* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ Considered Problem:
- Strict TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1} / {0/0,cons/2,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose,insert,sort} and constructors {0,cons,nil,s}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1} / {0/0,cons/2,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose,insert,sort} and constructors {0,cons,nil,s}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
choose(x,cons(y,z),u,v){u -> s(u),v -> s(v)} =
choose(x,cons(y,z),s(u),s(v)) ->^+ choose(x,cons(y,z),u,v)
= C[choose(x,cons(y,z),u,v) = choose(x,cons(y,z),u,v){}]
** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1} / {0/0,cons/2,nil/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose,insert,sort} and constructors {0,cons,nil,s}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
choose#(x,cons(v,w),y,0()) -> c_1()
choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
insert#(x,nil()) -> c_5()
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
sort#(nil()) -> c_7()
Weak DPs
and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
choose#(x,cons(v,w),y,0()) -> c_1()
choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
insert#(x,nil()) -> c_5()
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
sort#(nil()) -> c_7()
- Weak TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,5,7}
by application of
Pre({1,5,7}) = {2,3,4,6}.
Here rules are labelled as follows:
1: choose#(x,cons(v,w),y,0()) -> c_1()
2: choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
3: choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
4: insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
5: insert#(x,nil()) -> c_5()
6: sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
7: sort#(nil()) -> c_7()
** Step 1.b:3: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
- Weak DPs:
choose#(x,cons(v,w),y,0()) -> c_1()
insert#(x,nil()) -> c_5()
sort#(nil()) -> c_7()
- Weak TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
-->_1 insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)):3
-->_1 insert#(x,nil()) -> c_5():6
2:S:choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
-->_1 choose#(x,cons(v,w),y,0()) -> c_1():5
-->_1 choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)):2
-->_1 choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)):1
3:S:insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
-->_1 choose#(x,cons(v,w),y,0()) -> c_1():5
-->_1 choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)):2
-->_1 choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)):1
4:S:sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
-->_2 sort#(nil()) -> c_7():7
-->_1 insert#(x,nil()) -> c_5():6
-->_2 sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y)):4
-->_1 insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)):3
5:W:choose#(x,cons(v,w),y,0()) -> c_1()
6:W:insert#(x,nil()) -> c_5()
7:W:sort#(nil()) -> c_7()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
7: sort#(nil()) -> c_7()
6: insert#(x,nil()) -> c_5()
5: choose#(x,cons(v,w),y,0()) -> c_1()
** Step 1.b:4: Decompose WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
- Weak TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,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:
choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
- Weak DPs:
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
- Weak TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1
,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s}
Problem (S)
- Strict DPs:
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
- Weak DPs:
choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
- Weak TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1
,c_5/0,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s}
*** Step 1.b:4.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
- Weak DPs:
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
- Weak TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,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:
3: insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
The strictly oriented rules are moved into the weak component.
**** Step 1.b:4.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
- Weak DPs:
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
- Weak TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,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_3) = {1},
uargs(c_4) = {1},
uargs(c_6) = {1,2}
Following symbols are considered usable:
{choose,insert,sort,choose#,insert#,sort#}
TcT has computed the following interpretation:
p(0) = 0
p(choose) = 2 + x2
p(cons) = 1 + x2
p(insert) = 2 + x2
p(nil) = 0
p(s) = 0
p(sort) = 2*x1
p(choose#) = x2
p(insert#) = 1 + x2
p(sort#) = 2*x1 + 2*x1^2
p(c_1) = 1
p(c_2) = x1
p(c_3) = x1
p(c_4) = x1
p(c_5) = 1
p(c_6) = x1 + x2
p(c_7) = 0
Following rules are strictly oriented:
insert#(x,cons(v,w)) = 2 + w
> 1 + w
= c_4(choose#(x,cons(v,w),x,v))
Following rules are (at-least) weakly oriented:
choose#(x,cons(v,w),0(),s(z)) = 1 + w
>= 1 + w
= c_2(insert#(x,w))
choose#(x,cons(v,w),s(y),s(z)) = 1 + w
>= 1 + w
= c_3(choose#(x,cons(v,w),y,z))
sort#(cons(x,y)) = 4 + 6*y + 2*y^2
>= 1 + 4*y + 2*y^2
= c_6(insert#(x,sort(y)),sort#(y))
choose(x,cons(v,w),y,0()) = 3 + w
>= 2 + w
= cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) = 3 + w
>= 3 + w
= cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) = 3 + w
>= 3 + w
= choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) = 3 + w
>= 3 + w
= choose(x,cons(v,w),x,v)
insert(x,nil()) = 2
>= 1
= cons(x,nil())
sort(cons(x,y)) = 2 + 2*y
>= 2 + 2*y
= insert(x,sort(y))
sort(nil()) = 0
>= 0
= nil()
**** Step 1.b:4.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
- Weak DPs:
insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
- Weak TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 1.b:4.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
- Weak DPs:
insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
- Weak TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
The strictly oriented rules are moved into the weak component.
