WORST_CASE(Omega(n^1),O(n^2))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ Considered Problem:
- Strict TRS:
flatten(leaf()) -> nil()
flatten(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2)))
flattensort(t) -> insertionsort(flatten(t))
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())
insertionsort(cons(x,xs)) -> insert(x,insertionsort(xs))
insertionsort(nil()) -> nil()
- Weak TRS:
append(cons(x,xs),l2) -> cons(x,append(xs,l2))
append(nil(),l2) -> l2
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
- Signature:
{append/2,flatten/1,flattensort/1,if'insert/4,insert/2,insertionsort/1,leq/2} / {0/0,cons/2,false/0,leaf/0
,nil/0,node/3,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append,flatten,flattensort,if'insert,insert,insertionsort
,leq} and constructors {0,cons,false,leaf,nil,node,s,true}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
flatten(leaf()) -> nil()
flatten(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2)))
flattensort(t) -> insertionsort(flatten(t))
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())
insertionsort(cons(x,xs)) -> insert(x,insertionsort(xs))
insertionsort(nil()) -> nil()
- Weak TRS:
append(cons(x,xs),l2) -> cons(x,append(xs,l2))
append(nil(),l2) -> l2
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
- Signature:
{append/2,flatten/1,flattensort/1,if'insert/4,insert/2,insertionsort/1,leq/2} / {0/0,cons/2,false/0,leaf/0
,nil/0,node/3,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append,flatten,flattensort,if'insert,insert,insertionsort
,leq} and constructors {0,cons,false,leaf,nil,node,s,true}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
flatten(y){y -> node(x,y,z)} =
flatten(node(x,y,z)) ->^+ append(x,append(flatten(y),flatten(z)))
= C[flatten(y) = flatten(y){}]
** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
flatten(leaf()) -> nil()
flatten(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2)))
flattensort(t) -> insertionsort(flatten(t))
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())
insertionsort(cons(x,xs)) -> insert(x,insertionsort(xs))
insertionsort(nil()) -> nil()
- Weak TRS:
append(cons(x,xs),l2) -> cons(x,append(xs,l2))
append(nil(),l2) -> l2
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
- Signature:
{append/2,flatten/1,flattensort/1,if'insert/4,insert/2,insertionsort/1,leq/2} / {0/0,cons/2,false/0,leaf/0
,nil/0,node/3,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append,flatten,flattensort,if'insert,insert,insertionsort
,leq} and constructors {0,cons,false,leaf,nil,node,s,true}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
flatten#(leaf()) -> c_1()
flatten#(node(l,t1,t2)) -> c_2(append#(l,append(flatten(t1),flatten(t2)))
,append#(flatten(t1),flatten(t2))
,flatten#(t1)
,flatten#(t2))
flattensort#(t) -> c_3(insertionsort#(flatten(t)),flatten#(t))
if'insert#(false(),x,y,ys) -> c_4(insert#(x,ys))
if'insert#(true(),x,y,ys) -> c_5()
insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys),leq#(x,y))
insert#(x,nil()) -> c_7()
insertionsort#(cons(x,xs)) -> c_8(insert#(x,insertionsort(xs)),insertionsort#(xs))
insertionsort#(nil()) -> c_9()
Weak DPs
append#(cons(x,xs),l2) -> c_10(append#(xs,l2))
append#(nil(),l2) -> c_11()
leq#(0(),y) -> c_12()
leq#(s(x),0()) -> c_13()
leq#(s(x),s(y)) -> c_14(leq#(x,y))
and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
flatten#(leaf()) -> c_1()
flatten#(node(l,t1,t2)) -> c_2(append#(l,append(flatten(t1),flatten(t2)))
,append#(flatten(t1),flatten(t2))
,flatten#(t1)
,flatten#(t2))
flattensort#(t) -> c_3(insertionsort#(flatten(t)),flatten#(t))
if'insert#(false(),x,y,ys) -> c_4(insert#(x,ys))
if'insert#(true(),x,y,ys) -> c_5()
insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys),leq#(x,y))
insert#(x,nil()) -> c_7()
insertionsort#(cons(x,xs)) -> c_8(insert#(x,insertionsort(xs)),insertionsort#(xs))
insertionsort#(nil()) -> c_9()
- Weak DPs:
append#(cons(x,xs),l2) -> c_10(append#(xs,l2))
append#(nil(),l2) -> c_11()
leq#(0(),y) -> c_12()
leq#(s(x),0()) -> c_13()
leq#(s(x),s(y)) -> c_14(leq#(x,y))
- Weak TRS:
append(cons(x,xs),l2) -> cons(x,append(xs,l2))
append(nil(),l2) -> l2
flatten(leaf()) -> nil()
flatten(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2)))
flattensort(t) -> insertionsort(flatten(t))
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())
insertionsort(cons(x,xs)) -> insert(x,insertionsort(xs))
insertionsort(nil()) -> nil()
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
- Signature:
{append/2,flatten/1,flattensort/1,if'insert/4,insert/2,insertionsort/1,leq/2,append#/2,flatten#/1
,flattensort#/1,if'insert#/4,insert#/2,insertionsort#/1,leq#/2} / {0/0,cons/2,false/0,leaf/0,nil/0,node/3
,s/1,true/0,c_1/0,c_2/4,c_3/2,c_4/1,c_5/0,c_6/2,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,flatten#,flattensort#,if'insert#,insert#
,insertionsort#,leq#} and constructors {0,cons,false,leaf,nil,node,s,true}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,5,7,9}
by application of
Pre({1,5,7,9}) = {2,3,4,6,8}.
