* Step 1: Sum WORST_CASE(Omega(n^1),O(n^3))
+ Considered Problem:
- Strict TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
first(0(),X) -> nil()
first(s(X),cons(Y)) -> cons(Y)
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
terms(N) -> cons(recip(sqr(N)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1} / {0/0,cons/1,nil/0,recip/1,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add,dbl,first,sqr,terms} and constructors {0,cons,nil
,recip,s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
first(0(),X) -> nil()
first(s(X),cons(Y)) -> cons(Y)
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
terms(N) -> cons(recip(sqr(N)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1} / {0/0,cons/1,nil/0,recip/1,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add,dbl,first,sqr,terms} and constructors {0,cons,nil
,recip,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
add(x,y){x -> s(x)} =
add(s(x),y) ->^+ s(add(x,y))
= C[add(x,y) = add(x,y){}]
** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
first(0(),X) -> nil()
first(s(X),cons(Y)) -> cons(Y)
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
terms(N) -> cons(recip(sqr(N)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1} / {0/0,cons/1,nil/0,recip/1,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add,dbl,first,sqr,terms} and constructors {0,cons,nil
,recip,s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
add#(0(),X) -> c_1()
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(0()) -> c_3()
dbl#(s(X)) -> c_4(dbl#(X))
first#(0(),X) -> c_5()
first#(s(X),cons(Y)) -> c_6()
sqr#(0()) -> c_7()
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_9(sqr#(N))
Weak DPs
and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add#(0(),X) -> c_1()
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(0()) -> c_3()
dbl#(s(X)) -> c_4(dbl#(X))
first#(0(),X) -> c_5()
first#(s(X),cons(Y)) -> c_6()
sqr#(0()) -> c_7()
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_9(sqr#(N))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
first(0(),X) -> nil()
first(s(X),cons(Y)) -> cons(Y)
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
terms(N) -> cons(recip(sqr(N)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
add#(0(),X) -> c_1()
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(0()) -> c_3()
dbl#(s(X)) -> c_4(dbl#(X))
first#(0(),X) -> c_5()
first#(s(X),cons(Y)) -> c_6()
sqr#(0()) -> c_7()
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_9(sqr#(N))
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add#(0(),X) -> c_1()
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(0()) -> c_3()
dbl#(s(X)) -> c_4(dbl#(X))
first#(0(),X) -> c_5()
first#(s(X),cons(Y)) -> c_6()
sqr#(0()) -> c_7()
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_9(sqr#(N))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,3,5,6,7}
by application of
Pre({1,3,5,6,7}) = {2,4,8,9}.
Here rules are labelled as follows:
1: add#(0(),X) -> c_1()
2: add#(s(X),Y) -> c_2(add#(X,Y))
3: dbl#(0()) -> c_3()
4: dbl#(s(X)) -> c_4(dbl#(X))
5: first#(0(),X) -> c_5()
6: first#(s(X),cons(Y)) -> c_6()
7: sqr#(0()) -> c_7()
8: sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
9: terms#(N) -> c_9(sqr#(N))
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_9(sqr#(N))
- Weak DPs:
add#(0(),X) -> c_1()
dbl#(0()) -> c_3()
first#(0(),X) -> c_5()
first#(s(X),cons(Y)) -> c_6()
sqr#(0()) -> c_7()
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:add#(s(X),Y) -> c_2(add#(X,Y))
-->_1 add#(0(),X) -> c_1():5
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
2:S:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(0()) -> c_3():6
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):2
3:S:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_2 sqr#(0()) -> c_7():9
-->_3 dbl#(0()) -> c_3():6
-->_1 add#(0(),X) -> c_1():5
-->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3
-->_3 dbl#(s(X)) -> c_4(dbl#(X)):2
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
4:S:terms#(N) -> c_9(sqr#(N))
-->_1 sqr#(0()) -> c_7():9
-->_1 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3
5:W:add#(0(),X) -> c_1()
6:W:dbl#(0()) -> c_3()
7:W:first#(0(),X) -> c_5()
8:W:first#(s(X),cons(Y)) -> c_6()
9:W:sqr#(0()) -> c_7()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
8: first#(s(X),cons(Y)) -> c_6()
7: first#(0(),X) -> c_5()
9: sqr#(0()) -> c_7()
6: dbl#(0()) -> c_3()
5: add#(0(),X) -> c_1()
** Step 1.b:5: RemoveHeads WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
terms#(N) -> c_9(sqr#(N))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
RemoveHeads
+ Details:
Consider the dependency graph
1:S:add#(s(X),Y) -> c_2(add#(X,Y))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
2:S:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):2
3:S:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3
-->_3 dbl#(s(X)) -> c_4(dbl#(X)):2
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
4:S:terms#(N) -> c_9(sqr#(N))
-->_1 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):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).
