MAYBE
* Step 1: DependencyPairs MAYBE
+ Considered Problem:
- Strict TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
goal(x,y) -> power(x,y)
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2} / {Cons/2,Nil/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0,goal,mult,power} and constructors {Cons,Nil}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
add0#(x,Nil()) -> c_1()
add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs))
goal#(x,y) -> c_3(power#(x,y))
mult#(x,Nil()) -> c_4()
mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs))
power#(x,Nil()) -> c_6()
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
Weak DPs
and mark the set of starting terms.
* Step 2: UsableRules MAYBE
+ Considered Problem:
- Strict DPs:
add0#(x,Nil()) -> c_1()
add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs))
goal#(x,y) -> c_3(power#(x,y))
mult#(x,Nil()) -> c_4()
mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs))
power#(x,Nil()) -> c_6()
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
goal(x,y) -> power(x,y)
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/2,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
add0#(x,Nil()) -> c_1()
add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs))
goal#(x,y) -> c_3(power#(x,y))
mult#(x,Nil()) -> c_4()
mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs))
power#(x,Nil()) -> c_6()
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
* Step 3: PredecessorEstimation MAYBE
+ Considered Problem:
- Strict DPs:
add0#(x,Nil()) -> c_1()
add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs))
goal#(x,y) -> c_3(power#(x,y))
mult#(x,Nil()) -> c_4()
mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs))
power#(x,Nil()) -> c_6()
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/2,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,4,6}
by application of
Pre({1,4,6}) = {2,3,5,7}.
Here rules are labelled as follows:
1: add0#(x,Nil()) -> c_1()
2: add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs))
3: goal#(x,y) -> c_3(power#(x,y))
4: mult#(x,Nil()) -> c_4()
5: mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs))
6: power#(x,Nil()) -> c_6()
7: power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
* Step 4: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs))
goal#(x,y) -> c_3(power#(x,y))
mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs))
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak DPs:
add0#(x,Nil()) -> c_1()
mult#(x,Nil()) -> c_4()
power#(x,Nil()) -> c_6()
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/2,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs))
-->_1 add0#(x,Nil()) -> c_1():5
-->_1 add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs)):1
2:S:goal#(x,y) -> c_3(power#(x,y))
-->_1 power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs)):4
-->_1 power#(x,Nil()) -> c_6():7
3:S:mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs))
-->_2 mult#(x,Nil()) -> c_4():6
-->_1 add0#(x,Nil()) -> c_1():5
-->_2 mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs)):3
-->_1 add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs)):1
4:S:power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
-->_2 power#(x,Nil()) -> c_6():7
-->_1 mult#(x,Nil()) -> c_4():6
-->_2 power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs)):4
-->_1 mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs)):3
5:W:add0#(x,Nil()) -> c_1()
6:W:mult#(x,Nil()) -> c_4()
7:W:power#(x,Nil()) -> c_6()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: mult#(x,Nil()) -> c_4()
7: power#(x,Nil()) -> c_6()
5: add0#(x,Nil()) -> c_1()
* Step 5: RemoveHeads MAYBE
+ Considered Problem:
- Strict DPs:
add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs))
goal#(x,y) -> c_3(power#(x,y))
mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs))
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/2,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
RemoveHeads
+ Details:
Consider the dependency graph
1:S:add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs))
-->_1 add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs)):1
2:S:goal#(x,y) -> c_3(power#(x,y))
-->_1 power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs)):4
3:S:mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs))
-->_2 mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs)):3
-->_1 add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs)):1
4:S:power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
-->_2 power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs)):4
-->_1 mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs)):3
Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
[(2,goal#(x,y) -> c_3(power#(x,y)))]
* Step 6: Decompose MAYBE
+ Considered Problem:
- Strict DPs:
add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs))
mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs))
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/2,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ 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:
add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs))
- Weak DPs:
mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs))
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/2,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
Problem (S)
- Strict DPs:
mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs))
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak DPs:
add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/2,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
** Step 6.a:1: DecomposeDG MAYBE
+ Considered Problem:
- Strict DPs:
add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs))
- Weak DPs:
mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs))
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/2,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
and a lower component
add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs))
mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs))
Further, following extension rules are added to the lower component.
power#(x',Cons(x,xs)) -> mult#(x',power(x',xs))
power#(x',Cons(x,xs)) -> power#(x',xs)
*** Step 6.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/2,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ 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: power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
The strictly oriented rules are moved into the weak component.
