*** 1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
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()
Weak DP Rules:
Weak TRS Rules:
Signature:
{choose/4,insert/2,sort/1} / {0/0,cons/2,nil/0,s/1}
Obligation:
Innermost
basic terms: {choose,insert,sort}/{0,cons,nil,s}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
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.
*** 1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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()
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {choose#,insert#,sort#}/{0,cons,nil,s}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
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()
*** 1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
choose#(x,cons(v,w),y,0()) -> c_1()
insert#(x,nil()) -> c_5()
sort#(nil()) -> c_7()
Weak TRS Rules:
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
basic terms: {choose#,insert#,sort#}/{0,cons,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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()
*** 1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {choose#,insert#,sort#}/{0,cons,nil,s}
Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
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 DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
Weak TRS Rules:
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
basic terms: {choose#,insert#,sort#}/{0,cons,nil,s}
Problem (S)
Strict DP Rules:
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
Strict TRS Rules:
Weak DP Rules:
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 Rules:
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
basic terms: {choose#,insert#,sort#}/{0,cons,nil,s}
*** 1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
Weak TRS Rules:
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
basic terms: {choose#,insert#,sort#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} 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.
*** 1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
Weak TRS Rules:
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
basic terms: {choose#,insert#,sort#}/{0,cons,nil,s}
Applied Processor:
NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
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) = [1]
[2]
p(choose) = [2 0] x1 + [1 1] x2 + [0]
[0 0] [0 1] [1]
p(cons) = [1 0] x1 + [1 1] x2 + [0]
[0 0] [0 1] [1]
p(insert) = [2 0] x1 + [1 1] x2 + [0]
[0 0] [0 1] [1]
p(nil) = [0]
[0]
p(s) = [0 0] x1 + [0]
[0 1] [0]
p(sort) = [3 0] x1 + [0]
[0 1] [0]
p(choose#) = [0 0] x1 + [0 1] x2 + [0
0] x3 + [0 0] x4 + [0]
[0 3] [2 2] [2
0] [0 2] [2]
p(insert#) = [0 1] x2 + [0]
[0 0] [1]
p(sort#) = [1 2] x1 + [2]
[1 2] [0]
p(c_1) = [0]
[0]
p(c_2) = [1 0] x1 + [0]
[2 0] [1]
p(c_3) = [1 0] x1 + [0]
[0 0] [1]
p(c_4) = [1 0] x1 + [0]
[0 0] [0]
p(c_5) = [0]
[0]
p(c_6) = [1 2] x1 + [1 0] x2 + [0]
[1 1] [0 0] [0]
p(c_7) = [1]
[2]
Following rules are strictly oriented:
choose#(x,cons(v,w),0(),s(z)) = [0 0] v + [0 1] w + [0 0] x + [0
0] z + [1]
[2 0] [2 4] [0 3] [0
2] [6]
> [0 1] w + [0]
[0 2] [1]
= c_2(insert#(x,w))
Following rules are (at-least) weakly oriented:
choose#(x,cons(v,w),s(y),s(z)) = [0 0] v + [0 1] w + [0 0] x + [0
0] z + [1]
[2 0] [2 4] [0 3] [0
2] [4]
>= [0 1] w + [1]
[0 0] [1]
= c_3(choose#(x,cons(v,w),y,z))
insert#(x,cons(v,w)) = [0 1] w + [1]
[0 0] [1]
>= [0 1] w + [1]
[0 0] [0]
= c_4(choose#(x,cons(v,w),x,v))
sort#(cons(x,y)) = [1 0] x + [1 3] y + [4]
[1 0] [1 3] [2]
>= [1 3] y + [4]
[0 1] [1]
= c_6(insert#(x,sort(y)),sort#(y))
choose(x,cons(v,w),y,0()) = [1 0] v + [1 2] w + [2
0] x + [1]
[0 0] [0 1] [0
0] [2]
>= [1 0] v + [1 2] w + [1
0] x + [1]
[0 0] [0 1] [0
0] [2]
= cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) = [1 0] v + [1 2] w + [2
0] x + [1]
[0 0] [0 1] [0
0] [2]
>= [1 0] v + [1 2] w + [2
0] x + [1]