[(4,terms#(N) -> c_9(sqr#(N)))]
** Step 1.b:6: Decompose WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,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:
add#(s(X),Y) -> c_2(add#(X,Y))
- Weak DPs:
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
Problem (S)
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
- Weak DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
*** Step 1.b:6.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
- Weak DPs:
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:add#(s(X),Y) -> c_2(add#(X,Y))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
2:W:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):2
3:W:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
-->_3 dbl#(s(X)) -> c_4(dbl#(X)):2
-->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: dbl#(s(X)) -> c_4(dbl#(X))
*** Step 1.b:6.a:2: SimplifyRHS WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
- Weak DPs:
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:add#(s(X),Y) -> c_2(add#(X,Y))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
3:W:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
-->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):3
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X))
*** Step 1.b:6.a:3: DecomposeDG WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
- Weak DPs:
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ 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
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X))
and a lower component
add#(s(X),Y) -> c_2(add#(X,Y))
Further, following extension rules are added to the lower component.
sqr#(s(X)) -> add#(sqr(X),dbl(X))
sqr#(s(X)) -> sqr#(X)
**** Step 1.b:6.a:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,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: sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X))
The strictly oriented rules are moved into the weak component.
***** Step 1.b:6.a:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,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) = {1,2}
Following symbols are considered usable:
{add#,dbl#,first#,sqr#,terms#}
TcT has computed the following interpretation:
p(0) = [5]
p(add) = [4] x1 + [0]
p(cons) = [8]
p(dbl) = [3] x1 + [0]
p(first) = [1]
p(nil) = [0]
p(recip) = [2]
p(s) = [1] x1 + [4]
p(sqr) = [1] x1 + [3]
p(terms) = [1]
p(add#) = [0]
p(dbl#) = [0]
p(first#) = [2]
p(sqr#) = [4] x1 + [8]
p(terms#) = [2]
p(c_1) = [1]
p(c_2) = [1]
p(c_3) = [1]
p(c_4) = [1]
p(c_5) = [1]
p(c_6) = [1]
p(c_7) = [0]
p(c_8) = [8] x1 + [1] x2 + [12]
p(c_9) = [1] x1 + [0]
Following rules are strictly oriented:
sqr#(s(X)) = [4] X + [24]
> [4] X + [20]
= c_8(add#(sqr(X),dbl(X)),sqr#(X))
Following rules are (at-least) weakly oriented:
***** Step 1.b:6.a:3.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 1.b:6.a:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X))
-->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X))
***** Step 1.b:6.a:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
**** Step 1.b:6.a:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
- Weak DPs:
sqr#(s(X)) -> add#(sqr(X),dbl(X))
sqr#(s(X)) -> sqr#(X)
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,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: add#(s(X),Y) -> c_2(add#(X,Y))
The strictly oriented rules are moved into the weak component.
