WORST_CASE(?,O(n^3))
* Step 1: DependencyPairs WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
if'insert(false(),x,y,ys) -> cons(y,insert(x,ys))
if'insert(true(),x,y,ys) -> cons(x,cons(y,ys))
insert(x,cons(y,ys)) -> if'insert(leq(x,y),x,y,ys)
insert(x,nil()) -> cons(x,nil())
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
sort(cons(x,xs)) -> insert(x,sort(xs))
sort(nil()) -> nil()
- Signature:
{if'insert/4,insert/2,leq/2,sort/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if'insert,insert,leq,sort} and constructors {0,cons,false
,nil,s,true}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
if'insert#(false(),x,y,ys) -> c_1(insert#(x,ys))
if'insert#(true(),x,y,ys) -> c_2()
insert#(x,cons(y,ys)) -> c_3(if'insert#(leq(x,y),x,y,ys),leq#(x,y))
insert#(x,nil()) -> c_4()
leq#(0(),y) -> c_5()
leq#(s(x),0()) -> c_6()
leq#(s(x),s(y)) -> c_7(leq#(x,y))
sort#(cons(x,xs)) -> c_8(insert#(x,sort(xs)),sort#(xs))
sort#(nil()) -> c_9()
Weak DPs
and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
if'insert#(false(),x,y,ys) -> c_1(insert#(x,ys))
if'insert#(true(),x,y,ys) -> c_2()
insert#(x,cons(y,ys)) -> c_3(if'insert#(leq(x,y),x,y,ys),leq#(x,y))
insert#(x,nil()) -> c_4()
leq#(0(),y) -> c_5()
leq#(s(x),0()) -> c_6()
leq#(s(x),s(y)) -> c_7(leq#(x,y))
sort#(cons(x,xs)) -> c_8(insert#(x,sort(xs)),sort#(xs))
sort#(nil()) -> c_9()
- Weak TRS:
if'insert(false(),x,y,ys) -> cons(y,insert(x,ys))
if'insert(true(),x,y,ys) -> cons(x,cons(y,ys))
insert(x,cons(y,ys)) -> if'insert(leq(x,y),x,y,ys)
insert(x,nil()) -> cons(x,nil())
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
sort(cons(x,xs)) -> insert(x,sort(xs))
sort(nil()) -> nil()
- Signature:
{if'insert/4,insert/2,leq/2,sort/1,if'insert#/4,insert#/2,leq#/2,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1
,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/2,c_9/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if'insert#,insert#,leq#,sort#} and constructors {0,cons
,false,nil,s,true}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{2,4,5,6,9}
by application of
Pre({2,4,5,6,9}) = {1,3,7,8}.
Here rules are labelled as follows:
1: if'insert#(false(),x,y,ys) -> c_1(insert#(x,ys))
2: if'insert#(true(),x,y,ys) -> c_2()
3: insert#(x,cons(y,ys)) -> c_3(if'insert#(leq(x,y),x,y,ys),leq#(x,y))
4: insert#(x,nil()) -> c_4()
5: leq#(0(),y) -> c_5()
6: leq#(s(x),0()) -> c_6()
7: leq#(s(x),s(y)) -> c_7(leq#(x,y))
8: sort#(cons(x,xs)) -> c_8(insert#(x,sort(xs)),sort#(xs))
9: sort#(nil()) -> c_9()
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
if'insert#(false(),x,y,ys) -> c_1(insert#(x,ys))
insert#(x,cons(y,ys)) -> c_3(if'insert#(leq(x,y),x,y,ys),leq#(x,y))
leq#(s(x),s(y)) -> c_7(leq#(x,y))
sort#(cons(x,xs)) -> c_8(insert#(x,sort(xs)),sort#(xs))
- Weak DPs:
if'insert#(true(),x,y,ys) -> c_2()
insert#(x,nil()) -> c_4()
leq#(0(),y) -> c_5()
leq#(s(x),0()) -> c_6()
sort#(nil()) -> c_9()
- Weak TRS:
if'insert(false(),x,y,ys) -> cons(y,insert(x,ys))
if'insert(true(),x,y,ys) -> cons(x,cons(y,ys))
