WORST_CASE(?,O(n^2))
* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
-(x,0()) -> x
-(0(),y) -> 0()
-(s(x),s(y)) -> -(x,y)
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2} / {+/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*,-,exp} and constructors {+,0,s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
*#(0(),y) -> c_1()
*#(s(x),y) -> c_2(*#(x,y))
-#(x,0()) -> c_3()
-#(0(),y) -> c_4()
-#(s(x),s(y)) -> c_5(-#(x,y))
exp#(x,0()) -> c_6()
exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
Weak DPs
and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
*#(0(),y) -> c_1()
*#(s(x),y) -> c_2(*#(x,y))
-#(x,0()) -> c_3()
-#(0(),y) -> c_4()
-#(s(x),s(y)) -> c_5(-#(x,y))
exp#(x,0()) -> c_6()
exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
-(x,0()) -> x
-(0(),y) -> 0()
-(s(x),s(y)) -> -(x,y)
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
*#(0(),y) -> c_1()
*#(s(x),y) -> c_2(*#(x,y))
-#(x,0()) -> c_3()
-#(0(),y) -> c_4()
-#(s(x),s(y)) -> c_5(-#(x,y))
exp#(x,0()) -> c_6()
exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
*#(0(),y) -> c_1()
*#(s(x),y) -> c_2(*#(x,y))
-#(x,0()) -> c_3()
-#(0(),y) -> c_4()
-#(s(x),s(y)) -> c_5(-#(x,y))
exp#(x,0()) -> c_6()
exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,3,4,6}
by application of
Pre({1,3,4,6}) = {2,5,7}.
Here rules are labelled as follows:
1: *#(0(),y) -> c_1()
2: *#(s(x),y) -> c_2(*#(x,y))
3: -#(x,0()) -> c_3()
4: -#(0(),y) -> c_4()
5: -#(s(x),s(y)) -> c_5(-#(x,y))
6: exp#(x,0()) -> c_6()
7: exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
*#(s(x),y) -> c_2(*#(x,y))
-#(s(x),s(y)) -> c_5(-#(x,y))
exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
- Weak DPs:
*#(0(),y) -> c_1()
-#(x,0()) -> c_3()
-#(0(),y) -> c_4()
exp#(x,0()) -> c_6()
- Weak TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:*#(s(x),y) -> c_2(*#(x,y))
-->_1 *#(0(),y) -> c_1():4
-->_1 *#(s(x),y) -> c_2(*#(x,y)):1
2:S:-#(s(x),s(y)) -> c_5(-#(x,y))
-->_1 -#(0(),y) -> c_4():6
-->_1 -#(x,0()) -> c_3():5
-->_1 -#(s(x),s(y)) -> c_5(-#(x,y)):2
3:S:exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
-->_2 exp#(x,0()) -> c_6():7
-->_1 *#(0(),y) -> c_1():4
-->_2 exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y)):3
-->_1 *#(s(x),y) -> c_2(*#(x,y)):1
4:W:*#(0(),y) -> c_1()
5:W:-#(x,0()) -> c_3()
6:W:-#(0(),y) -> c_4()
7:W:exp#(x,0()) -> c_6()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
7: exp#(x,0()) -> c_6()
5: -#(x,0()) -> c_3()
6: -#(0(),y) -> c_4()
4: *#(0(),y) -> c_1()
* Step 5: Decompose WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
*#(s(x),y) -> c_2(*#(x,y))
-#(s(x),s(y)) -> c_5(-#(x,y))
exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,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:
*#(s(x),y) -> c_2(*#(x,y))
- Weak DPs:
-#(s(x),s(y)) -> c_5(-#(x,y))
exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
Problem (S)
- Strict DPs:
-#(s(x),s(y)) -> c_5(-#(x,y))
exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
- Weak DPs:
*#(s(x),y) -> c_2(*#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
** Step 5.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
*#(s(x),y) -> c_2(*#(x,y))
- Weak DPs:
-#(s(x),s(y)) -> c_5(-#(x,y))
exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:*#(s(x),y) -> c_2(*#(x,y))
-->_1 *#(s(x),y) -> c_2(*#(x,y)):1
2:W:-#(s(x),s(y)) -> c_5(-#(x,y))
-->_1 -#(s(x),s(y)) -> c_5(-#(x,y)):2
3:W:exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
-->_1 *#(s(x),y) -> c_2(*#(x,y)):1
-->_2 exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: -#(s(x),s(y)) -> c_5(-#(x,y))
** Step 5.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
*#(s(x),y) -> c_2(*#(x,y))
- Weak DPs:
exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,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: *#(s(x),y) -> c_2(*#(x,y))
The strictly oriented rules are moved into the weak component.