***** Step 1.b:6.a:3.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
- Weak DPs:
sqr#(s(X)) -> add#(sqr(X),dbl(X))
sqr#(s(X)) -> sqr#(X)
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,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_2) = {1}
Following symbols are considered usable:
{add,dbl,sqr,add#,dbl#,first#,sqr#,terms#}
TcT has computed the following interpretation:
p(0) = 0
p(add) = x1 + x2
p(cons) = 0
p(dbl) = 2*x1
p(first) = x1^2 + x2 + 2*x2^2
p(nil) = 1
p(recip) = 1 + x1
p(s) = 1 + x1
p(sqr) = x1^2
p(terms) = 1 + x1
p(add#) = 6 + x1
p(dbl#) = 1 + x1^2
p(first#) = 0
p(sqr#) = 5 + 2*x1^2
p(terms#) = 1 + x1 + x1^2
p(c_1) = 0
p(c_2) = x1
p(c_3) = 0
p(c_4) = 1 + x1
p(c_5) = 0
p(c_6) = 0
p(c_7) = 1
p(c_8) = 1
p(c_9) = 1
Following rules are strictly oriented:
add#(s(X),Y) = 7 + X
> 6 + X
= c_2(add#(X,Y))
Following rules are (at-least) weakly oriented:
sqr#(s(X)) = 7 + 4*X + 2*X^2
>= 6 + X^2
= add#(sqr(X),dbl(X))
sqr#(s(X)) = 7 + 4*X + 2*X^2
>= 5 + 2*X^2
= sqr#(X)
add(0(),X) = X
>= X
= X
add(s(X),Y) = 1 + X + Y
>= 1 + X + Y
= s(add(X,Y))
dbl(0()) = 0
>= 0
= 0()
dbl(s(X)) = 2 + 2*X
>= 2 + 2*X
= s(s(dbl(X)))
sqr(0()) = 0
>= 0
= 0()
sqr(s(X)) = 1 + 2*X + X^2
>= 1 + 2*X + X^2
= s(add(sqr(X),dbl(X)))
***** Step 1.b:6.a:3.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
sqr#(s(X)) -> add#(sqr(X),dbl(X))
sqr#(s(X)) -> sqr#(X)
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 1.b:6.a:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
sqr#(s(X)) -> add#(sqr(X),dbl(X))
sqr#(s(X)) -> sqr#(X)
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:add#(s(X),Y) -> c_2(add#(X,Y))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
2:W:sqr#(s(X)) -> add#(sqr(X),dbl(X))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):1
3:W:sqr#(s(X)) -> sqr#(X)
-->_1 sqr#(s(X)) -> sqr#(X):3
-->_1 sqr#(s(X)) -> add#(sqr(X),dbl(X)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: sqr#(s(X)) -> sqr#(X)
2: sqr#(s(X)) -> add#(sqr(X),dbl(X))
1: add#(s(X),Y) -> c_2(add#(X,Y))
***** Step 1.b:6.a:3.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 1.b:6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
- Weak DPs:
add#(s(X),Y) -> c_2(add#(X,Y))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):1
2:S:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):3
-->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):2
-->_3 dbl#(s(X)) -> c_4(dbl#(X)):1
3:W:add#(s(X),Y) -> c_2(add#(X,Y))
-->_1 add#(s(X),Y) -> c_2(add#(X,Y)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: add#(s(X),Y) -> c_2(add#(X,Y))
*** Step 1.b:6.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/3,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):1
2:S:sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X))
-->_2 sqr#(s(X)) -> c_8(add#(sqr(X),dbl(X)),sqr#(X),dbl#(X)):2
-->_3 dbl#(s(X)) -> c_4(dbl#(X)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
*** Step 1.b:6.b:3: UsableRules WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
- Weak TRS:
add(0(),X) -> X
add(s(X),Y) -> s(add(X,Y))
dbl(0()) -> 0()
dbl(s(X)) -> s(s(dbl(X)))
sqr(0()) -> 0()
sqr(s(X)) -> s(add(sqr(X),dbl(X)))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
*** Step 1.b:6.b:4: Decompose WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,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:
dbl#(s(X)) -> c_4(dbl#(X))
- Weak DPs:
sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
Problem (S)
- Strict DPs:
sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
- Weak DPs:
dbl#(s(X)) -> c_4(dbl#(X))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
**** Step 1.b:6.b:4.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
- Weak DPs:
sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,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: dbl#(s(X)) -> c_4(dbl#(X))
The strictly oriented rules are moved into the weak component.
