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