*** Step 5.a:2.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
*#(s(x),y) -> c_2(*#(x,y))
- Weak DPs:
exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,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},
uargs(c_7) = {1,2}
Following symbols are considered usable:
{*#,-#,exp#}
TcT has computed the following interpretation:
p(*) = 2*x1^2 + 2*x2 + 2*x2^2
p(+) = x2
p(-) = 2 + x1 + x1*x2 + x1^2 + 2*x2^2
p(0) = 0
p(exp) = 0
p(s) = 1 + x1
p(*#) = 1 + x1
p(-#) = 1 + x1 + x2 + x2^2
p(exp#) = 6 + 2*x1 + x1*x2 + 3*x2 + 2*x2^2
p(c_1) = 0
p(c_2) = x1
p(c_3) = 4
p(c_4) = 4
p(c_5) = 0
p(c_6) = 1
p(c_7) = x1 + x2
Following rules are strictly oriented:
*#(s(x),y) = 2 + x
> 1 + x
= c_2(*#(x,y))
Following rules are (at-least) weakly oriented:
exp#(x,s(y)) = 11 + 3*x + x*y + 7*y + 2*y^2
>= 7 + 3*x + x*y + 3*y + 2*y^2
= c_7(*#(x,exp(x,y)),exp#(x,y))
*** Step 5.a:2.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
*#(s(x),y) -> c_2(*#(x,y))
exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
*** Step 5.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
*#(s(x),y) -> c_2(*#(x,y))
exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:*#(s(x),y) -> c_2(*#(x,y))
-->_1 *#(s(x),y) -> c_2(*#(x,y)):1
2:W:exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
-->_2 exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y)):2
-->_1 *#(s(x),y) -> c_2(*#(x,y)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
1: *#(s(x),y) -> c_2(*#(x,y))
*** Step 5.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
-#(s(x),s(y)) -> c_5(-#(x,y))
exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
- Weak DPs:
*#(s(x),y) -> c_2(*#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:-#(s(x),s(y)) -> c_5(-#(x,y))
-->_1 -#(s(x),s(y)) -> c_5(-#(x,y)):1
2:S:exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
-->_1 *#(s(x),y) -> c_2(*#(x,y)):3
-->_2 exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y)):2
3:W:*#(s(x),y) -> c_2(*#(x,y))
-->_1 *#(s(x),y) -> c_2(*#(x,y)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: *#(s(x),y) -> c_2(*#(x,y))
** Step 5.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
-#(s(x),s(y)) -> c_5(-#(x,y))
exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:-#(s(x),s(y)) -> c_5(-#(x,y))
-->_1 -#(s(x),s(y)) -> c_5(-#(x,y)):1
2:S:exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y))
-->_2 exp#(x,s(y)) -> c_7(*#(x,exp(x,y)),exp#(x,y)):2
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
exp#(x,s(y)) -> c_7(exp#(x,y))
** Step 5.b:3: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
-#(s(x),s(y)) -> c_5(-#(x,y))
exp#(x,s(y)) -> c_7(exp#(x,y))
- Weak TRS:
*(0(),y) -> 0()
*(s(x),y) -> +(y,*(x,y))
exp(x,0()) -> s(0())
exp(x,s(y)) -> *(x,exp(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
-#(s(x),s(y)) -> c_5(-#(x,y))
exp#(x,s(y)) -> c_7(exp#(x,y))
** Step 5.b:4: Decompose WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
-#(s(x),s(y)) -> c_5(-#(x,y))
exp#(x,s(y)) -> c_7(exp#(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,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:
-#(s(x),s(y)) -> c_5(-#(x,y))
- Weak DPs:
exp#(x,s(y)) -> c_7(exp#(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
Problem (S)
- Strict DPs:
exp#(x,s(y)) -> c_7(exp#(x,y))
- Weak DPs:
-#(s(x),s(y)) -> c_5(-#(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
*** Step 5.b:4.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
-#(s(x),s(y)) -> c_5(-#(x,y))
- Weak DPs:
exp#(x,s(y)) -> c_7(exp#(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:-#(s(x),s(y)) -> c_5(-#(x,y))
-->_1 -#(s(x),s(y)) -> c_5(-#(x,y)):1
2:W:exp#(x,s(y)) -> c_7(exp#(x,y))
-->_1 exp#(x,s(y)) -> c_7(exp#(x,y)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: exp#(x,s(y)) -> c_7(exp#(x,y))
*** Step 5.b:4.a:2: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
-#(s(x),s(y)) -> c_5(-#(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,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: -#(s(x),s(y)) -> c_5(-#(x,y))
The strictly oriented rules are moved into the weak component.