Here rules are labelled as follows:
1: flatten#(leaf()) -> c_1()
2: flatten#(node(l,t1,t2)) -> c_2(append#(l,append(flatten(t1),flatten(t2)))
,append#(flatten(t1),flatten(t2))
,flatten#(t1)
,flatten#(t2))
3: flattensort#(t) -> c_3(insertionsort#(flatten(t)),flatten#(t))
4: if'insert#(false(),x,y,ys) -> c_4(insert#(x,ys))
5: if'insert#(true(),x,y,ys) -> c_5()
6: insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys),leq#(x,y))
7: insert#(x,nil()) -> c_7()
8: insertionsort#(cons(x,xs)) -> c_8(insert#(x,insertionsort(xs)),insertionsort#(xs))
9: insertionsort#(nil()) -> c_9()
10: append#(cons(x,xs),l2) -> c_10(append#(xs,l2))
11: append#(nil(),l2) -> c_11()
12: leq#(0(),y) -> c_12()
13: leq#(s(x),0()) -> c_13()
14: leq#(s(x),s(y)) -> c_14(leq#(x,y))
** Step 1.b:3: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
flatten#(node(l,t1,t2)) -> c_2(append#(l,append(flatten(t1),flatten(t2)))
,append#(flatten(t1),flatten(t2))
,flatten#(t1)
,flatten#(t2))
flattensort#(t) -> c_3(insertionsort#(flatten(t)),flatten#(t))
if'insert#(false(),x,y,ys) -> c_4(insert#(x,ys))
insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys),leq#(x,y))
insertionsort#(cons(x,xs)) -> c_8(insert#(x,insertionsort(xs)),insertionsort#(xs))
- Weak DPs:
append#(cons(x,xs),l2) -> c_10(append#(xs,l2))
append#(nil(),l2) -> c_11()
flatten#(leaf()) -> c_1()
if'insert#(true(),x,y,ys) -> c_5()
insert#(x,nil()) -> c_7()
insertionsort#(nil()) -> c_9()
leq#(0(),y) -> c_12()
leq#(s(x),0()) -> c_13()
leq#(s(x),s(y)) -> c_14(leq#(x,y))
- Weak TRS:
append(cons(x,xs),l2) -> cons(x,append(xs,l2))
append(nil(),l2) -> l2
flatten(leaf()) -> nil()
flatten(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2)))
flattensort(t) -> insertionsort(flatten(t))
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())
insertionsort(cons(x,xs)) -> insert(x,insertionsort(xs))
insertionsort(nil()) -> nil()
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
- Signature:
{append/2,flatten/1,flattensort/1,if'insert/4,insert/2,insertionsort/1,leq/2,append#/2,flatten#/1
,flattensort#/1,if'insert#/4,insert#/2,insertionsort#/1,leq#/2} / {0/0,cons/2,false/0,leaf/0,nil/0,node/3
,s/1,true/0,c_1/0,c_2/4,c_3/2,c_4/1,c_5/0,c_6/2,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,flatten#,flattensort#,if'insert#,insert#
,insertionsort#,leq#} and constructors {0,cons,false,leaf,nil,node,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:flatten#(node(l,t1,t2)) -> c_2(append#(l,append(flatten(t1),flatten(t2)))
,append#(flatten(t1),flatten(t2))
,flatten#(t1)
,flatten#(t2))
-->_2 append#(cons(x,xs),l2) -> c_10(append#(xs,l2)):6
-->_1 append#(cons(x,xs),l2) -> c_10(append#(xs,l2)):6
-->_4 flatten#(leaf()) -> c_1():8
-->_3 flatten#(leaf()) -> c_1():8
-->_2 append#(nil(),l2) -> c_11():7
-->_1 append#(nil(),l2) -> c_11():7
-->_4 flatten#(node(l,t1,t2)) -> c_2(append#(l,append(flatten(t1),flatten(t2)))
,append#(flatten(t1),flatten(t2))
,flatten#(t1)
,flatten#(t2)):1
-->_3 flatten#(node(l,t1,t2)) -> c_2(append#(l,append(flatten(t1),flatten(t2)))
,append#(flatten(t1),flatten(t2))
,flatten#(t1)
,flatten#(t2)):1
2:S:flattensort#(t) -> c_3(insertionsort#(flatten(t)),flatten#(t))
-->_1 insertionsort#(cons(x,xs)) -> c_8(insert#(x,insertionsort(xs)),insertionsort#(xs)):5
-->_1 insertionsort#(nil()) -> c_9():11
-->_2 flatten#(leaf()) -> c_1():8
-->_2 flatten#(node(l,t1,t2)) -> c_2(append#(l,append(flatten(t1),flatten(t2)))
,append#(flatten(t1),flatten(t2))
,flatten#(t1)
,flatten#(t2)):1
3:S:if'insert#(false(),x,y,ys) -> c_4(insert#(x,ys))
-->_1 insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys),leq#(x,y)):4
-->_1 insert#(x,nil()) -> c_7():10
4:S:insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys),leq#(x,y))
-->_2 leq#(s(x),s(y)) -> c_14(leq#(x,y)):14
-->_2 leq#(s(x),0()) -> c_13():13
-->_2 leq#(0(),y) -> c_12():12
-->_1 if'insert#(true(),x,y,ys) -> c_5():9
-->_1 if'insert#(false(),x,y,ys) -> c_4(insert#(x,ys)):3
5:S:insertionsort#(cons(x,xs)) -> c_8(insert#(x,insertionsort(xs)),insertionsort#(xs))
-->_2 insertionsort#(nil()) -> c_9():11
-->_1 insert#(x,nil()) -> c_7():10
-->_2 insertionsort#(cons(x,xs)) -> c_8(insert#(x,insertionsort(xs)),insertionsort#(xs)):5
-->_1 insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys),leq#(x,y)):4
6:W:append#(cons(x,xs),l2) -> c_10(append#(xs,l2))
-->_1 append#(nil(),l2) -> c_11():7
-->_1 append#(cons(x,xs),l2) -> c_10(append#(xs,l2)):6
7:W:append#(nil(),l2) -> c_11()
8:W:flatten#(leaf()) -> c_1()
9:W:if'insert#(true(),x,y,ys) -> c_5()
10:W:insert#(x,nil()) -> c_7()
11:W:insertionsort#(nil()) -> c_9()
12:W:leq#(0(),y) -> c_12()
13:W:leq#(s(x),0()) -> c_13()
14:W:leq#(s(x),s(y)) -> c_14(leq#(x,y))
-->_1 leq#(s(x),s(y)) -> c_14(leq#(x,y)):14
-->_1 leq#(s(x),0()) -> c_13():13
-->_1 leq#(0(),y) -> c_12():12
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
9: if'insert#(true(),x,y,ys) -> c_5()
14: leq#(s(x),s(y)) -> c_14(leq#(x,y))
12: leq#(0(),y) -> c_12()
13: leq#(s(x),0()) -> c_13()
10: insert#(x,nil()) -> c_7()
11: insertionsort#(nil()) -> c_9()
8: flatten#(leaf()) -> c_1()
6: append#(cons(x,xs),l2) -> c_10(append#(xs,l2))
7: append#(nil(),l2) -> c_11()
** Step 1.b:4: SimplifyRHS WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
flatten#(node(l,t1,t2)) -> c_2(append#(l,append(flatten(t1),flatten(t2)))
,append#(flatten(t1),flatten(t2))
,flatten#(t1)
,flatten#(t2))
flattensort#(t) -> c_3(insertionsort#(flatten(t)),flatten#(t))
if'insert#(false(),x,y,ys) -> c_4(insert#(x,ys))
insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys),leq#(x,y))
insertionsort#(cons(x,xs)) -> c_8(insert#(x,insertionsort(xs)),insertionsort#(xs))
- Weak TRS:
append(cons(x,xs),l2) -> cons(x,append(xs,l2))
append(nil(),l2) -> l2
flatten(leaf()) -> nil()
flatten(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2)))