insert(x,cons(y,ys)) -> if'insert(leq(x,y),x,y,ys)
insert(x,nil()) -> cons(x,nil())
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
sort(cons(x,xs)) -> insert(x,sort(xs))
sort(nil()) -> nil()
- Signature:
{if'insert/4,insert/2,leq/2,sort/1,if'insert#/4,insert#/2,leq#/2,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1
,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/2,c_9/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if'insert#,insert#,leq#,sort#} and constructors {0,cons
,false,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:if'insert#(false(),x,y,ys) -> c_1(insert#(x,ys))
-->_1 insert#(x,cons(y,ys)) -> c_3(if'insert#(leq(x,y),x,y,ys),leq#(x,y)):2
-->_1 insert#(x,nil()) -> c_4():6
2:S:insert#(x,cons(y,ys)) -> c_3(if'insert#(leq(x,y),x,y,ys),leq#(x,y))
-->_2 leq#(s(x),s(y)) -> c_7(leq#(x,y)):3
-->_2 leq#(s(x),0()) -> c_6():8
-->_2 leq#(0(),y) -> c_5():7
-->_1 if'insert#(true(),x,y,ys) -> c_2():5
-->_1 if'insert#(false(),x,y,ys) -> c_1(insert#(x,ys)):1
3:S:leq#(s(x),s(y)) -> c_7(leq#(x,y))
-->_1 leq#(s(x),0()) -> c_6():8
-->_1 leq#(0(),y) -> c_5():7
-->_1 leq#(s(x),s(y)) -> c_7(leq#(x,y)):3
4:S:sort#(cons(x,xs)) -> c_8(insert#(x,sort(xs)),sort#(xs))
-->_2 sort#(nil()) -> c_9():9
-->_1 insert#(x,nil()) -> c_4():6
-->_2 sort#(cons(x,xs)) -> c_8(insert#(x,sort(xs)),sort#(xs)):4
-->_1 insert#(x,cons(y,ys)) -> c_3(if'insert#(leq(x,y),x,y,ys),leq#(x,y)):2
5:W:if'insert#(true(),x,y,ys) -> c_2()
6:W:insert#(x,nil()) -> c_4()
7:W:leq#(0(),y) -> c_5()
8:W:leq#(s(x),0()) -> c_6()
9:W:sort#(nil()) -> c_9()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
9: sort#(nil()) -> c_9()
6: insert#(x,nil()) -> c_4()
5: if'insert#(true(),x,y,ys) -> c_2()
7: leq#(0(),y) -> c_5()
8: leq#(s(x),0()) -> c_6()
* Step 4: DecomposeDG WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
if'insert#(false(),x,y,ys) -> c_1(insert#(x,ys))
insert#(x,cons(y,ys)) -> c_3(if'insert#(leq(x,y),x,y,ys),leq#(x,y))
leq#(s(x),s(y)) -> c_7(leq#(x,y))
sort#(cons(x,xs)) -> c_8(insert#(x,sort(xs)),sort#(xs))
- Weak TRS:
if'insert(false(),x,y,ys) -> cons(y,insert(x,ys))
if'insert(true(),x,y,ys) -> cons(x,cons(y,ys))
insert(x,cons(y,ys)) -> if'insert(leq(x,y),x,y,ys)
insert(x,nil()) -> cons(x,nil())
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
sort(cons(x,xs)) -> insert(x,sort(xs))
sort(nil()) -> nil()
- Signature:
{if'insert/4,insert/2,leq/2,sort/1,if'insert#/4,insert#/2,leq#/2,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1
,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/2,c_9/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if'insert#,insert#,leq#,sort#} and constructors {0,cons
,false,nil,s,true}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
sort#(cons(x,xs)) -> c_8(insert#(x,sort(xs)),sort#(xs))
and a lower component
if'insert#(false(),x,y,ys) -> c_1(insert#(x,ys))
insert#(x,cons(y,ys)) -> c_3(if'insert#(leq(x,y),x,y,ys),leq#(x,y))
leq#(s(x),s(y)) -> c_7(leq#(x,y))
Further, following extension rules are added to the lower component.