***** Step 1.b:4.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
- Weak DPs:
insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
- Weak TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s}
+ Applied Processor:
NaturalMI {miDimension = 2, miDegree = 2, 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_2) = {1},
uargs(c_3) = {1},
uargs(c_4) = {1},
uargs(c_6) = {1,2}
Following symbols are considered usable:
{choose,insert,sort,choose#,insert#,sort#}
TcT has computed the following interpretation:
p(0) = [2]
[0]
p(choose) = [1 1] x2 + [1]
[0 1] [1]
p(cons) = [1 1] x2 + [1]
[0 1] [1]
p(insert) = [1 1] x2 + [1]
[0 1] [1]
p(nil) = [0]
[0]
p(s) = [0 2] x1 + [0]
[0 0] [0]
p(sort) = [1 2] x1 + [0]
[0 1] [2]
p(choose#) = [0 0] x1 + [0 2] x2 + [0 0] x3 + [0 0] x4 + [2]
[0 1] [2 0] [1 0] [2 0] [3]
p(insert#) = [0 0] x1 + [0 2] x2 + [2]
[0 1] [0 0] [1]
p(sort#) = [3 3] x1 + [0]
[1 3] [2]
p(c_1) = [2]
[0]
p(c_2) = [1 0] x1 + [1]
[0 1] [0]
p(c_3) = [1 0] x1 + [0]
[1 0] [1]
p(c_4) = [1 0] x1 + [0]
[0 0] [0]
p(c_5) = [0]
[1]
p(c_6) = [1 0] x1 + [1 0] x2 + [0]
[1 0] [0 0] [0]
p(c_7) = [2]
[1]
Following rules are strictly oriented:
choose#(x,cons(v,w),0(),s(z)) = [0 2] w + [0 0] x + [0 0] z + [4]
[2 2] [0 1] [0 4] [7]
> [0 2] w + [0 0] x + [3]
[0 0] [0 1] [1]
= c_2(insert#(x,w))
Following rules are (at-least) weakly oriented:
choose#(x,cons(v,w),s(y),s(z)) = [0 2] w + [0 0] x + [0 0] y + [0 0] z + [4]
[2 2] [0 1] [0 2] [0 4] [5]
>= [0 2] w + [4]
[0 2] [5]
= c_3(choose#(x,cons(v,w),y,z))
insert#(x,cons(v,w)) = [0 2] w + [0 0] x + [4]
[0 0] [0 1] [1]
>= [0 2] w + [4]
[0 0] [0]
= c_4(choose#(x,cons(v,w),x,v))
sort#(cons(x,y)) = [3 6] y + [6]
[1 4] [6]
>= [3 5] y + [6]
[0 2] [6]
= c_6(insert#(x,sort(y)),sort#(y))
choose(x,cons(v,w),y,0()) = [1 2] w + [3]
[0 1] [2]
>= [1 2] w + [3]
[0 1] [2]
= cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) = [1 2] w + [3]
[0 1] [2]
>= [1 2] w + [3]
[0 1] [2]
= cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) = [1 2] w + [3]
[0 1] [2]
>= [1 2] w + [3]
[0 1] [2]
= choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) = [1 2] w + [3]
[0 1] [2]
>= [1 2] w + [3]
[0 1] [2]
= choose(x,cons(v,w),x,v)
insert(x,nil()) = [1]
[1]
>= [1]
[1]
= cons(x,nil())
sort(cons(x,y)) = [1 3] y + [3]
[0 1] [3]
>= [1 3] y + [3]
[0 1] [3]
= insert(x,sort(y))
sort(nil()) = [0]
[2]
>= [0]
[0]
= nil()
***** Step 1.b:4.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
- Weak DPs:
choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
- Weak TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 1.b:4.a:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
- Weak DPs:
choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
- Weak TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,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: choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
The strictly oriented rules are moved into the weak component.