flattensort(t) -> insertionsort(flatten(t))
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())
insertionsort(cons(x,xs)) -> insert(x,insertionsort(xs))
insertionsort(nil()) -> nil()
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
- Signature:
{append/2,flatten/1,flattensort/1,if'insert/4,insert/2,insertionsort/1,leq/2,append#/2,flatten#/1
,flattensort#/1,if'insert#/4,insert#/2,insertionsort#/1,leq#/2} / {0/0,cons/2,false/0,leaf/0,nil/0,node/3
,s/1,true/0,c_1/0,c_2/4,c_3/2,c_4/1,c_5/0,c_6/2,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,flatten#,flattensort#,if'insert#,insert#
,insertionsort#,leq#} and constructors {0,cons,false,leaf,nil,node,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:flatten#(node(l,t1,t2)) -> c_2(append#(l,append(flatten(t1),flatten(t2)))
,append#(flatten(t1),flatten(t2))
,flatten#(t1)
,flatten#(t2))
-->_4 flatten#(node(l,t1,t2)) -> c_2(append#(l,append(flatten(t1),flatten(t2)))
,append#(flatten(t1),flatten(t2))
,flatten#(t1)
,flatten#(t2)):1
-->_3 flatten#(node(l,t1,t2)) -> c_2(append#(l,append(flatten(t1),flatten(t2)))
,append#(flatten(t1),flatten(t2))
,flatten#(t1)
,flatten#(t2)):1
2:S:flattensort#(t) -> c_3(insertionsort#(flatten(t)),flatten#(t))
-->_1 insertionsort#(cons(x,xs)) -> c_8(insert#(x,insertionsort(xs)),insertionsort#(xs)):5
-->_2 flatten#(node(l,t1,t2)) -> c_2(append#(l,append(flatten(t1),flatten(t2)))
,append#(flatten(t1),flatten(t2))
,flatten#(t1)
,flatten#(t2)):1
3:S:if'insert#(false(),x,y,ys) -> c_4(insert#(x,ys))
-->_1 insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys),leq#(x,y)):4
4:S:insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys),leq#(x,y))
-->_1 if'insert#(false(),x,y,ys) -> c_4(insert#(x,ys)):3
5:S:insertionsort#(cons(x,xs)) -> c_8(insert#(x,insertionsort(xs)),insertionsort#(xs))
-->_2 insertionsort#(cons(x,xs)) -> c_8(insert#(x,insertionsort(xs)),insertionsort#(xs)):5
-->_1 insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys),leq#(x,y)):4
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
flatten#(node(l,t1,t2)) -> c_2(flatten#(t1),flatten#(t2))
insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys))
** Step 1.b:5: UsableRules WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
flatten#(node(l,t1,t2)) -> c_2(flatten#(t1),flatten#(t2))
flattensort#(t) -> c_3(insertionsort#(flatten(t)),flatten#(t))
if'insert#(false(),x,y,ys) -> c_4(insert#(x,ys))
insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys))
insertionsort#(cons(x,xs)) -> c_8(insert#(x,insertionsort(xs)),insertionsort#(xs))
- Weak TRS:
append(cons(x,xs),l2) -> cons(x,append(xs,l2))
append(nil(),l2) -> l2
flatten(leaf()) -> nil()
flatten(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2)))
flattensort(t) -> insertionsort(flatten(t))
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())
insertionsort(cons(x,xs)) -> insert(x,insertionsort(xs))
insertionsort(nil()) -> nil()
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
- Signature:
{append/2,flatten/1,flattensort/1,if'insert/4,insert/2,insertionsort/1,leq/2,append#/2,flatten#/1
,flattensort#/1,if'insert#/4,insert#/2,insertionsort#/1,leq#/2} / {0/0,cons/2,false/0,leaf/0,nil/0,node/3
,s/1,true/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/0,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,flatten#,flattensort#,if'insert#,insert#
,insertionsort#,leq#} and constructors {0,cons,false,leaf,nil,node,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
append(cons(x,xs),l2) -> cons(x,append(xs,l2))
append(nil(),l2) -> l2
flatten(leaf()) -> nil()
flatten(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2)))
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())
insertionsort(cons(x,xs)) -> insert(x,insertionsort(xs))
insertionsort(nil()) -> nil()
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
flatten#(node(l,t1,t2)) -> c_2(flatten#(t1),flatten#(t2))
flattensort#(t) -> c_3(insertionsort#(flatten(t)),flatten#(t))
if'insert#(false(),x,y,ys) -> c_4(insert#(x,ys))
insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys))
insertionsort#(cons(x,xs)) -> c_8(insert#(x,insertionsort(xs)),insertionsort#(xs))
** Step 1.b:6: DecomposeDG WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
flatten#(node(l,t1,t2)) -> c_2(flatten#(t1),flatten#(t2))
flattensort#(t) -> c_3(insertionsort#(flatten(t)),flatten#(t))
if'insert#(false(),x,y,ys) -> c_4(insert#(x,ys))
insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys))
insertionsort#(cons(x,xs)) -> c_8(insert#(x,insertionsort(xs)),insertionsort#(xs))
- Weak TRS:
append(cons(x,xs),l2) -> cons(x,append(xs,l2))
append(nil(),l2) -> l2
flatten(leaf()) -> nil()
flatten(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2)))
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())
insertionsort(cons(x,xs)) -> insert(x,insertionsort(xs))
insertionsort(nil()) -> nil()
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
- Signature:
{append/2,flatten/1,flattensort/1,if'insert/4,insert/2,insertionsort/1,leq/2,append#/2,flatten#/1
,flattensort#/1,if'insert#/4,insert#/2,insertionsort#/1,leq#/2} / {0/0,cons/2,false/0,leaf/0,nil/0,node/3
,s/1,true/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/0,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,flatten#,flattensort#,if'insert#,insert#
,insertionsort#,leq#} and constructors {0,cons,false,leaf,nil,node,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
flatten#(node(l,t1,t2)) -> c_2(flatten#(t1),flatten#(t2))
flattensort#(t) -> c_3(insertionsort#(flatten(t)),flatten#(t))
insertionsort#(cons(x,xs)) -> c_8(insert#(x,insertionsort(xs)),insertionsort#(xs))
and a lower component
if'insert#(false(),x,y,ys) -> c_4(insert#(x,ys))
insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys))
Further, following extension rules are added to the lower component.