sort#(cons(x,xs)) -> insert#(x,sort(xs))
sort#(cons(x,xs)) -> sort#(xs)
** Step 4.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sort#(cons(x,xs)) -> c_8(insert#(x,sort(xs)),sort#(xs))
- Weak TRS:
if'insert(false(),x,y,ys) -> cons(y,insert(x,ys))
if'insert(true(),x,y,ys) -> cons(x,cons(y,ys))
insert(x,cons(y,ys)) -> if'insert(leq(x,y),x,y,ys)
insert(x,nil()) -> cons(x,nil())
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
sort(cons(x,xs)) -> insert(x,sort(xs))
sort(nil()) -> nil()
- Signature:
{if'insert/4,insert/2,leq/2,sort/1,if'insert#/4,insert#/2,leq#/2,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1
,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/2,c_9/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if'insert#,insert#,leq#,sort#} and constructors {0,cons
,false,nil,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:sort#(cons(x,xs)) -> c_8(insert#(x,sort(xs)),sort#(xs))
-->_2 sort#(cons(x,xs)) -> c_8(insert#(x,sort(xs)),sort#(xs)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
sort#(cons(x,xs)) -> c_8(sort#(xs))
** Step 4.a:2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sort#(cons(x,xs)) -> c_8(sort#(xs))
- Weak TRS:
if'insert(false(),x,y,ys) -> cons(y,insert(x,ys))
if'insert(true(),x,y,ys) -> cons(x,cons(y,ys))
insert(x,cons(y,ys)) -> if'insert(leq(x,y),x,y,ys)
insert(x,nil()) -> cons(x,nil())
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
sort(cons(x,xs)) -> insert(x,sort(xs))
sort(nil()) -> nil()
- Signature:
{if'insert/4,insert/2,leq/2,sort/1,if'insert#/4,insert#/2,leq#/2,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1
,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if'insert#,insert#,leq#,sort#} and constructors {0,cons
,false,nil,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
sort#(cons(x,xs)) -> c_8(sort#(xs))
** Step 4.a:3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
sort#(cons(x,xs)) -> c_8(sort#(xs))
- Signature:
{if'insert/4,insert/2,leq/2,sort/1,if'insert#/4,insert#/2,leq#/2,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1
,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if'insert#,insert#,leq#,sort#} and constructors {0,cons
,false,nil,s,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_8) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(cons) = [1] x1 + [1] x2 + [9]
p(false) = [0]
p(if'insert) = [2]
p(insert) = [0]
p(leq) = [0]
p(nil) = [0]
p(s) = [1] x1 + [0]
p(sort) = [0]
p(true) = [0]
p(if'insert#) = [0]
p(insert#) = [0]
p(leq#) = [0]
p(sort#) = [2] x1 + [11]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [1] x1 + [5]
p(c_9) = [0]
Following rules are strictly oriented:
sort#(cons(x,xs)) = [2] x + [2] xs + [29]
> [2] xs + [16]
= c_8(sort#(xs))
Following rules are (at-least) weakly oriented:
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 4.a:4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
sort#(cons(x,xs)) -> c_8(sort#(xs))
- Signature:
{if'insert/4,insert/2,leq/2,sort/1,if'insert#/4,insert#/2,leq#/2,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1
,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/1,c_9/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if'insert#,insert#,leq#,sort#} and constructors {0,cons
,false,nil,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
** Step 4.b:1: DecomposeDG WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
if'insert#(false(),x,y,ys) -> c_1(insert#(x,ys))
insert#(x,cons(y,ys)) -> c_3(if'insert#(leq(x,y),x,y,ys),leq#(x,y))
leq#(s(x),s(y)) -> c_7(leq#(x,y))
- Weak DPs:
sort#(cons(x,xs)) -> insert#(x,sort(xs))
sort#(cons(x,xs)) -> sort#(xs)
- Weak TRS:
if'insert(false(),x,y,ys) -> cons(y,insert(x,ys))
if'insert(true(),x,y,ys) -> cons(x,cons(y,ys))
insert(x,cons(y,ys)) -> if'insert(leq(x,y),x,y,ys)
insert(x,nil()) -> cons(x,nil())
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
sort(cons(x,xs)) -> insert(x,sort(xs))
sort(nil()) -> nil()
- Signature:
{if'insert/4,insert/2,leq/2,sort/1,if'insert#/4,insert#/2,leq#/2,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1
,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/2,c_9/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if'insert#,insert#,leq#,sort#} and constructors {0,cons
,false,nil,s,true}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
if'insert#(false(),x,y,ys) -> c_1(insert#(x,ys))
insert#(x,cons(y,ys)) -> c_3(if'insert#(leq(x,y),x,y,ys),leq#(x,y))
sort#(cons(x,xs)) -> insert#(x,sort(xs))
sort#(cons(x,xs)) -> sort#(xs)
and a lower component
leq#(s(x),s(y)) -> c_7(leq#(x,y))
Further, following extension rules are added to the lower component.