**** Step 6.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/2,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ 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) = {1,2}
Following symbols are considered usable:
{add0#,goal#,mult#,power#}
TcT has computed the following interpretation:
p(Cons) = [1] x2 + [8]
p(Nil) = [0]
p(add0) = [1] x1 + [0]
p(goal) = [0]
p(mult) = [8] x1 + [1]
p(power) = [1] x1 + [0]
p(add0#) = [1] x1 + [2] x2 + [0]
p(goal#) = [1]
p(mult#) = [0]
p(power#) = [1] x2 + [0]
p(c_1) = [1]
p(c_2) = [2] x1 + [1]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [1] x1 + [1]
p(c_6) = [0]
p(c_7) = [4] x1 + [1] x2 + [0]
Following rules are strictly oriented:
power#(x',Cons(x,xs)) = [1] xs + [8]
> [1] xs + [0]
= c_7(mult#(x',power(x',xs)),power#(x',xs))
Following rules are (at-least) weakly oriented:
**** Step 6.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/2,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 6.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/2,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
-->_2 power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
**** Step 6.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/2,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 6.a:1.b:1: Failure MAYBE
+ Considered Problem:
- Strict DPs:
add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs))
- Weak DPs:
mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs))
power#(x',Cons(x,xs)) -> mult#(x',power(x',xs))
power#(x',Cons(x,xs)) -> power#(x',xs)
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/2,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
** Step 6.b:1: RemoveWeakSuffixes MAYBE
+ Considered Problem:
- Strict DPs:
mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs))
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak DPs:
add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/2,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs))
-->_1 add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs)):3
-->_2 mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs)):1
2:S:power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
-->_2 power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs)):2
-->_1 mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs)):1
3:W:add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs))
-->_1 add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: add0#(x',Cons(x,xs)) -> c_2(add0#(x',xs))
** Step 6.b:2: SimplifyRHS MAYBE
+ Considered Problem:
- Strict DPs:
mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs))
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/2,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs))
-->_2 mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs)):1
2:S:power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
-->_2 power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs)):2
-->_1 mult#(x',Cons(x,xs)) -> c_5(add0#(x',mult(x',xs)),mult#(x',xs)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
mult#(x',Cons(x,xs)) -> c_5(mult#(x',xs))
** Step 6.b:3: Decompose MAYBE
+ Considered Problem:
- Strict DPs:
mult#(x',Cons(x,xs)) -> c_5(mult#(x',xs))
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ 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:
mult#(x',Cons(x,xs)) -> c_5(mult#(x',xs))
- Weak DPs:
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
Problem (S)
- Strict DPs:
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak DPs:
mult#(x',Cons(x,xs)) -> c_5(mult#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
*** Step 6.b:3.a:1: DecomposeDG MAYBE
+ Considered Problem:
- Strict DPs:
mult#(x',Cons(x,xs)) -> c_5(mult#(x',xs))
- Weak DPs:
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
and a lower component
mult#(x',Cons(x,xs)) -> c_5(mult#(x',xs))
Further, following extension rules are added to the lower component.
power#(x',Cons(x,xs)) -> mult#(x',power(x',xs))
power#(x',Cons(x,xs)) -> power#(x',xs)
**** Step 6.b:3.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ 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: power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
The strictly oriented rules are moved into the weak component.