flatten#(node(l,t1,t2)) -> flatten#(t1)
flatten#(node(l,t1,t2)) -> flatten#(t2)
flattensort#(t) -> flatten#(t)
flattensort#(t) -> insertionsort#(flatten(t))
insertionsort#(cons(x,xs)) -> insert#(x,insertionsort(xs))
insertionsort#(cons(x,xs)) -> insertionsort#(xs)
*** Step 1.b:6.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
flatten#(node(l,t1,t2)) -> c_2(flatten#(t1),flatten#(t2))
flattensort#(t) -> c_3(insertionsort#(flatten(t)),flatten#(t))
insertionsort#(cons(x,xs)) -> c_8(insert#(x,insertionsort(xs)),insertionsort#(xs))
- Weak TRS:
append(cons(x,xs),l2) -> cons(x,append(xs,l2))
append(nil(),l2) -> l2
flatten(leaf()) -> nil()
flatten(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2)))
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())
insertionsort(cons(x,xs)) -> insert(x,insertionsort(xs))
insertionsort(nil()) -> nil()
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
- Signature:
{append/2,flatten/1,flattensort/1,if'insert/4,insert/2,insertionsort/1,leq/2,append#/2,flatten#/1
,flattensort#/1,if'insert#/4,insert#/2,insertionsort#/1,leq#/2} / {0/0,cons/2,false/0,leaf/0,nil/0,node/3
,s/1,true/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/0,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,flatten#,flattensort#,if'insert#,insert#
,insertionsort#,leq#} and constructors {0,cons,false,leaf,nil,node,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:flatten#(node(l,t1,t2)) -> c_2(flatten#(t1),flatten#(t2))
-->_2 flatten#(node(l,t1,t2)) -> c_2(flatten#(t1),flatten#(t2)):1
-->_1 flatten#(node(l,t1,t2)) -> c_2(flatten#(t1),flatten#(t2)):1
2:S:flattensort#(t) -> c_3(insertionsort#(flatten(t)),flatten#(t))
-->_1 insertionsort#(cons(x,xs)) -> c_8(insert#(x,insertionsort(xs)),insertionsort#(xs)):3
-->_2 flatten#(node(l,t1,t2)) -> c_2(flatten#(t1),flatten#(t2)):1
3:S:insertionsort#(cons(x,xs)) -> c_8(insert#(x,insertionsort(xs)),insertionsort#(xs))
-->_2 insertionsort#(cons(x,xs)) -> c_8(insert#(x,insertionsort(xs)),insertionsort#(xs)):3
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
insertionsort#(cons(x,xs)) -> c_8(insertionsort#(xs))
*** Step 1.b:6.a:2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
flatten#(node(l,t1,t2)) -> c_2(flatten#(t1),flatten#(t2))
flattensort#(t) -> c_3(insertionsort#(flatten(t)),flatten#(t))
insertionsort#(cons(x,xs)) -> c_8(insertionsort#(xs))
- Weak TRS:
append(cons(x,xs),l2) -> cons(x,append(xs,l2))
append(nil(),l2) -> l2
flatten(leaf()) -> nil()
flatten(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2)))
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())
insertionsort(cons(x,xs)) -> insert(x,insertionsort(xs))
insertionsort(nil()) -> nil()
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
- Signature:
{append/2,flatten/1,flattensort/1,if'insert/4,insert/2,insertionsort/1,leq/2,append#/2,flatten#/1
,flattensort#/1,if'insert#/4,insert#/2,insertionsort#/1,leq#/2} / {0/0,cons/2,false/0,leaf/0,nil/0,node/3
,s/1,true/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,flatten#,flattensort#,if'insert#,insert#
,insertionsort#,leq#} and constructors {0,cons,false,leaf,nil,node,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
append(cons(x,xs),l2) -> cons(x,append(xs,l2))
append(nil(),l2) -> l2
flatten(leaf()) -> nil()
flatten(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2)))
flatten#(node(l,t1,t2)) -> c_2(flatten#(t1),flatten#(t2))
flattensort#(t) -> c_3(insertionsort#(flatten(t)),flatten#(t))
insertionsort#(cons(x,xs)) -> c_8(insertionsort#(xs))
*** Step 1.b:6.a:3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
flatten#(node(l,t1,t2)) -> c_2(flatten#(t1),flatten#(t2))
flattensort#(t) -> c_3(insertionsort#(flatten(t)),flatten#(t))
insertionsort#(cons(x,xs)) -> c_8(insertionsort#(xs))
- Weak TRS:
append(cons(x,xs),l2) -> cons(x,append(xs,l2))
append(nil(),l2) -> l2
flatten(leaf()) -> nil()
flatten(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2)))
- Signature:
{append/2,flatten/1,flattensort/1,if'insert/4,insert/2,insertionsort/1,leq/2,append#/2,flatten#/1
,flattensort#/1,if'insert#/4,insert#/2,insertionsort#/1,leq#/2} / {0/0,cons/2,false/0,leaf/0,nil/0,node/3
,s/1,true/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,flatten#,flattensort#,if'insert#,insert#
,insertionsort#,leq#} and constructors {0,cons,false,leaf,nil,node,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(append) = {1,2},
uargs(cons) = {2},
uargs(insertionsort#) = {1},
uargs(c_2) = {1,2},
uargs(c_3) = {1,2},
uargs(c_8) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(append) = [1] x1 + [1] x2 + [0]