if'insert#(false(),x,y,ys) -> insert#(x,ys)
insert#(x,cons(y,ys)) -> if'insert#(leq(x,y),x,y,ys)
insert#(x,cons(y,ys)) -> leq#(x,y)
sort#(cons(x,xs)) -> insert#(x,sort(xs))
sort#(cons(x,xs)) -> sort#(xs)
*** Step 4.b:1.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
if'insert#(false(),x,y,ys) -> c_1(insert#(x,ys))
insert#(x,cons(y,ys)) -> c_3(if'insert#(leq(x,y),x,y,ys),leq#(x,y))
- Weak DPs:
sort#(cons(x,xs)) -> insert#(x,sort(xs))
sort#(cons(x,xs)) -> sort#(xs)
- Weak TRS:
if'insert(false(),x,y,ys) -> cons(y,insert(x,ys))
if'insert(true(),x,y,ys) -> cons(x,cons(y,ys))
insert(x,cons(y,ys)) -> if'insert(leq(x,y),x,y,ys)
insert(x,nil()) -> cons(x,nil())
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
sort(cons(x,xs)) -> insert(x,sort(xs))
sort(nil()) -> nil()
- Signature:
{if'insert/4,insert/2,leq/2,sort/1,if'insert#/4,insert#/2,leq#/2,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1
,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/2,c_9/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if'insert#,insert#,leq#,sort#} and constructors {0,cons
,false,nil,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:if'insert#(false(),x,y,ys) -> c_1(insert#(x,ys))
-->_1 insert#(x,cons(y,ys)) -> c_3(if'insert#(leq(x,y),x,y,ys),leq#(x,y)):2
2:S:insert#(x,cons(y,ys)) -> c_3(if'insert#(leq(x,y),x,y,ys),leq#(x,y))
-->_1 if'insert#(false(),x,y,ys) -> c_1(insert#(x,ys)):1
3:W:sort#(cons(x,xs)) -> insert#(x,sort(xs))
-->_1 insert#(x,cons(y,ys)) -> c_3(if'insert#(leq(x,y),x,y,ys),leq#(x,y)):2
4:W:sort#(cons(x,xs)) -> sort#(xs)
-->_1 sort#(cons(x,xs)) -> sort#(xs):4
-->_1 sort#(cons(x,xs)) -> insert#(x,sort(xs)):3
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
insert#(x,cons(y,ys)) -> c_3(if'insert#(leq(x,y),x,y,ys))
*** Step 4.b:1.a:2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
if'insert#(false(),x,y,ys) -> c_1(insert#(x,ys))
insert#(x,cons(y,ys)) -> c_3(if'insert#(leq(x,y),x,y,ys))
- Weak DPs:
sort#(cons(x,xs)) -> insert#(x,sort(xs))
sort#(cons(x,xs)) -> sort#(xs)
- Weak TRS:
if'insert(false(),x,y,ys) -> cons(y,insert(x,ys))
if'insert(true(),x,y,ys) -> cons(x,cons(y,ys))
insert(x,cons(y,ys)) -> if'insert(leq(x,y),x,y,ys)
insert(x,nil()) -> cons(x,nil())
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
sort(cons(x,xs)) -> insert(x,sort(xs))
sort(nil()) -> nil()
- Signature:
{if'insert/4,insert/2,leq/2,sort/1,if'insert#/4,insert#/2,leq#/2,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1
,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/2,c_9/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if'insert#,insert#,leq#,sort#} and constructors {0,cons
,false,nil,s,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(cons) = {2},
uargs(if'insert) = {1},
uargs(insert) = {2},
uargs(if'insert#) = {1},
uargs(insert#) = {2},
uargs(c_1) = {1},