***** Step 6.b:3.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ 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) = {1,2}
Following symbols are considered usable:
{add0#,goal#,mult#,power#}
TcT has computed the following interpretation:
p(Cons) = [1] x2 + [5]
p(Nil) = [1]
p(add0) = [8] x1 + [4]
p(goal) = [2]
p(mult) = [4]
p(power) = [0]
p(add0#) = [4] x2 + [2]
p(goal#) = [8] x2 + [2]
p(mult#) = [8]
p(power#) = [5] x2 + [6]
p(c_1) = [8]
p(c_2) = [1] x1 + [0]
p(c_3) = [1] x1 + [2]
p(c_4) = [1]
p(c_5) = [2] x1 + [1]
p(c_6) = [2]
p(c_7) = [1] x1 + [1] x2 + [7]
Following rules are strictly oriented:
power#(x',Cons(x,xs)) = [5] xs + [31]
> [5] xs + [21]
= c_7(mult#(x',power(x',xs)),power#(x',xs))
Following rules are (at-least) weakly oriented:
***** Step 6.b:3.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 6.b:3.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
-->_2 power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
***** Step 6.b:3.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
**** Step 6.b:3.a:1.b:1: Failure MAYBE
+ Considered Problem:
- Strict DPs:
mult#(x',Cons(x,xs)) -> c_5(mult#(x',xs))
- Weak DPs:
power#(x',Cons(x,xs)) -> mult#(x',power(x',xs))
power#(x',Cons(x,xs)) -> power#(x',xs)
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is still open.
*** Step 6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak DPs:
mult#(x',Cons(x,xs)) -> c_5(mult#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
-->_1 mult#(x',Cons(x,xs)) -> c_5(mult#(x',xs)):2
-->_2 power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs)):1
2:W:mult#(x',Cons(x,xs)) -> c_5(mult#(x',xs))
-->_1 mult#(x',Cons(x,xs)) -> c_5(mult#(x',xs)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: mult#(x',Cons(x,xs)) -> c_5(mult#(x',xs))
*** Step 6.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs))
-->_2 power#(x',Cons(x,xs)) -> c_7(mult#(x',power(x',xs)),power#(x',xs)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
power#(x',Cons(x,xs)) -> c_7(power#(x',xs))
*** Step 6.b:3.b:3: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
power#(x',Cons(x,xs)) -> c_7(power#(x',xs))
- Weak TRS:
add0(x,Nil()) -> x
add0(x',Cons(x,xs)) -> Cons(Cons(Nil(),Nil()),add0(x',xs))
mult(x,Nil()) -> Nil()
mult(x',Cons(x,xs)) -> add0(x',mult(x',xs))
power(x,Nil()) -> Cons(Nil(),Nil())
power(x',Cons(x,xs)) -> mult(x',power(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
power#(x',Cons(x,xs)) -> c_7(power#(x',xs))
*** Step 6.b:3.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
power#(x',Cons(x,xs)) -> c_7(power#(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ 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: power#(x',Cons(x,xs)) -> c_7(power#(x',xs))
The strictly oriented rules are moved into the weak component.
**** Step 6.b:3.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
power#(x',Cons(x,xs)) -> c_7(power#(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ 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:
{add0#,goal#,mult#,power#}
TcT has computed the following interpretation:
p(Cons) = [1] x1 + [1] x2 + [4]
p(Nil) = [0]
p(add0) = [0]
p(goal) = [1] x1 + [1] x2 + [0]
p(mult) = [1] x1 + [1] x2 + [0]
p(power) = [1] x1 + [2] x2 + [1]
p(add0#) = [1] x2 + [8]
p(goal#) = [2] x1 + [2] x2 + [8]
p(mult#) = [2] x1 + [1]
p(power#) = [4] x2 + [0]
p(c_1) = [1]
p(c_2) = [1]
p(c_3) = [1]
p(c_4) = [4]
p(c_5) = [1] x1 + [1]
p(c_6) = [1]
p(c_7) = [1] x1 + [14]
Following rules are strictly oriented:
power#(x',Cons(x,xs)) = [4] x + [4] xs + [16]
> [4] xs + [14]
= c_7(power#(x',xs))
Following rules are (at-least) weakly oriented:
**** Step 6.b:3.b:4.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
power#(x',Cons(x,xs)) -> c_7(power#(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 6.b:3.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
power#(x',Cons(x,xs)) -> c_7(power#(x',xs))
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:power#(x',Cons(x,xs)) -> c_7(power#(x',xs))
-->_1 power#(x',Cons(x,xs)) -> c_7(power#(x',xs)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: power#(x',Cons(x,xs)) -> c_7(power#(x',xs))
**** Step 6.b:3.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{add0/2,goal/2,mult/2,power/2,add0#/2,goal#/2,mult#/2,power#/2} / {Cons/2,Nil/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add0#,goal#,mult#,power#} and constructors {Cons,Nil}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
MAYBE