = [0]
p(x) = [4] x2 + [1]
p(activate#) = [2] x1 + [0]
p(and#) = [2] x1 + [1] x2 + [8]
p(plus#) = [1] x1 + [8] x2 + [0]
p(x#) = [8] x1 + [4] x2 + [0]
p(c_1) = [2]
p(c_2) = [1] x1 + [0]
p(c_3) = [1]
p(c_4) = [2] x1 + [4]
p(c_5) = [1]
p(c_6) = [1] x1 + [15]
Following rules are strictly oriented:
x#(N,s(M)) = [4] M + [8] N + [16]
> [4] M + [8] N + [15]
= c_6(x#(N,M))
Following rules are (at-least) weakly oriented:
*** Step 6.b:4.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
x#(N,s(M)) -> c_6(x#(N,M))
- Signature:
{activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
*** Step 6.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
x#(N,s(M)) -> c_6(x#(N,M))
- Signature:
{activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:x#(N,s(M)) -> c_6(x#(N,M))
-->_1 x#(N,s(M)) -> c_6(x#(N,M)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: x#(N,s(M)) -> c_6(x#(N,M))
*** Step 6.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{activate/1,and/2,plus/2,x/2,activate#/1,and#/2,plus#/2,x#/2} / {0/0,s/1,tt/0,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0
,c_6/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,and#,plus#,x#} and constructors {0,s,tt}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^2))