p(cons) = [1] x1 + [1] x2 + [0]
p(false) = [0]
p(flatten) = [1] x1 + [0]
p(flattensort) = [0]
p(if'insert) = [0]
p(insert) = [0]
p(insertionsort) = [0]
p(leaf) = [0]
p(leq) = [0]
p(nil) = [0]
p(node) = [1] x1 + [1] x2 + [1] x3 + [0]
p(s) = [1] x1 + [0]
p(true) = [0]
p(append#) = [0]
p(flatten#) = [5] x1 + [0]
p(flattensort#) = [6] x1 + [9]
p(if'insert#) = [0]
p(insert#) = [0]
p(insertionsort#) = [1] x1 + [0]
p(leq#) = [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [1] x1 + [1] x2 + [0]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [1] x1 + [0]
p(c_9) = [0]
p(c_10) = [0]
p(c_11) = [0]
p(c_12) = [0]
p(c_13) = [0]
p(c_14) = [0]
Following rules are strictly oriented:
flattensort#(t) = [6] t + [9]
> [6] t + [0]
= c_3(insertionsort#(flatten(t)),flatten#(t))
Following rules are (at-least) weakly oriented:
flatten#(node(l,t1,t2)) = [5] l + [5] t1 + [5] t2 + [0]
>= [5] t1 + [5] t2 + [0]
= c_2(flatten#(t1),flatten#(t2))
insertionsort#(cons(x,xs)) = [1] x + [1] xs + [0]
>= [1] xs + [0]
= c_8(insertionsort#(xs))
append(cons(x,xs),l2) = [1] l2 + [1] x + [1] xs + [0]
>= [1] l2 + [1] x + [1] xs + [0]
= cons(x,append(xs,l2))
append(nil(),l2) = [1] l2 + [0]
>= [1] l2 + [0]
= l2
flatten(leaf()) = [0]
>= [0]
= nil()
flatten(node(l,t1,t2)) = [1] l + [1] t1 + [1] t2 + [0]
>= [1] l + [1] t1 + [1] t2 + [0]
= append(l,append(flatten(t1),flatten(t2)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 1.b:6.a:4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
flatten#(node(l,t1,t2)) -> c_2(flatten#(t1),flatten#(t2))
insertionsort#(cons(x,xs)) -> c_8(insertionsort#(xs))
- Weak DPs:
flattensort#(t) -> c_3(insertionsort#(flatten(t)),flatten#(t))
- Weak TRS:
append(cons(x,xs),l2) -> cons(x,append(xs,l2))
append(nil(),l2) -> l2
flatten(leaf()) -> nil()
flatten(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2)))
- Signature:
{append/2,flatten/1,flattensort/1,if'insert/4,insert/2,insertionsort/1,leq/2,append#/2,flatten#/1
,flattensort#/1,if'insert#/4,insert#/2,insertionsort#/1,leq#/2} / {0/0,cons/2,false/0,leaf/0,nil/0,node/3
,s/1,true/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,flatten#,flattensort#,if'insert#,insert#
,insertionsort#,leq#} and constructors {0,cons,false,leaf,nil,node,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(append) = {1,2},
uargs(cons) = {2},
uargs(insertionsort#) = {1},
uargs(c_2) = {1,2},
uargs(c_3) = {1,2},
uargs(c_8) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(append) = [1] x1 + [1] x2 + [0]
p(cons) = [1] x1 + [1] x2 + [15]
p(false) = [0]
p(flatten) = [1] x1 + [0]
p(flattensort) = [0]
p(if'insert) = [0]
p(insert) = [0]
p(insertionsort) = [0]
p(leaf) = [0]
p(leq) = [0]
p(nil) = [0]
p(node) = [1] x1 + [1] x2 + [1] x3 + [0]
p(s) = [1] x1 + [0]
p(true) = [0]
p(append#) = [0]
p(flatten#) = [5] x1 + [0]
p(flattensort#) = [6] x1 + [0]
p(if'insert#) = [0]
p(insert#) = [0]
p(insertionsort#) = [1] x1 + [0]
p(leq#) = [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [0]
p(c_3) = [1] x1 + [1] x2 + [0]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [0]
p(c_7) = [0]
p(c_8) = [1] x1 + [0]
p(c_9) = [0]
p(c_10) = [0]
p(c_11) = [0]
p(c_12) = [0]
p(c_13) = [0]
p(c_14) = [0]
Following rules are strictly oriented:
insertionsort#(cons(x,xs)) = [1] x + [1] xs + [15]
> [1] xs + [0]
= c_8(insertionsort#(xs))
Following rules are (at-least) weakly oriented:
flatten#(node(l,t1,t2)) = [5] l + [5] t1 + [5] t2 + [0]
>= [5] t1 + [5] t2 + [0]
= c_2(flatten#(t1),flatten#(t2))
flattensort#(t) = [6] t + [0]
>= [6] t + [0]
= c_3(insertionsort#(flatten(t)),flatten#(t))
append(cons(x,xs),l2) = [1] l2 + [1] x + [1] xs + [15]
>= [1] l2 + [1] x + [1] xs + [15]
= cons(x,append(xs,l2))
append(nil(),l2) = [1] l2 + [0]
>= [1] l2 + [0]
= l2
flatten(leaf()) = [0]
>= [0]
= nil()
flatten(node(l,t1,t2)) = [1] l + [1] t1 + [1] t2 + [0]
>= [1] l + [1] t1 + [1] t2 + [0]
= append(l,append(flatten(t1),flatten(t2)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 1.b:6.a:5: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
flatten#(node(l,t1,t2)) -> c_2(flatten#(t1),flatten#(t2))
- Weak DPs:
flattensort#(t) -> c_3(insertionsort#(flatten(t)),flatten#(t))
insertionsort#(cons(x,xs)) -> c_8(insertionsort#(xs))
- Weak TRS:
append(cons(x,xs),l2) -> cons(x,append(xs,l2))
append(nil(),l2) -> l2
flatten(leaf()) -> nil()
flatten(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2)))
- Signature:
{append/2,flatten/1,flattensort/1,if'insert/4,insert/2,insertionsort/1,leq/2,append#/2,flatten#/1
,flattensort#/1,if'insert#/4,insert#/2,insertionsort#/1,leq#/2} / {0/0,cons/2,false/0,leaf/0,nil/0,node/3