uargs(c_3) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(cons) = [1] x2 + [0]
p(false) = [0]
p(if'insert) = [1] x1 + [1] x4 + [0]
p(insert) = [1] x2 + [0]
p(leq) = [0]
p(nil) = [0]
p(s) = [1] x1 + [0]
p(sort) = [0]
p(true) = [0]
p(if'insert#) = [1] x1 + [1] x4 + [0]
p(insert#) = [1] x2 + [7]
p(leq#) = [0]
p(sort#) = [7]
p(c_1) = [1] x1 + [0]
p(c_2) = [0]
p(c_3) = [1] x1 + [1]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [0]
p(c_9) = [0]
Following rules are strictly oriented:
insert#(x,cons(y,ys)) = [1] ys + [7]
> [1] ys + [1]
= c_3(if'insert#(leq(x,y),x,y,ys))
Following rules are (at-least) weakly oriented:
if'insert#(false(),x,y,ys) = [1] ys + [0]
>= [1] ys + [7]
= c_1(insert#(x,ys))
sort#(cons(x,xs)) = [7]
>= [7]
= insert#(x,sort(xs))
sort#(cons(x,xs)) = [7]
>= [7]
= sort#(xs)
if'insert(false(),x,y,ys) = [1] ys + [0]
>= [1] ys + [0]
= cons(y,insert(x,ys))
if'insert(true(),x,y,ys) = [1] ys + [0]
>= [1] ys + [0]
= cons(x,cons(y,ys))
insert(x,cons(y,ys)) = [1] ys + [0]
>= [1] ys + [0]
= if'insert(leq(x,y),x,y,ys)
insert(x,nil()) = [0]
>= [0]
= cons(x,nil())
leq(0(),y) = [0]
>= [0]
= true()
leq(s(x),0()) = [0]
>= [0]
= false()
leq(s(x),s(y)) = [0]
>= [0]
= leq(x,y)
sort(cons(x,xs)) = [0]
>= [0]
= insert(x,sort(xs))
sort(nil()) = [0]
>= [0]
= nil()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 4.b:1.a:3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
if'insert#(false(),x,y,ys) -> c_1(insert#(x,ys))
- Weak DPs:
insert#(x,cons(y,ys)) -> c_3(if'insert#(leq(x,y),x,y,ys))
sort#(cons(x,xs)) -> insert#(x,sort(xs))
sort#(cons(x,xs)) -> sort#(xs)
- Weak TRS:
if'insert(false(),x,y,ys) -> cons(y,insert(x,ys))
if'insert(true(),x,y,ys) -> cons(x,cons(y,ys))
insert(x,cons(y,ys)) -> if'insert(leq(x,y),x,y,ys)
insert(x,nil()) -> cons(x,nil())
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
sort(cons(x,xs)) -> insert(x,sort(xs))
sort(nil()) -> nil()
- Signature:
{if'insert/4,insert/2,leq/2,sort/1,if'insert#/4,insert#/2,leq#/2,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1
,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/2,c_9/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if'insert#,insert#,leq#,sort#} and constructors {0,cons
,false,nil,s,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(cons) = {2},
uargs(if'insert) = {1},
uargs(insert) = {2},
uargs(if'insert#) = {1},
uargs(insert#) = {2},
uargs(c_1) = {1},
uargs(c_3) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(cons) = [1] x2 + [2]
p(false) = [0]
p(if'insert) = [1] x1 + [1] x4 + [4]
p(insert) = [1] x2 + [2]
p(leq) = [0]
p(nil) = [0]
p(s) = [1]
p(sort) = [1] x1 + [0]
p(true) = [0]
p(if'insert#) = [1] x1 + [1] x4 + [1]
p(insert#) = [1] x2 + [0]
p(leq#) = [2] x1 + [1] x2 + [0]
p(sort#) = [2] x1 + [0]
p(c_1) = [1] x1 + [0]
p(c_2) = [1]
p(c_3) = [1] x1 + [1]
p(c_4) = [1]
p(c_5) = [0]
p(c_6) = [1]
p(c_7) = [1] x1 + [2]
