WORST_CASE(?,O(n^3))
* Step 1: DependencyPairs WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
append(add(N,X),Y) -> add(N,append(X,Y))
append(nil(),Y) -> Y
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
f_3(pair(Y,Z),N,X) -> append(qsort(Y),add(X,qsort(Z)))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
qsort(add(N,X)) -> f_3(split(N,X),N,X)
qsort(nil()) -> nil()
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
- Signature:
{append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append,f_1,f_2,f_3,lt,qsort,split} and constructors {0
,add,false,nil,pair,s,true}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
append#(add(N,X),Y) -> c_1(append#(X,Y))
append#(nil(),Y) -> c_2()
f_1#(pair(X,Z),N,M,Y) -> c_3(f_2#(lt(N,M),N,M,Y,X,Z),lt#(N,M))
f_2#(false(),N,M,Y,X,Z) -> c_4()
f_2#(true(),N,M,Y,X,Z) -> c_5()
f_3#(pair(Y,Z),N,X) -> c_6(append#(qsort(Y),add(X,qsort(Z))),qsort#(Y),qsort#(Z))
lt#(0(),s(X)) -> c_7()
lt#(s(X),0()) -> c_8()
lt#(s(X),s(Y)) -> c_9(lt#(X,Y))
qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X),split#(N,X))
qsort#(nil()) -> c_11()
split#(N,add(M,Y)) -> c_12(f_1#(split(N,Y),N,M,Y),split#(N,Y))
split#(N,nil()) -> c_13()
Weak DPs
and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
append#(add(N,X),Y) -> c_1(append#(X,Y))
append#(nil(),Y) -> c_2()
f_1#(pair(X,Z),N,M,Y) -> c_3(f_2#(lt(N,M),N,M,Y,X,Z),lt#(N,M))
f_2#(false(),N,M,Y,X,Z) -> c_4()
f_2#(true(),N,M,Y,X,Z) -> c_5()
f_3#(pair(Y,Z),N,X) -> c_6(append#(qsort(Y),add(X,qsort(Z))),qsort#(Y),qsort#(Z))
lt#(0(),s(X)) -> c_7()
lt#(s(X),0()) -> c_8()
lt#(s(X),s(Y)) -> c_9(lt#(X,Y))
qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X),split#(N,X))
qsort#(nil()) -> c_11()
split#(N,add(M,Y)) -> c_12(f_1#(split(N,Y),N,M,Y),split#(N,Y))
split#(N,nil()) -> c_13()
- Weak TRS:
append(add(N,X),Y) -> add(N,append(X,Y))
append(nil(),Y) -> Y
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
f_3(pair(Y,Z),N,X) -> append(qsort(Y),add(X,qsort(Z)))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
qsort(add(N,X)) -> f_3(split(N,X),N,X)
qsort(nil()) -> nil()
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
- Signature:
{append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2,append#/2,f_1#/4,f_2#/6,f_3#/3,lt#/2,qsort#/1
,split#/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/3,c_7/0,c_8/0
,c_9/1,c_10/2,c_11/0,c_12/2,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,f_1#,f_2#,f_3#,lt#,qsort#
,split#} and constructors {0,add,false,nil,pair,s,true}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{2,4,5,7,8,11,13}
by application of
Pre({2,4,5,7,8,11,13}) = {1,3,6,9,10,12}.
Here rules are labelled as follows:
1: append#(add(N,X),Y) -> c_1(append#(X,Y))
2: append#(nil(),Y) -> c_2()
3: f_1#(pair(X,Z),N,M,Y) -> c_3(f_2#(lt(N,M),N,M,Y,X,Z),lt#(N,M))
4: f_2#(false(),N,M,Y,X,Z) -> c_4()
5: f_2#(true(),N,M,Y,X,Z) -> c_5()
6: f_3#(pair(Y,Z),N,X) -> c_6(append#(qsort(Y),add(X,qsort(Z))),qsort#(Y),qsort#(Z))
7: lt#(0(),s(X)) -> c_7()
8: lt#(s(X),0()) -> c_8()
9: lt#(s(X),s(Y)) -> c_9(lt#(X,Y))
10: qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X),split#(N,X))
11: qsort#(nil()) -> c_11()
12: split#(N,add(M,Y)) -> c_12(f_1#(split(N,Y),N,M,Y),split#(N,Y))
13: split#(N,nil()) -> c_13()
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
append#(add(N,X),Y) -> c_1(append#(X,Y))
f_1#(pair(X,Z),N,M,Y) -> c_3(f_2#(lt(N,M),N,M,Y,X,Z),lt#(N,M))
f_3#(pair(Y,Z),N,X) -> c_6(append#(qsort(Y),add(X,qsort(Z))),qsort#(Y),qsort#(Z))
lt#(s(X),s(Y)) -> c_9(lt#(X,Y))
qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X),split#(N,X))
split#(N,add(M,Y)) -> c_12(f_1#(split(N,Y),N,M,Y),split#(N,Y))
- Weak DPs:
append#(nil(),Y) -> c_2()
f_2#(false(),N,M,Y,X,Z) -> c_4()
f_2#(true(),N,M,Y,X,Z) -> c_5()
lt#(0(),s(X)) -> c_7()
lt#(s(X),0()) -> c_8()
qsort#(nil()) -> c_11()
split#(N,nil()) -> c_13()
- Weak TRS:
append(add(N,X),Y) -> add(N,append(X,Y))