,s/1,true/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,flatten#,flattensort#,if'insert#,insert#
,insertionsort#,leq#} and constructors {0,cons,false,leaf,nil,node,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(append) = {1,2},
uargs(cons) = {2},
uargs(insertionsort#) = {1},
uargs(c_2) = {1,2},
uargs(c_3) = {1,2},
uargs(c_8) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(append) = [1] x1 + [1] x2 + [1]
p(cons) = [1] x1 + [1] x2 + [2]
p(false) = [0]
p(flatten) = [3] x1 + [5]
p(flattensort) = [0]
p(if'insert) = [0]
p(insert) = [0]
p(insertionsort) = [0]
p(leaf) = [2]
p(leq) = [1] x2 + [8]
p(nil) = [0]
p(node) = [1] x1 + [1] x2 + [1] x3 + [4]
p(s) = [2]
p(true) = [1]
p(append#) = [1] x2 + [2]
p(flatten#) = [4] x1 + [0]
p(flattensort#) = [9] x1 + [8]
p(if'insert#) = [1] x1 + [1] x2 + [1] x3 + [8] x4 + [1]
p(insert#) = [1] x1 + [1] x2 + [1]
p(insertionsort#) = [1] x1 + [2]
p(leq#) = [1] x1 + [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [1] x2 + [12]
p(c_3) = [1] x1 + [1] x2 + [1]
p(c_4) = [1] x1 + [1]
p(c_5) = [1]
p(c_6) = [1]
p(c_7) = [0]
p(c_8) = [1] x1 + [0]
p(c_9) = [2]
p(c_10) = [1] x1 + [8]
p(c_11) = [1]
p(c_12) = [0]
p(c_13) = [1]
p(c_14) = [8] x1 + [2]
Following rules are strictly oriented:
flatten#(node(l,t1,t2)) = [4] l + [4] t1 + [4] t2 + [16]
> [4] t1 + [4] t2 + [12]
= c_2(flatten#(t1),flatten#(t2))
Following rules are (at-least) weakly oriented:
flattensort#(t) = [9] t + [8]
>= [7] t + [8]
= c_3(insertionsort#(flatten(t)),flatten#(t))
insertionsort#(cons(x,xs)) = [1] x + [1] xs + [4]
>= [1] xs + [2]
= c_8(insertionsort#(xs))
append(cons(x,xs),l2) = [1] l2 + [1] x + [1] xs + [3]
>= [1] l2 + [1] x + [1] xs + [3]
= cons(x,append(xs,l2))
append(nil(),l2) = [1] l2 + [1]
>= [1] l2 + [0]
= l2
flatten(leaf()) = [11]
>= [0]
= nil()
flatten(node(l,t1,t2)) = [3] l + [3] t1 + [3] t2 + [17]
>= [1] l + [3] t1 + [3] t2 + [12]
= append(l,append(flatten(t1),flatten(t2)))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 1.b:6.a:6: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
flatten#(node(l,t1,t2)) -> c_2(flatten#(t1),flatten#(t2))
flattensort#(t) -> c_3(insertionsort#(flatten(t)),flatten#(t))
insertionsort#(cons(x,xs)) -> c_8(insertionsort#(xs))
- Weak TRS:
append(cons(x,xs),l2) -> cons(x,append(xs,l2))
append(nil(),l2) -> l2
flatten(leaf()) -> nil()
flatten(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2)))
- Signature:
{append/2,flatten/1,flattensort/1,if'insert/4,insert/2,insertionsort/1,leq/2,append#/2,flatten#/1
,flattensort#/1,if'insert#/4,insert#/2,insertionsort#/1,leq#/2} / {0/0,cons/2,false/0,leaf/0,nil/0,node/3
,s/1,true/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/0,c_6/1,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,flatten#,flattensort#,if'insert#,insert#
,insertionsort#,leq#} and constructors {0,cons,false,leaf,nil,node,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 1.b:6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
if'insert#(false(),x,y,ys) -> c_4(insert#(x,ys))
insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys))
- Weak DPs:
flatten#(node(l,t1,t2)) -> flatten#(t1)
flatten#(node(l,t1,t2)) -> flatten#(t2)
flattensort#(t) -> flatten#(t)
flattensort#(t) -> insertionsort#(flatten(t))
insertionsort#(cons(x,xs)) -> insert#(x,insertionsort(xs))
insertionsort#(cons(x,xs)) -> insertionsort#(xs)
- Weak TRS:
append(cons(x,xs),l2) -> cons(x,append(xs,l2))
append(nil(),l2) -> l2
flatten(leaf()) -> nil()
flatten(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2)))
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())
insertionsort(cons(x,xs)) -> insert(x,insertionsort(xs))
insertionsort(nil()) -> nil()
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
- Signature:
{append/2,flatten/1,flattensort/1,if'insert/4,insert/2,insertionsort/1,leq/2,append#/2,flatten#/1
,flattensort#/1,if'insert#/4,insert#/2,insertionsort#/1,leq#/2} / {0/0,cons/2,false/0,leaf/0,nil/0,node/3
,s/1,true/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/0,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,flatten#,flattensort#,if'insert#,insert#
,insertionsort#,leq#} and constructors {0,cons,false,leaf,nil,node,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:if'insert#(false(),x,y,ys) -> c_4(insert#(x,ys))
-->_1 insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys)):2
2:S:insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys))
-->_1 if'insert#(false(),x,y,ys) -> c_4(insert#(x,ys)):1
3:W:flatten#(node(l,t1,t2)) -> flatten#(t1)
-->_1 flatten#(node(l,t1,t2)) -> flatten#(t2):4
-->_1 flatten#(node(l,t1,t2)) -> flatten#(t1):3