***** Step 1.b:6.b:4.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
dbl#(s(X)) -> c_4(dbl#(X))
- Weak DPs:
sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,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_4) = {1},
uargs(c_8) = {1,2}
Following symbols are considered usable:
{add#,dbl#,first#,sqr#,terms#}
TcT has computed the following interpretation:
p(0) = 0
p(add) = 1 + x1 + x1*x2 + x1^2 + x2^2
p(cons) = 0
p(dbl) = 1 + 8*x1
p(first) = 2*x1 + x1*x2 + x1^2 + x2
p(nil) = 0
p(recip) = 0
p(s) = 1 + x1
p(sqr) = 1
p(terms) = 1 + x1 + 8*x1^2
p(add#) = x1 + 2*x1^2 + x2 + x2^2
p(dbl#) = 8 + 8*x1
p(first#) = 4*x1 + x1^2 + x2
p(sqr#) = 2 + 10*x1 + 9*x1^2
p(terms#) = 0
p(c_1) = 0
p(c_2) = 1 + x1
p(c_3) = 1
p(c_4) = x1
p(c_5) = 0
p(c_6) = 0
p(c_7) = 0
p(c_8) = x1 + x2
p(c_9) = 0
Following rules are strictly oriented:
dbl#(s(X)) = 16 + 8*X
> 8 + 8*X
= c_4(dbl#(X))
Following rules are (at-least) weakly oriented:
sqr#(s(X)) = 21 + 28*X + 9*X^2
>= 10 + 18*X + 9*X^2
= c_8(sqr#(X),dbl#(X))
***** Step 1.b:6.b:4.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 1.b:6.b:4.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
dbl#(s(X)) -> c_4(dbl#(X))
sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):1
2:W:sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
-->_1 sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)):2
-->_2 dbl#(s(X)) -> c_4(dbl#(X)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
1: dbl#(s(X)) -> c_4(dbl#(X))
***** Step 1.b:6.b:4.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
**** Step 1.b:6.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
- Weak DPs:
dbl#(s(X)) -> c_4(dbl#(X))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
-->_2 dbl#(s(X)) -> c_4(dbl#(X)):2
-->_1 sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)):1
2:W:dbl#(s(X)) -> c_4(dbl#(X))
-->_1 dbl#(s(X)) -> c_4(dbl#(X)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: dbl#(s(X)) -> c_4(dbl#(X))
**** Step 1.b:6.b:4.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/2,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:sqr#(s(X)) -> c_8(sqr#(X),dbl#(X))
-->_1 sqr#(s(X)) -> c_8(sqr#(X),dbl#(X)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
sqr#(s(X)) -> c_8(sqr#(X))
**** Step 1.b:6.b:4.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sqr#(s(X)) -> c_8(sqr#(X))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,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: sqr#(s(X)) -> c_8(sqr#(X))
The strictly oriented rules are moved into the weak component.
***** Step 1.b:6.b:4.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sqr#(s(X)) -> c_8(sqr#(X))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,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:
{add#,dbl#,first#,sqr#,terms#}
TcT has computed the following interpretation:
p(0) = [0]
p(add) = [0]
p(cons) = [2]
p(dbl) = [8]
p(first) = [1]
p(nil) = [4]
p(recip) = [8]
p(s) = [1] x1 + [2]
p(sqr) = [8]
p(terms) = [8] x1 + [0]
p(add#) = [1] x1 + [1] x2 + [2]
p(dbl#) = [1]
p(first#) = [1] x1 + [2] x2 + [2]
p(sqr#) = [1] x1 + [1]
p(terms#) = [2]
p(c_1) = [1]
p(c_2) = [1]
p(c_3) = [1]
p(c_4) = [1] x1 + [1]
p(c_5) = [0]
p(c_6) = [1]
p(c_7) = [4]
p(c_8) = [1] x1 + [0]
p(c_9) = [8]
Following rules are strictly oriented:
sqr#(s(X)) = [1] X + [3]
> [1] X + [1]
= c_8(sqr#(X))
Following rules are (at-least) weakly oriented:
***** Step 1.b:6.b:4.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
sqr#(s(X)) -> c_8(sqr#(X))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 1.b:6.b:4.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
sqr#(s(X)) -> c_8(sqr#(X))
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:sqr#(s(X)) -> c_8(sqr#(X))
-->_1 sqr#(s(X)) -> c_8(sqr#(X)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: sqr#(s(X)) -> c_8(sqr#(X))
***** Step 1.b:6.b:4.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{add/2,dbl/1,first/2,sqr/1,terms/1,add#/2,dbl#/1,first#/2,sqr#/1,terms#/1} / {0/0,cons/1,nil/0,recip/1,s/1
,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/0,c_8/1,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {add#,dbl#,first#,sqr#,terms#} and constructors {0,cons
,nil,recip,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^3))