****** Step 1.b:4.a:1.b:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
- Weak DPs:
choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
- Weak TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,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_3) = {1},
uargs(c_4) = {1},
uargs(c_6) = {1,2}
Following symbols are considered usable:
{choose,insert,sort,choose#,insert#,sort#}
TcT has computed the following interpretation:
p(0) = 0
p(choose) = 1 + x1 + x2
p(cons) = 1 + x1 + x2
p(insert) = 1 + x1 + x2
p(nil) = 0
p(s) = 1 + x1
p(sort) = 2*x1
p(choose#) = 2 + x1*x2 + x1^2 + 2*x2 + x3
p(insert#) = 2 + x1 + x1*x2 + x1^2 + 2*x2
p(sort#) = 2*x1^2
p(c_1) = 1
p(c_2) = x1
p(c_3) = x1
p(c_4) = x1
p(c_5) = 0
p(c_6) = x1 + x2
p(c_7) = 0
Following rules are strictly oriented:
choose#(x,cons(v,w),s(y),s(z)) = 5 + 2*v + v*x + 2*w + w*x + x + x^2 + y
> 4 + 2*v + v*x + 2*w + w*x + x + x^2 + y
= c_3(choose#(x,cons(v,w),y,z))
Following rules are (at-least) weakly oriented:
choose#(x,cons(v,w),0(),s(z)) = 4 + 2*v + v*x + 2*w + w*x + x + x^2
>= 2 + 2*w + w*x + x + x^2
= c_2(insert#(x,w))
insert#(x,cons(v,w)) = 4 + 2*v + v*x + 2*w + w*x + 2*x + x^2
>= 4 + 2*v + v*x + 2*w + w*x + 2*x + x^2
= c_4(choose#(x,cons(v,w),x,v))
sort#(cons(x,y)) = 2 + 4*x + 4*x*y + 2*x^2 + 4*y + 2*y^2
>= 2 + x + 2*x*y + x^2 + 4*y + 2*y^2
= c_6(insert#(x,sort(y)),sort#(y))
choose(x,cons(v,w),y,0()) = 2 + v + w + x
>= 2 + v + w + x
= cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) = 2 + v + w + x
>= 2 + v + w + x
= cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) = 2 + v + w + x
>= 2 + v + w + x
= choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) = 2 + v + w + x
>= 2 + v + w + x
= choose(x,cons(v,w),x,v)
insert(x,nil()) = 1 + x
>= 1 + x
= cons(x,nil())
sort(cons(x,y)) = 2 + 2*x + 2*y
>= 1 + x + 2*y
= insert(x,sort(y))
sort(nil()) = 0
>= 0
= nil()
****** Step 1.b:4.a:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
- Weak TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
****** Step 1.b:4.a:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
- Weak TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
-->_1 insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)):3
2:W:choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
-->_1 choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)):2
-->_1 choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)):1
3:W:insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
-->_1 choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)):2
-->_1 choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)):1
4:W:sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
-->_2 sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y)):4
-->_1 insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
1: choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
3: insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
2: choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
****** Step 1.b:4.a:1.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 1.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
- Weak DPs:
choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
- Weak TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
-->_1 insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)):4
-->_2 sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y)):1
2:W:choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
-->_1 insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)):4
3:W:choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
-->_1 choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)):3
-->_1 choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)):2
4:W:insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
-->_1 choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)):3
-->_1 choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v))
2: choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
3: choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
*** Step 1.b:4.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
- Weak TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/2,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
-->_2 sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
sort#(cons(x,y)) -> c_6(sort#(y))
*** Step 1.b:4.b:3: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sort#(cons(x,y)) -> c_6(sort#(y))
- Weak TRS:
choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v)
insert(x,nil()) -> cons(x,nil())
sort(cons(x,y)) -> insert(x,sort(y))
sort(nil()) -> nil()
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/1,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
sort#(cons(x,y)) -> c_6(sort#(y))
*** Step 1.b:4.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sort#(cons(x,y)) -> c_6(sort#(y))
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/1,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,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: sort#(cons(x,y)) -> c_6(sort#(y))
The strictly oriented rules are moved into the weak component.
**** Step 1.b:4.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sort#(cons(x,y)) -> c_6(sort#(y))
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/1,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,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_6) = {1}
Following symbols are considered usable:
{choose#,insert#,sort#}
TcT has computed the following interpretation:
p(0) = [1]
p(choose) = [2] x2 + [0]
p(cons) = [1] x1 + [1] x2 + [8]
p(insert) = [1]
p(nil) = [1]
p(s) = [4]
p(sort) = [2] x1 + [1]
p(choose#) = [1] x1 + [4] x3 + [1]
p(insert#) = [1] x2 + [0]
p(sort#) = [2] x1 + [1]
p(c_1) = [1]
p(c_2) = [1]
p(c_3) = [1] x1 + [0]
p(c_4) = [2]
p(c_5) = [0]
p(c_6) = [1] x1 + [9]
p(c_7) = [0]
Following rules are strictly oriented:
sort#(cons(x,y)) = [2] x + [2] y + [17]
> [2] y + [10]
= c_6(sort#(y))
Following rules are (at-least) weakly oriented:
**** Step 1.b:4.b:4.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
sort#(cons(x,y)) -> c_6(sort#(y))
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/1,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 1.b:4.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
sort#(cons(x,y)) -> c_6(sort#(y))
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/1,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:sort#(cons(x,y)) -> c_6(sort#(y))
-->_1 sort#(cons(x,y)) -> c_6(sort#(y)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: sort#(cons(x,y)) -> c_6(sort#(y))
**** Step 1.b:4.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0
,c_6/1,c_7/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^2))