4:W:flatten#(node(l,t1,t2)) -> flatten#(t2)
-->_1 flatten#(node(l,t1,t2)) -> flatten#(t2):4
-->_1 flatten#(node(l,t1,t2)) -> flatten#(t1):3
5:W:flattensort#(t) -> flatten#(t)
-->_1 flatten#(node(l,t1,t2)) -> flatten#(t2):4
-->_1 flatten#(node(l,t1,t2)) -> flatten#(t1):3
6:W:flattensort#(t) -> insertionsort#(flatten(t))
-->_1 insertionsort#(cons(x,xs)) -> insertionsort#(xs):8
-->_1 insertionsort#(cons(x,xs)) -> insert#(x,insertionsort(xs)):7
7:W:insertionsort#(cons(x,xs)) -> insert#(x,insertionsort(xs))
-->_1 insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys)):2
8:W:insertionsort#(cons(x,xs)) -> insertionsort#(xs)
-->_1 insertionsort#(cons(x,xs)) -> insertionsort#(xs):8
-->_1 insertionsort#(cons(x,xs)) -> insert#(x,insertionsort(xs)):7
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: flattensort#(t) -> flatten#(t)
3: flatten#(node(l,t1,t2)) -> flatten#(t1)
4: flatten#(node(l,t1,t2)) -> flatten#(t2)
*** Step 1.b:6.b:2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
if'insert#(false(),x,y,ys) -> c_4(insert#(x,ys))
insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys))
- Weak DPs:
flattensort#(t) -> insertionsort#(flatten(t))
insertionsort#(cons(x,xs)) -> insert#(x,insertionsort(xs))
insertionsort#(cons(x,xs)) -> insertionsort#(xs)
- Weak TRS:
append(cons(x,xs),l2) -> cons(x,append(xs,l2))
append(nil(),l2) -> l2
flatten(leaf()) -> nil()
flatten(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2)))
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())
insertionsort(cons(x,xs)) -> insert(x,insertionsort(xs))
insertionsort(nil()) -> nil()
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
- Signature:
{append/2,flatten/1,flattensort/1,if'insert/4,insert/2,insertionsort/1,leq/2,append#/2,flatten#/1
,flattensort#/1,if'insert#/4,insert#/2,insertionsort#/1,leq#/2} / {0/0,cons/2,false/0,leaf/0,nil/0,node/3
,s/1,true/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/0,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,flatten#,flattensort#,if'insert#,insert#
,insertionsort#,leq#} and constructors {0,cons,false,leaf,nil,node,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(append) = {1,2},
uargs(cons) = {2},
uargs(if'insert) = {1},
uargs(insert) = {2},
uargs(if'insert#) = {1},
uargs(insert#) = {2},
uargs(insertionsort#) = {1},
uargs(c_4) = {1},
uargs(c_6) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(append) = [1] x1 + [1] x2 + [0]
p(cons) = [1] x2 + [2]
p(false) = [0]
p(flatten) = [4] x1 + [0]
p(flattensort) = [4] x1 + [1]
p(if'insert) = [1] x1 + [1] x4 + [4]
p(insert) = [1] x2 + [2]
p(insertionsort) = [1] x1 + [3]
p(leaf) = [2]
p(leq) = [0]
p(nil) = [1]
p(node) = [1] x1 + [1] x2 + [1] x3 + [0]
p(s) = [1] x1 + [0]
p(true) = [0]
p(append#) = [2] x2 + [0]
p(flatten#) = [1] x1 + [2]
p(flattensort#) = [7] x1 + [1]
p(if'insert#) = [1] x1 + [1] x4 + [5]
p(insert#) = [1] x2 + [0]
p(insertionsort#) = [1] x1 + [1]
p(leq#) = [0]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [0]
p(c_4) = [1] x1 + [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [0]
p(c_8) = [0]
p(c_9) = [0]
p(c_10) = [0]
p(c_11) = [0]
p(c_12) = [0]
p(c_13) = [1]
p(c_14) = [0]
Following rules are strictly oriented:
if'insert#(false(),x,y,ys) = [1] ys + [5]
> [1] ys + [0]
= c_4(insert#(x,ys))
Following rules are (at-least) weakly oriented:
flattensort#(t) = [7] t + [1]
>= [4] t + [1]
= insertionsort#(flatten(t))
insert#(x,cons(y,ys)) = [1] ys + [2]
>= [1] ys + [5]
= c_6(if'insert#(leq(x,y),x,y,ys))
insertionsort#(cons(x,xs)) = [1] xs + [3]
>= [1] xs + [3]
= insert#(x,insertionsort(xs))
insertionsort#(cons(x,xs)) = [1] xs + [3]
>= [1] xs + [1]
= insertionsort#(xs)
append(cons(x,xs),l2) = [1] l2 + [1] xs + [2]
>= [1] l2 + [1] xs + [2]
= cons(x,append(xs,l2))
append(nil(),l2) = [1] l2 + [1]
>= [1] l2 + [0]
= l2
flatten(leaf()) = [8]
>= [1]
= nil()
flatten(node(l,t1,t2)) = [4] l + [4] t1 + [4] t2 + [0]
>= [1] l + [4] t1 + [4] t2 + [0]
= append(l,append(flatten(t1),flatten(t2)))
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()) = [3]
>= [3]
= cons(x,nil())
insertionsort(cons(x,xs)) = [1] xs + [5]
>= [1] xs + [5]
= insert(x,insertionsort(xs))
insertionsort(nil()) = [4]
>= [1]
= nil()
leq(0(),y) = [0]
>= [0]
= true()
leq(s(x),0()) = [0]
>= [0]
= false()
leq(s(x),s(y)) = [0]
>= [0]
= leq(x,y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 1.b:6.b:3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys))
- Weak DPs:
flattensort#(t) -> insertionsort#(flatten(t))
if'insert#(false(),x,y,ys) -> c_4(insert#(x,ys))
insertionsort#(cons(x,xs)) -> insert#(x,insertionsort(xs))