**** Step 5.b:4.a:2.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
-#(s(x),s(y)) -> c_5(-#(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,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_5) = {1}
Following symbols are considered usable:
{*#,-#,exp#}
TcT has computed the following interpretation:
p(*) = [1] x1 + [2]
p(+) = [0]
p(-) = [1] x2 + [0]
p(0) = [2]
p(exp) = [2] x1 + [1] x2 + [2]
p(s) = [1] x1 + [2]
p(*#) = [1] x1 + [4] x2 + [2]
p(-#) = [8] x1 + [12]
p(exp#) = [1] x1 + [2] x2 + [1]
p(c_1) = [4]
p(c_2) = [4] x1 + [0]
p(c_3) = [1]
p(c_4) = [0]
p(c_5) = [1] x1 + [1]
p(c_6) = [1]
p(c_7) = [1]
Following rules are strictly oriented:
-#(s(x),s(y)) = [8] x + [28]
> [8] x + [13]
= c_5(-#(x,y))
Following rules are (at-least) weakly oriented:
**** Step 5.b:4.a:2.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
-#(s(x),s(y)) -> c_5(-#(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 5.b:4.a:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
-#(s(x),s(y)) -> c_5(-#(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:-#(s(x),s(y)) -> c_5(-#(x,y))
-->_1 -#(s(x),s(y)) -> c_5(-#(x,y)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: -#(s(x),s(y)) -> c_5(-#(x,y))
**** Step 5.b:4.a:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 5.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
exp#(x,s(y)) -> c_7(exp#(x,y))
- Weak DPs:
-#(s(x),s(y)) -> c_5(-#(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:exp#(x,s(y)) -> c_7(exp#(x,y))
-->_1 exp#(x,s(y)) -> c_7(exp#(x,y)):1
2:W:-#(s(x),s(y)) -> c_5(-#(x,y))
-->_1 -#(s(x),s(y)) -> c_5(-#(x,y)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: -#(s(x),s(y)) -> c_5(-#(x,y))
*** Step 5.b:4.b:2: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
exp#(x,s(y)) -> c_7(exp#(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,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: exp#(x,s(y)) -> c_7(exp#(x,y))
The strictly oriented rules are moved into the weak component.
**** Step 5.b:4.b:2.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
exp#(x,s(y)) -> c_7(exp#(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_7) = {1}
Following symbols are considered usable:
{*#,-#,exp#}
TcT has computed the following interpretation:
p(*) = [0]
p(+) = [1] x1 + [1] x2 + [0]
p(-) = [1] x1 + [1] x2 + [8]
p(0) = [2]
p(exp) = [2] x2 + [1]
p(s) = [1] x1 + [2]
p(*#) = [2]
p(-#) = [1] x2 + [0]
p(exp#) = [8] x2 + [12]
p(c_1) = [0]
p(c_2) = [1]
p(c_3) = [1]
p(c_4) = [0]
p(c_5) = [1] x1 + [0]
p(c_6) = [1]
p(c_7) = [1] x1 + [15]
Following rules are strictly oriented:
exp#(x,s(y)) = [8] y + [28]
> [8] y + [27]
= c_7(exp#(x,y))
Following rules are (at-least) weakly oriented:
**** Step 5.b:4.b:2.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
exp#(x,s(y)) -> c_7(exp#(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 5.b:4.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
exp#(x,s(y)) -> c_7(exp#(x,y))
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:exp#(x,s(y)) -> c_7(exp#(x,y))
-->_1 exp#(x,s(y)) -> c_7(exp#(x,y)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: exp#(x,s(y)) -> c_7(exp#(x,y))
**** Step 5.b:4.b:2.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{*/2,-/2,exp/2,*#/2,-#/2,exp#/2} / {+/2,0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {*#,-#,exp#} and constructors {+,0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^2))