p(c_8) = [2] x1 + [1] x2 + [0]
p(c_9) = [4]
Following rules are strictly oriented:
if'insert#(false(),x,y,ys) = [1] ys + [1]
> [1] ys + [0]
= c_1(insert#(x,ys))
Following rules are (at-least) weakly oriented:
insert#(x,cons(y,ys)) = [1] ys + [2]
>= [1] ys + [2]
= c_3(if'insert#(leq(x,y),x,y,ys))
sort#(cons(x,xs)) = [2] xs + [4]
>= [1] xs + [0]
= insert#(x,sort(xs))
sort#(cons(x,xs)) = [2] xs + [4]
>= [2] xs + [0]
= sort#(xs)
if'insert(false(),x,y,ys) = [1] ys + [4]
>= [1] ys + [4]
= cons(y,insert(x,ys))
if'insert(true(),x,y,ys) = [1] ys + [4]
>= [1] ys + [4]
= cons(x,cons(y,ys))
insert(x,cons(y,ys)) = [1] ys + [4]
>= [1] ys + [4]
= if'insert(leq(x,y),x,y,ys)
insert(x,nil()) = [2]
>= [2]
= cons(x,nil())
leq(0(),y) = [0]
>= [0]
= true()
leq(s(x),0()) = [0]
>= [0]
= false()
leq(s(x),s(y)) = [0]
>= [0]
= leq(x,y)
sort(cons(x,xs)) = [1] xs + [2]
>= [1] xs + [2]
= insert(x,sort(xs))
sort(nil()) = [0]
>= [0]
= nil()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 4.b:1.a:4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
if'insert#(false(),x,y,ys) -> c_1(insert#(x,ys))
insert#(x,cons(y,ys)) -> c_3(if'insert#(leq(x,y),x,y,ys))
sort#(cons(x,xs)) -> insert#(x,sort(xs))
sort#(cons(x,xs)) -> sort#(xs)
- Weak TRS:
if'insert(false(),x,y,ys) -> cons(y,insert(x,ys))
if'insert(true(),x,y,ys) -> cons(x,cons(y,ys))
insert(x,cons(y,ys)) -> if'insert(leq(x,y),x,y,ys)
insert(x,nil()) -> cons(x,nil())
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
sort(cons(x,xs)) -> insert(x,sort(xs))
sort(nil()) -> nil()
- Signature:
{if'insert/4,insert/2,leq/2,sort/1,if'insert#/4,insert#/2,leq#/2,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1
,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/2,c_9/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if'insert#,insert#,leq#,sort#} and constructors {0,cons
,false,nil,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 4.b:1.b:1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
leq#(s(x),s(y)) -> c_7(leq#(x,y))
- Weak DPs:
if'insert#(false(),x,y,ys) -> insert#(x,ys)
insert#(x,cons(y,ys)) -> if'insert#(leq(x,y),x,y,ys)
insert#(x,cons(y,ys)) -> leq#(x,y)
sort#(cons(x,xs)) -> insert#(x,sort(xs))
sort#(cons(x,xs)) -> sort#(xs)
- Weak TRS:
if'insert(false(),x,y,ys) -> cons(y,insert(x,ys))
if'insert(true(),x,y,ys) -> cons(x,cons(y,ys))
insert(x,cons(y,ys)) -> if'insert(leq(x,y),x,y,ys)
insert(x,nil()) -> cons(x,nil())
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
sort(cons(x,xs)) -> insert(x,sort(xs))
sort(nil()) -> nil()
- Signature:
{if'insert/4,insert/2,leq/2,sort/1,if'insert#/4,insert#/2,leq#/2,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1
,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/2,c_9/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if'insert#,insert#,leq#,sort#} and constructors {0,cons
,false,nil,s,true}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(cons) = {2},