insertionsort#(cons(x,xs)) -> insertionsort#(xs)
- Weak TRS:
append(cons(x,xs),l2) -> cons(x,append(xs,l2))
append(nil(),l2) -> l2
flatten(leaf()) -> nil()
flatten(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2)))
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())
insertionsort(cons(x,xs)) -> insert(x,insertionsort(xs))
insertionsort(nil()) -> nil()
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
- Signature:
{append/2,flatten/1,flattensort/1,if'insert/4,insert/2,insertionsort/1,leq/2,append#/2,flatten#/1
,flattensort#/1,if'insert#/4,insert#/2,insertionsort#/1,leq#/2} / {0/0,cons/2,false/0,leaf/0,nil/0,node/3
,s/1,true/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/0,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,flatten#,flattensort#,if'insert#,insert#
,insertionsort#,leq#} and constructors {0,cons,false,leaf,nil,node,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(append) = {1,2},
uargs(cons) = {2},
uargs(if'insert) = {1},
uargs(insert) = {2},
uargs(if'insert#) = {1},
uargs(insert#) = {2},
uargs(insertionsort#) = {1},
uargs(c_4) = {1},
uargs(c_6) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(append) = [1] x1 + [1] x2 + [1]
p(cons) = [1] x2 + [4]
p(false) = [4]
p(flatten) = [1] x1 + [1]
p(flattensort) = [1] x1 + [0]
p(if'insert) = [1] x1 + [1] x4 + [4]
p(insert) = [1] x2 + [4]
p(insertionsort) = [1] x1 + [2]
p(leaf) = [4]
p(leq) = [4]
p(nil) = [4]
p(node) = [1] x1 + [1] x2 + [1] x3 + [4]
p(s) = [1]
p(true) = [4]
p(append#) = [1] x1 + [4] x2 + [1]
p(flatten#) = [2] x1 + [1]
p(flattensort#) = [4] x1 + [7]
p(if'insert#) = [1] x1 + [1] x4 + [2]
p(insert#) = [1] x2 + [5]
p(insertionsort#) = [1] x1 + [6]
p(leq#) = [0]
p(c_1) = [0]
p(c_2) = [4] x1 + [0]
p(c_3) = [1] x1 + [4] x2 + [1]
p(c_4) = [1] x1 + [1]
p(c_5) = [0]
p(c_6) = [1] x1 + [2]
p(c_7) = [1]
p(c_8) = [2] x1 + [1] x2 + [0]
p(c_9) = [1]
p(c_10) = [2] x1 + [0]
p(c_11) = [2]
p(c_12) = [1]
p(c_13) = [0]
p(c_14) = [4] x1 + [1]
Following rules are strictly oriented:
insert#(x,cons(y,ys)) = [1] ys + [9]
> [1] ys + [8]
= c_6(if'insert#(leq(x,y),x,y,ys))
Following rules are (at-least) weakly oriented:
flattensort#(t) = [4] t + [7]
>= [1] t + [7]
= insertionsort#(flatten(t))
if'insert#(false(),x,y,ys) = [1] ys + [6]
>= [1] ys + [6]
= c_4(insert#(x,ys))
insertionsort#(cons(x,xs)) = [1] xs + [10]
>= [1] xs + [7]
= insert#(x,insertionsort(xs))
insertionsort#(cons(x,xs)) = [1] xs + [10]
>= [1] xs + [6]
= insertionsort#(xs)
append(cons(x,xs),l2) = [1] l2 + [1] xs + [5]
>= [1] l2 + [1] xs + [5]
= cons(x,append(xs,l2))
append(nil(),l2) = [1] l2 + [5]
>= [1] l2 + [0]
= l2
flatten(leaf()) = [5]
>= [4]
= nil()
flatten(node(l,t1,t2)) = [1] l + [1] t1 + [1] t2 + [5]
>= [1] l + [1] t1 + [1] t2 + [4]
= append(l,append(flatten(t1),flatten(t2)))
if'insert(false(),x,y,ys) = [1] ys + [8]
>= [1] ys + [8]
= cons(y,insert(x,ys))
if'insert(true(),x,y,ys) = [1] ys + [8]
>= [1] ys + [8]
= cons(x,cons(y,ys))
insert(x,cons(y,ys)) = [1] ys + [8]
>= [1] ys + [8]
= if'insert(leq(x,y),x,y,ys)
insert(x,nil()) = [8]
>= [8]
= cons(x,nil())
insertionsort(cons(x,xs)) = [1] xs + [6]
>= [1] xs + [6]
= insert(x,insertionsort(xs))
insertionsort(nil()) = [6]
>= [4]
= nil()
leq(0(),y) = [4]
>= [4]
= true()
leq(s(x),0()) = [4]
>= [4]
= false()
leq(s(x),s(y)) = [4]
>= [4]
= leq(x,y)
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 1.b:6.b:4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
flattensort#(t) -> insertionsort#(flatten(t))
if'insert#(false(),x,y,ys) -> c_4(insert#(x,ys))
insert#(x,cons(y,ys)) -> c_6(if'insert#(leq(x,y),x,y,ys))
insertionsort#(cons(x,xs)) -> insert#(x,insertionsort(xs))
insertionsort#(cons(x,xs)) -> insertionsort#(xs)
- Weak TRS:
append(cons(x,xs),l2) -> cons(x,append(xs,l2))
append(nil(),l2) -> l2
flatten(leaf()) -> nil()
flatten(node(l,t1,t2)) -> append(l,append(flatten(t1),flatten(t2)))
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())
insertionsort(cons(x,xs)) -> insert(x,insertionsort(xs))
insertionsort(nil()) -> nil()
leq(0(),y) -> true()
leq(s(x),0()) -> false()
leq(s(x),s(y)) -> leq(x,y)
- Signature:
{append/2,flatten/1,flattensort/1,if'insert/4,insert/2,insertionsort/1,leq/2,append#/2,flatten#/1
,flattensort#/1,if'insert#/4,insert#/2,insertionsort#/1,leq#/2} / {0/0,cons/2,false/0,leaf/0,nil/0,node/3
,s/1,true/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/0,c_6/1,c_7/0,c_8/2,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,flatten#,flattensort#,if'insert#,insert#
,insertionsort#,leq#} and constructors {0,cons,false,leaf,nil,node,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^2))