[0 0] [0 1] [0
0] [2]
= cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) = [1 0] v + [1 2] w + [2
0] x + [1]
[0 0] [0 1] [0
0] [2]
>= [1 0] v + [1 2] w + [2
0] x + [1]
[0 0] [0 1] [0
0] [2]
= choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) = [1 0] v + [1 2] w + [2
0] x + [1]
[0 0] [0 1] [0
0] [2]
>= [1 0] v + [1 2] w + [2
0] x + [1]
[0 0] [0 1] [0
0] [2]
= choose(x,cons(v,w),x,v)
insert(x,nil()) = [2 0] x + [0]
[0 0] [1]
>= [1 0] x + [0]
[0 0] [1]
= cons(x,nil())
sort(cons(x,y)) = [3 0] x + [3 3] y + [0]
[0 0] [0 1] [1]
>= [2 0] x + [3 1] y + [0]
[0 0] [0 1] [1]
= insert(x,sort(y))
sort(nil()) = [0]
[0]
>= [0]
[0]
= nil()
*** 1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
Weak TRS Rules:
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
basic terms: {choose#,insert#,sort#}/{0,cons,nil,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
Weak TRS Rules:
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
basic terms: {choose#,insert#,sort#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
2: insert#(x,cons(v,w)) ->
c_4(choose#(x,cons(v,w),x,v))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w))
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
Weak TRS Rules:
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
basic terms: {choose#,insert#,sort#}/{0,cons,nil,s}
Applied Processor:
NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
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) = [0]
[0]
p(choose) = [1 1] x2 + [0]
[0 1] [1]
p(cons) = [1 1] x2 + [0]
[0 1] [1]
p(insert) = [1 1] x2 + [0]
[0 1] [1]
p(nil) = [0]
[0]
p(s) = [0 1] x1 + [1]
[0 0] [1]
p(sort) = [2 2] x1 + [0]
[0 2] [0]
p(choose#) = [0 1] x2 + [0 0] x3 + [0
0] x4 + [1]
[2 2] [1 0] [2
1] [0]
p(insert#) = [0 1] x2 + [2]
[0 0] [1]
p(sort#) = [2 2] x1 + [0]
[0 1] [0]
p(c_1) = [0]
[0]
p(c_2) = [1 0] x1 + [0]
[1 3] [0]
p(c_3) = [1 0] x1 + [0]
[2 0] [0]
p(c_4) = [1 0] x1 + [0]
[0 0] [1]
p(c_5) = [0]
[1]
p(c_6) = [1 0] x1 + [1 0] x2 + [0]
[0 0] [0 1] [0]
p(c_7) = [2]
[0]
Following rules are strictly oriented:
insert#(x,cons(v,w)) = [0 1] w + [3]
[0 0] [1]
> [0 1] w + [2]
[0 0] [1]
= c_4(choose#(x,cons(v,w),x,v))
Following rules are (at-least) weakly oriented:
choose#(x,cons(v,w),0(),s(z)) = [0 1] w + [0 0] z + [2]
[2 4] [0 2] [5]
>= [0 1] w + [2]
[0 1] [5]
= c_2(insert#(x,w))
choose#(x,cons(v,w),s(y),s(z)) = [0 1] w + [0 0] y + [0
0] z + [2]
[2 4] [0 1] [0
2] [6]
>= [0 1] w + [2]
[0 2] [4]
= c_3(choose#(x,cons(v,w),y,z))
sort#(cons(x,y)) = [2 4] y + [2]
[0 1] [1]
>= [2 4] y + [2]
[0 1] [0]
= c_6(insert#(x,sort(y)),sort#(y))
choose(x,cons(v,w),y,0()) = [1 2] w + [1]
[0 1] [2]
>= [1 2] w + [1]
[0 1] [2]
= cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) = [1 2] w + [1]
[0 1] [2]
>= [1 2] w + [1]
[0 1] [2]
= cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) = [1 2] w + [1]
[0 1] [2]
>= [1 2] w + [1]
[0 1] [2]
= choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) = [1 2] w + [1]
[0 1] [2]
>= [1 2] w + [1]
[0 1] [2]
= choose(x,cons(v,w),x,v)
insert(x,nil()) = [0]
[1]
>= [0]
[1]
= cons(x,nil())
sort(cons(x,y)) = [2 4] y + [2]
[0 2] [2]
>= [2 4] y + [0]
[0 2] [1]
= insert(x,sort(y))
sort(nil()) = [0]
[0]
>= [0]
[0]
= nil()
*** 1.1.1.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
Strict TRS Rules:
Weak DP Rules:
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 Rules:
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
basic terms: {choose#,insert#,sort#}/{0,cons,nil,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.2.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
Strict TRS Rules:
Weak DP Rules:
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 Rules:
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
basic terms: {choose#,insert#,sort#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} 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.