uargs(if'insert) = {1},
uargs(insert) = {2},
uargs(if'insert#) = {1},
uargs(insert#) = {2},
uargs(c_7) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(cons) = [1] x1 + [1] x2 + [2]
p(false) = [4]
p(if'insert) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [4]
p(insert) = [1] x1 + [1] x2 + [6]
p(leq) = [4]
p(nil) = [2]
p(s) = [1] x1 + [1]
p(sort) = [4] x1 + [0]
p(true) = [1]
p(if'insert#) = [1] x1 + [1] x4 + [4]
p(insert#) = [1] x2 + [6]
p(leq#) = [1] x2 + [0]
p(sort#) = [4] x1 + [1]
p(c_1) = [4] x1 + [0]
p(c_2) = [0]
p(c_3) = [1] x2 + [0]
p(c_4) = [0]
p(c_5) = [4]
p(c_6) = [2]
p(c_7) = [1] x1 + [0]
p(c_8) = [2]
p(c_9) = [2]
Following rules are strictly oriented:
leq#(s(x),s(y)) = [1] y + [1]
> [1] y + [0]
= c_7(leq#(x,y))
Following rules are (at-least) weakly oriented:
if'insert#(false(),x,y,ys) = [1] ys + [8]
>= [1] ys + [6]
= insert#(x,ys)
insert#(x,cons(y,ys)) = [1] y + [1] ys + [8]
>= [1] ys + [8]
= if'insert#(leq(x,y),x,y,ys)
insert#(x,cons(y,ys)) = [1] y + [1] ys + [8]
>= [1] y + [0]
= leq#(x,y)
sort#(cons(x,xs)) = [4] x + [4] xs + [9]
>= [4] xs + [6]
= insert#(x,sort(xs))
sort#(cons(x,xs)) = [4] x + [4] xs + [9]
>= [4] xs + [1]
= sort#(xs)
if'insert(false(),x,y,ys) = [1] x + [1] y + [1] ys + [8]
>= [1] x + [1] y + [1] ys + [8]
= cons(y,insert(x,ys))
if'insert(true(),x,y,ys) = [1] x + [1] y + [1] ys + [5]
>= [1] x + [1] y + [1] ys + [4]
= cons(x,cons(y,ys))
insert(x,cons(y,ys)) = [1] x + [1] y + [1] ys + [8]
>= [1] x + [1] y + [1] ys + [8]
= if'insert(leq(x,y),x,y,ys)
insert(x,nil()) = [1] x + [8]
>= [1] x + [4]
= cons(x,nil())
leq(0(),y) = [4]
>= [1]
= true()
leq(s(x),0()) = [4]
>= [4]
= false()
leq(s(x),s(y)) = [4]
>= [4]
= leq(x,y)
sort(cons(x,xs)) = [4] x + [4] xs + [8]
>= [1] x + [4] xs + [6]
= insert(x,sort(xs))
sort(nil()) = [8]
>= [2]
= nil()
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 4.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
if'insert#(false(),x,y,ys) -> insert#(x,ys)
insert#(x,cons(y,ys)) -> if'insert#(leq(x,y),x,y,ys)
insert#(x,cons(y,ys)) -> leq#(x,y)
leq#(s(x),s(y)) -> c_7(leq#(x,y))
sort#(cons(x,xs)) -> insert#(x,sort(xs))
sort#(cons(x,xs)) -> sort#(xs)
- Weak TRS:
if'insert(false(),x,y,ys) -> cons(y,insert(x,ys))
if'insert(true(),x,y,ys) -> cons(x,cons(y,ys))
insert(x,cons(y,ys)) -> if'insert(leq(x,y),x,y,ys)
insert(x,nil()) -> cons(x,nil())
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
sort(cons(x,xs)) -> insert(x,sort(xs))
sort(nil()) -> nil()
- Signature:
{if'insert/4,insert/2,leq/2,sort/1,if'insert#/4,insert#/2,leq#/2,sort#/1} / {0/0,cons/2,false/0,nil/0,s/1
,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/2,c_9/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {if'insert#,insert#,leq#,sort#} and constructors {0,cons
,false,nil,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^3))