* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
+ Considered Problem:
- Strict TRS:
a__c() -> b()
a__c() -> c()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(b(),X,c()) -> a__f(X,a__c(),X)
mark(b()) -> b()
mark(c()) -> a__c()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__c/0,a__f/3,mark/1} / {b/0,c/0,f/3}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {b,c,f}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
a__c() -> b()
a__c() -> c()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(b(),X,c()) -> a__f(X,a__c(),X)
mark(b()) -> b()
mark(c()) -> a__c()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__c/0,a__f/3,mark/1} / {b/0,c/0,f/3}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {b,c,f}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
mark(y){y -> f(x,y,z)} =
mark(f(x,y,z)) ->^+ a__f(x,mark(y),z)
= C[mark(y) = mark(y){}]
** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
a__c() -> b()
a__c() -> c()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(b(),X,c()) -> a__f(X,a__c(),X)
mark(b()) -> b()
mark(c()) -> a__c()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__c/0,a__f/3,mark/1} / {b/0,c/0,f/3}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {b,c,f}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
a__c#() -> c_1()
a__c#() -> c_2()
a__f#(X1,X2,X3) -> c_3()
a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#())
mark#(b()) -> c_5()
mark#(c()) -> c_6(a__c#())
mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
Weak DPs
and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
a__c#() -> c_1()
a__c#() -> c_2()
a__f#(X1,X2,X3) -> c_3()
a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#())
mark#(b()) -> c_5()
mark#(c()) -> c_6(a__c#())
mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
- Weak TRS:
a__c() -> b()
a__c() -> c()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(b(),X,c()) -> a__f(X,a__c(),X)
mark(b()) -> b()
mark(c()) -> a__c()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {b,c,f}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2,3,5}
by application of
Pre({1,2,3,5}) = {4,6,7}.
Here rules are labelled as follows:
1: a__c#() -> c_1()
2: a__c#() -> c_2()
3: a__f#(X1,X2,X3) -> c_3()
4: a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#())
5: mark#(b()) -> c_5()
6: mark#(c()) -> c_6(a__c#())
7: mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#())
mark#(c()) -> c_6(a__c#())
mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
- Weak DPs:
a__c#() -> c_1()
a__c#() -> c_2()
a__f#(X1,X2,X3) -> c_3()
mark#(b()) -> c_5()
- Weak TRS:
a__c() -> b()
a__c() -> c()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(b(),X,c()) -> a__f(X,a__c(),X)
mark(b()) -> b()
mark(c()) -> a__c()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {b,c,f}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2}
by application of
Pre({1,2}) = {3}.
Here rules are labelled as follows:
1: a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#())
2: mark#(c()) -> c_6(a__c#())
3: mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
4: a__c#() -> c_1()
5: a__c#() -> c_2()
6: a__f#(X1,X2,X3) -> c_3()
7: mark#(b()) -> c_5()
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
- Weak DPs:
a__c#() -> c_1()
a__c#() -> c_2()
a__f#(X1,X2,X3) -> c_3()
a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#())
mark#(b()) -> c_5()
mark#(c()) -> c_6(a__c#())
- Weak TRS:
a__c() -> b()
a__c() -> c()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(b(),X,c()) -> a__f(X,a__c(),X)
mark(b()) -> b()
mark(c()) -> a__c()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {b,c,f}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
-->_2 mark#(c()) -> c_6(a__c#()):7
-->_1 a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#()):5
-->_2 mark#(b()) -> c_5():6
-->_1 a__f#(X1,X2,X3) -> c_3():4
-->_2 mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2)):1
2:W:a__c#() -> c_1()
3:W:a__c#() -> c_2()
4:W:a__f#(X1,X2,X3) -> c_3()
5:W:a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#())
-->_1 a__f#(X1,X2,X3) -> c_3():4
-->_2 a__c#() -> c_2():3
-->_2 a__c#() -> c_1():2
6:W:mark#(b()) -> c_5()
7:W:mark#(c()) -> c_6(a__c#())
-->_1 a__c#() -> c_2():3
-->_1 a__c#() -> c_1():2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: mark#(b()) -> c_5()
5: a__f#(b(),X,c()) -> c_4(a__f#(X,a__c(),X),a__c#())
4: a__f#(X1,X2,X3) -> c_3()
7: mark#(c()) -> c_6(a__c#())
2: a__c#() -> c_1()
3: a__c#() -> c_2()
** Step 1.b:5: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
- Weak TRS:
a__c() -> b()
a__c() -> c()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(b(),X,c()) -> a__f(X,a__c(),X)
mark(b()) -> b()
mark(c()) -> a__c()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {b,c,f}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2))
-->_2 mark#(f(X1,X2,X3)) -> c_7(a__f#(X1,mark(X2),X3),mark#(X2)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
** Step 1.b:6: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
- Weak TRS:
a__c() -> b()
a__c() -> c()
a__f(X1,X2,X3) -> f(X1,X2,X3)
a__f(b(),X,c()) -> a__f(X,a__c(),X)
mark(b()) -> b()
mark(c()) -> a__c()
mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3)
- Signature:
{a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {b,c,f}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
** Step 1.b:7: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
- Signature:
{a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {b,c,f}
+ 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: mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
The strictly oriented rules are moved into the weak component.
*** Step 1.b:7.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
- Signature:
{a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {b,c,f}
+ 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:
{a__c#,a__f#,mark#}
TcT has computed the following interpretation:
p(a__c) = [0]
p(a__f) = [1] x1 + [1]
p(b) = [2]
p(c) = [0]
p(f) = [1] x2 + [8]
p(mark) = [0]
p(a__c#) = [0]
p(a__f#) = [8] x1 + [1]
p(mark#) = [2] x1 + [6]
p(c_1) = [1]
p(c_2) = [1]
p(c_3) = [1]
p(c_4) = [2] x2 + [0]
p(c_5) = [4]
p(c_6) = [1]
p(c_7) = [1] x1 + [8]
Following rules are strictly oriented:
mark#(f(X1,X2,X3)) = [2] X2 + [22]
> [2] X2 + [14]
= c_7(mark#(X2))
Following rules are (at-least) weakly oriented:
*** Step 1.b:7.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
- Signature:
{a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {b,c,f}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
*** Step 1.b:7.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
- Signature:
{a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {b,c,f}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
-->_1 mark#(f(X1,X2,X3)) -> c_7(mark#(X2)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: mark#(f(X1,X2,X3)) -> c_7(mark#(X2))
*** Step 1.b:7.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{a__c/0,a__f/3,mark/1,a__c#/0,a__f#/3,mark#/1} / {b/0,c/0,f/3,c_1/0,c_2/0,c_3/0,c_4/2,c_5/0,c_6/1,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {a__c#,a__f#,mark#} and constructors {b,c,f}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^1))