*** 1.1.1.1.1.2.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z))
Strict TRS Rules:
Weak DP Rules:
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 Rules:
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
basic terms: {choose#,insert#,sort#}/{0,cons,nil,s}
Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
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) = 1
p(choose) = 1 + 2*x1 + x2
p(cons) = 1 + x1 + x2
p(insert) = 1 + 2*x1 + x2
p(nil) = 0
p(s) = 1 + x1
p(sort) = 1 + 2*x1
p(choose#) = 2*x1*x2 + x2 + x3
p(insert#) = 1 + 2*x1 + 2*x1*x2 + x2
p(sort#) = 2*x1 + 2*x1^2
p(c_1) = 0
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:
choose#(x,cons(v,w),s(y),s(z)) = 2 + v + 2*v*x + w + 2*w*x + 2*x + y
> 1 + v + 2*v*x + w + 2*w*x + 2*x + 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)) = 2 + v + 2*v*x + w + 2*w*x + 2*x
>= 1 + w + 2*w*x + 2*x
= c_2(insert#(x,w))
insert#(x,cons(v,w)) = 2 + v + 2*v*x + w + 2*w*x + 4*x
>= 1 + v + 2*v*x + w + 2*w*x + 3*x
= c_4(choose#(x,cons(v,w),x,v))
sort#(cons(x,y)) = 4 + 6*x + 4*x*y + 2*x^2 + 6*y + 2*y^2
>= 2 + 4*x + 4*x*y + 4*y + 2*y^2
= c_6(insert#(x,sort(y)),sort#(y))
choose(x,cons(v,w),y,0()) = 2 + v + w + 2*x
>= 2 + v + w + x
= cons(x,cons(v,w))
choose(x,cons(v,w),0(),s(z)) = 2 + v + w + 2*x
>= 2 + v + w + 2*x
= cons(v,insert(x,w))
choose(x,cons(v,w),s(y),s(z)) = 2 + v + w + 2*x
>= 2 + v + w + 2*x
= choose(x,cons(v,w),y,z)
insert(x,cons(v,w)) = 2 + v + w + 2*x
>= 2 + v + w + 2*x
= choose(x,cons(v,w),x,v)
insert(x,nil()) = 1 + 2*x
>= 1 + x
= cons(x,nil())
sort(cons(x,y)) = 3 + 2*x + 2*y
>= 2 + 2*x + 2*y
= insert(x,sort(y))
sort(nil()) = 1
>= 0
= nil()
*** 1.1.1.1.1.2.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
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 Rules:
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
basic terms: {choose#,insert#,sort#}/{0,cons,nil,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.2.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
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 Rules:
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
basic terms: {choose#,insert#,sort#}/{0,cons,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.2.2.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {choose#,insert#,sort#}/{0,cons,nil,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
Strict TRS Rules:
Weak DP Rules:
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 Rules:
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
basic terms: {choose#,insert#,sort#}/{0,cons,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {choose#,insert#,sort#}/{0,cons,nil,s}
Applied Processor:
SimplifyRHS
Proof:
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))
*** 1.1.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sort#(cons(x,y)) -> c_6(sort#(y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {choose#,insert#,sort#}/{0,cons,nil,s}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
sort#(cons(x,y)) -> c_6(sort#(y))
*** 1.1.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sort#(cons(x,y)) -> c_6(sort#(y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {choose#,insert#,sort#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: sort#(cons(x,y)) ->
c_6(sort#(y))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.2.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sort#(cons(x,y)) -> c_6(sort#(y))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {choose#,insert#,sort#}/{0,cons,nil,s}
Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
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) = [8]
p(choose) = [8] x4 + [2]
p(cons) = [1] x2 + [5]
p(insert) = [1] x1 + [1] x2 + [1]
p(nil) = [1]
p(s) = [0]
p(sort) = [1]
p(choose#) = [1] x4 + [2]
p(insert#) = [2] x1 + [1] x2 + [1]
p(sort#) = [4] x1 + [8]
p(c_1) = [0]
p(c_2) = [2] x1 + [0]
p(c_3) = [1]
p(c_4) = [2]
p(c_5) = [0]
p(c_6) = [1] x1 + [15]
p(c_7) = [1]
Following rules are strictly oriented:
sort#(cons(x,y)) = [4] y + [28]
> [4] y + [23]
= c_6(sort#(y))
Following rules are (at-least) weakly oriented:
*** 1.1.1.1.2.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
sort#(cons(x,y)) -> c_6(sort#(y))
Weak TRS Rules:
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
basic terms: {choose#,insert#,sort#}/{0,cons,nil,s}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.2.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
sort#(cons(x,y)) -> c_6(sort#(y))
Weak TRS Rules:
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
basic terms: {choose#,insert#,sort#}/{0,cons,nil,s}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.2.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {choose#,insert#,sort#}/{0,cons,nil,s}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).