* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
D(+(x,y)) -> +(D(x),D(y))
D(-(x,y)) -> -(D(x),D(y))
D(constant()) -> 0()
D(t()) -> 1()
- Signature:
{D/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0}
- Obligation:
runtime complexity wrt. defined symbols {D} and constructors {*,+,-,0,1,constant,t}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak dependency pairs:
Strict DPs
D#(*(x,y)) -> c_1(y,D#(x),x,D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(constant()) -> c_4()
D#(t()) -> c_5()
Weak DPs
and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
D#(*(x,y)) -> c_1(y,D#(x),x,D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(constant()) -> c_4()
D#(t()) -> c_5()
- Strict TRS:
D(*(x,y)) -> +(*(y,D(x)),*(x,D(y)))
D(+(x,y)) -> +(D(x),D(y))
D(-(x,y)) -> -(D(x),D(y))
D(constant()) -> 0()
D(t()) -> 1()
- Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/4,c_2/2,c_3/2,c_4/0,c_5/0}
- Obligation:
runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
D#(*(x,y)) -> c_1(y,D#(x),x,D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(constant()) -> c_4()
D#(t()) -> c_5()
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
D#(*(x,y)) -> c_1(y,D#(x),x,D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
D#(constant()) -> c_4()
D#(t()) -> c_5()
- Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/4,c_2/2,c_3/2,c_4/0,c_5/0}
- Obligation:
runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{4,5}
by application of
Pre({4,5}) = {1,2,3}.
Here rules are labelled as follows:
1: D#(*(x,y)) -> c_1(y,D#(x),x,D#(y))
2: D#(+(x,y)) -> c_2(D#(x),D#(y))
3: D#(-(x,y)) -> c_3(D#(x),D#(y))
4: D#(constant()) -> c_4()
5: D#(t()) -> c_5()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
D#(*(x,y)) -> c_1(y,D#(x),x,D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
- Weak DPs:
D#(constant()) -> c_4()
D#(t()) -> c_5()
- Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/4,c_2/2,c_3/2,c_4/0,c_5/0}
- Obligation:
runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:D#(*(x,y)) -> c_1(y,D#(x),x,D#(y))
-->_4 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_3 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_4 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_3 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_4 D#(t()) -> c_5():5
-->_3 D#(t()) -> c_5():5
-->_2 D#(t()) -> c_5():5
-->_1 D#(t()) -> c_5():5
-->_4 D#(constant()) -> c_4():4
-->_3 D#(constant()) -> c_4():4
-->_2 D#(constant()) -> c_4():4
-->_1 D#(constant()) -> c_4():4
-->_4 D#(*(x,y)) -> c_1(y,D#(x),x,D#(y)):1
-->_3 D#(*(x,y)) -> c_1(y,D#(x),x,D#(y)):1
-->_2 D#(*(x,y)) -> c_1(y,D#(x),x,D#(y)):1
-->_1 D#(*(x,y)) -> c_1(y,D#(x),x,D#(y)):1
2:S:D#(+(x,y)) -> c_2(D#(x),D#(y))
-->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_2 D#(t()) -> c_5():5
-->_1 D#(t()) -> c_5():5
-->_2 D#(constant()) -> c_4():4
-->_1 D#(constant()) -> c_4():4
-->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_2 D#(*(x,y)) -> c_1(y,D#(x),x,D#(y)):1
-->_1 D#(*(x,y)) -> c_1(y,D#(x),x,D#(y)):1
3:S:D#(-(x,y)) -> c_3(D#(x),D#(y))
-->_2 D#(t()) -> c_5():5
-->_1 D#(t()) -> c_5():5
-->_2 D#(constant()) -> c_4():4
-->_1 D#(constant()) -> c_4():4
-->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_2 D#(*(x,y)) -> c_1(y,D#(x),x,D#(y)):1
-->_1 D#(*(x,y)) -> c_1(y,D#(x),x,D#(y)):1
4:W:D#(constant()) -> c_4()
5:W:D#(t()) -> c_5()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: D#(constant()) -> c_4()
5: D#(t()) -> c_5()
* Step 5: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
D#(*(x,y)) -> c_1(y,D#(x),x,D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
- Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/4,c_2/2,c_3/2,c_4/0,c_5/0}
- Obligation:
runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
+ 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:
2: D#(+(x,y)) -> c_2(D#(x),D#(y))
The strictly oriented rules are moved into the weak component.
** Step 5.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
D#(*(x,y)) -> c_1(y,D#(x),x,D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
- Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/4,c_2/2,c_3/2,c_4/0,c_5/0}
- Obligation:
runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
+ 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_1) = {2,4},
uargs(c_2) = {1,2},
uargs(c_3) = {1,2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(*) = [1] x1 + [1] x2 + [0]
p(+) = [1] x1 + [1] x2 + [4]
p(-) = [1] x1 + [1] x2 + [0]
p(0) = [1]
p(1) = [4]
p(D) = [1] x1 + [4]
p(constant) = [1]
p(t) = [0]
p(D#) = [4] x1 + [0]
p(c_1) = [1] x2 + [1] x4 + [0]
p(c_2) = [1] x1 + [1] x2 + [12]
p(c_3) = [1] x1 + [1] x2 + [0]
p(c_4) = [0]
p(c_5) = [1]
Following rules are strictly oriented:
D#(+(x,y)) = [4] x + [4] y + [16]
> [4] x + [4] y + [12]
= c_2(D#(x),D#(y))
Following rules are (at-least) weakly oriented:
D#(*(x,y)) = [4] x + [4] y + [0]
>= [4] x + [4] y + [0]
= c_1(y,D#(x),x,D#(y))
D#(-(x,y)) = [4] x + [4] y + [0]
>= [4] x + [4] y + [0]
= c_3(D#(x),D#(y))
** Step 5.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
D#(*(x,y)) -> c_1(y,D#(x),x,D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
- Weak DPs:
D#(+(x,y)) -> c_2(D#(x),D#(y))
- Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/4,c_2/2,c_3/2,c_4/0,c_5/0}
- Obligation:
runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
** Step 5.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
D#(*(x,y)) -> c_1(y,D#(x),x,D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
- Weak DPs:
D#(+(x,y)) -> c_2(D#(x),D#(y))
- Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/4,c_2/2,c_3/2,c_4/0,c_5/0}
- Obligation:
runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
+ 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: D#(*(x,y)) -> c_1(y,D#(x),x,D#(y))
The strictly oriented rules are moved into the weak component.
*** Step 5.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
D#(*(x,y)) -> c_1(y,D#(x),x,D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
- Weak DPs:
D#(+(x,y)) -> c_2(D#(x),D#(y))
- Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/4,c_2/2,c_3/2,c_4/0,c_5/0}
- Obligation:
runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
+ 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_1) = {2,4},
uargs(c_2) = {1,2},
uargs(c_3) = {1,2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(*) = [1] x1 + [1] x2 + [8]
p(+) = [1] x1 + [1] x2 + [0]
p(-) = [1] x1 + [1] x2 + [0]
p(0) = [0]
p(1) = [1]
p(D) = [2]
p(constant) = [0]
p(t) = [0]
p(D#) = [2] x1 + [0]
p(c_1) = [1] x2 + [1] x4 + [13]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [1] x1 + [1] x2 + [0]
p(c_4) = [1]
p(c_5) = [0]
Following rules are strictly oriented:
D#(*(x,y)) = [2] x + [2] y + [16]
> [2] x + [2] y + [13]
= c_1(y,D#(x),x,D#(y))
Following rules are (at-least) weakly oriented:
D#(+(x,y)) = [2] x + [2] y + [0]
>= [2] x + [2] y + [0]
= c_2(D#(x),D#(y))
D#(-(x,y)) = [2] x + [2] y + [0]
>= [2] x + [2] y + [0]
= c_3(D#(x),D#(y))
*** Step 5.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
D#(-(x,y)) -> c_3(D#(x),D#(y))
- Weak DPs:
D#(*(x,y)) -> c_1(y,D#(x),x,D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
- Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/4,c_2/2,c_3/2,c_4/0,c_5/0}
- Obligation:
runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
*** Step 5.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
D#(-(x,y)) -> c_3(D#(x),D#(y))
- Weak DPs:
D#(*(x,y)) -> c_1(y,D#(x),x,D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
- Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/4,c_2/2,c_3/2,c_4/0,c_5/0}
- Obligation:
runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
+ 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: D#(-(x,y)) -> c_3(D#(x),D#(y))
The strictly oriented rules are moved into the weak component.
**** Step 5.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
D#(-(x,y)) -> c_3(D#(x),D#(y))
- Weak DPs:
D#(*(x,y)) -> c_1(y,D#(x),x,D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
- Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/4,c_2/2,c_3/2,c_4/0,c_5/0}
- Obligation:
runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
+ 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_1) = {2,4},
uargs(c_2) = {1,2},
uargs(c_3) = {1,2}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(*) = [1] x1 + [1] x2 + [1]
p(+) = [1] x1 + [1] x2 + [1]
p(-) = [1] x1 + [1] x2 + [1]
p(0) = [1]
p(1) = [0]
p(D) = [0]
p(constant) = [1]
p(t) = [1]
p(D#) = [2] x1 + [0]
p(c_1) = [1] x2 + [1] x4 + [2]
p(c_2) = [1] x1 + [1] x2 + [2]
p(c_3) = [1] x1 + [1] x2 + [0]
p(c_4) = [1]
p(c_5) = [1]
Following rules are strictly oriented:
D#(-(x,y)) = [2] x + [2] y + [2]
> [2] x + [2] y + [0]
= c_3(D#(x),D#(y))
Following rules are (at-least) weakly oriented:
D#(*(x,y)) = [2] x + [2] y + [2]
>= [2] x + [2] y + [2]
= c_1(y,D#(x),x,D#(y))
D#(+(x,y)) = [2] x + [2] y + [2]
>= [2] x + [2] y + [2]
= c_2(D#(x),D#(y))
**** Step 5.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
D#(*(x,y)) -> c_1(y,D#(x),x,D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
- Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/4,c_2/2,c_3/2,c_4/0,c_5/0}
- Obligation:
runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 5.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
D#(*(x,y)) -> c_1(y,D#(x),x,D#(y))
D#(+(x,y)) -> c_2(D#(x),D#(y))
D#(-(x,y)) -> c_3(D#(x),D#(y))
- Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/4,c_2/2,c_3/2,c_4/0,c_5/0}
- Obligation:
runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:D#(*(x,y)) -> c_1(y,D#(x),x,D#(y))
-->_4 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_3 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_4 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_3 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_4 D#(*(x,y)) -> c_1(y,D#(x),x,D#(y)):1
-->_3 D#(*(x,y)) -> c_1(y,D#(x),x,D#(y)):1
-->_2 D#(*(x,y)) -> c_1(y,D#(x),x,D#(y)):1
-->_1 D#(*(x,y)) -> c_1(y,D#(x),x,D#(y)):1
2:W:D#(+(x,y)) -> c_2(D#(x),D#(y))
-->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_2 D#(*(x,y)) -> c_1(y,D#(x),x,D#(y)):1
-->_1 D#(*(x,y)) -> c_1(y,D#(x),x,D#(y)):1
3:W:D#(-(x,y)) -> c_3(D#(x),D#(y))
-->_2 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_1 D#(-(x,y)) -> c_3(D#(x),D#(y)):3
-->_2 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_1 D#(+(x,y)) -> c_2(D#(x),D#(y)):2
-->_2 D#(*(x,y)) -> c_1(y,D#(x),x,D#(y)):1
-->_1 D#(*(x,y)) -> c_1(y,D#(x),x,D#(y)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: D#(*(x,y)) -> c_1(y,D#(x),x,D#(y))
3: D#(-(x,y)) -> c_3(D#(x),D#(y))
2: D#(+(x,y)) -> c_2(D#(x),D#(y))
**** Step 5.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{D/1,D#/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0,c_1/4,c_2/2,c_3/2,c_4/0,c_5/0}
- Obligation:
runtime complexity wrt. defined symbols {D#} and constructors {*,+,-,0,1,constant,t}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))