append(nil(),Y) -> Y
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
f_3(pair(Y,Z),N,X) -> append(qsort(Y),add(X,qsort(Z)))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
qsort(add(N,X)) -> f_3(split(N,X),N,X)
qsort(nil()) -> nil()
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
- Signature:
{append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2,append#/2,f_1#/4,f_2#/6,f_3#/3,lt#/2,qsort#/1
,split#/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/3,c_7/0,c_8/0
,c_9/1,c_10/2,c_11/0,c_12/2,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,f_1#,f_2#,f_3#,lt#,qsort#
,split#} and constructors {0,add,false,nil,pair,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:append#(add(N,X),Y) -> c_1(append#(X,Y))
-->_1 append#(nil(),Y) -> c_2():7
-->_1 append#(add(N,X),Y) -> c_1(append#(X,Y)):1
2:S:f_1#(pair(X,Z),N,M,Y) -> c_3(f_2#(lt(N,M),N,M,Y,X,Z),lt#(N,M))
-->_2 lt#(s(X),s(Y)) -> c_9(lt#(X,Y)):4
-->_2 lt#(s(X),0()) -> c_8():11
-->_2 lt#(0(),s(X)) -> c_7():10
-->_1 f_2#(true(),N,M,Y,X,Z) -> c_5():9
-->_1 f_2#(false(),N,M,Y,X,Z) -> c_4():8
3:S:f_3#(pair(Y,Z),N,X) -> c_6(append#(qsort(Y),add(X,qsort(Z))),qsort#(Y),qsort#(Z))
-->_3 qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X),split#(N,X)):5
-->_2 qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X),split#(N,X)):5
-->_3 qsort#(nil()) -> c_11():12
-->_2 qsort#(nil()) -> c_11():12
-->_1 append#(nil(),Y) -> c_2():7
-->_1 append#(add(N,X),Y) -> c_1(append#(X,Y)):1
4:S:lt#(s(X),s(Y)) -> c_9(lt#(X,Y))
-->_1 lt#(s(X),0()) -> c_8():11
-->_1 lt#(0(),s(X)) -> c_7():10
-->_1 lt#(s(X),s(Y)) -> c_9(lt#(X,Y)):4
5:S:qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X),split#(N,X))
-->_2 split#(N,add(M,Y)) -> c_12(f_1#(split(N,Y),N,M,Y),split#(N,Y)):6
-->_2 split#(N,nil()) -> c_13():13
-->_1 f_3#(pair(Y,Z),N,X) -> c_6(append#(qsort(Y),add(X,qsort(Z))),qsort#(Y),qsort#(Z)):3
6:S:split#(N,add(M,Y)) -> c_12(f_1#(split(N,Y),N,M,Y),split#(N,Y))
-->_2 split#(N,nil()) -> c_13():13
-->_2 split#(N,add(M,Y)) -> c_12(f_1#(split(N,Y),N,M,Y),split#(N,Y)):6
-->_1 f_1#(pair(X,Z),N,M,Y) -> c_3(f_2#(lt(N,M),N,M,Y,X,Z),lt#(N,M)):2
7:W:append#(nil(),Y) -> c_2()
8:W:f_2#(false(),N,M,Y,X,Z) -> c_4()
9:W:f_2#(true(),N,M,Y,X,Z) -> c_5()
10:W:lt#(0(),s(X)) -> c_7()
11:W:lt#(s(X),0()) -> c_8()
12:W:qsort#(nil()) -> c_11()
13:W:split#(N,nil()) -> c_13()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
12: qsort#(nil()) -> c_11()
13: split#(N,nil()) -> c_13()
8: f_2#(false(),N,M,Y,X,Z) -> c_4()
9: f_2#(true(),N,M,Y,X,Z) -> c_5()
10: lt#(0(),s(X)) -> c_7()
11: lt#(s(X),0()) -> c_8()
7: append#(nil(),Y) -> c_2()
* Step 4: SimplifyRHS WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
append#(add(N,X),Y) -> c_1(append#(X,Y))
f_1#(pair(X,Z),N,M,Y) -> c_3(f_2#(lt(N,M),N,M,Y,X,Z),lt#(N,M))
f_3#(pair(Y,Z),N,X) -> c_6(append#(qsort(Y),add(X,qsort(Z))),qsort#(Y),qsort#(Z))
lt#(s(X),s(Y)) -> c_9(lt#(X,Y))
qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X),split#(N,X))
split#(N,add(M,Y)) -> c_12(f_1#(split(N,Y),N,M,Y),split#(N,Y))
- Weak TRS:
append(add(N,X),Y) -> add(N,append(X,Y))
append(nil(),Y) -> Y
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
f_3(pair(Y,Z),N,X) -> append(qsort(Y),add(X,qsort(Z)))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
qsort(add(N,X)) -> f_3(split(N,X),N,X)
qsort(nil()) -> nil()
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
- Signature:
{append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2,append#/2,f_1#/4,f_2#/6,f_3#/3,lt#/2,qsort#/1
,split#/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/2,c_4/0,c_5/0,c_6/3,c_7/0,c_8/0
,c_9/1,c_10/2,c_11/0,c_12/2,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,f_1#,f_2#,f_3#,lt#,qsort#
,split#} and constructors {0,add,false,nil,pair,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:append#(add(N,X),Y) -> c_1(append#(X,Y))
-->_1 append#(add(N,X),Y) -> c_1(append#(X,Y)):1
2:S:f_1#(pair(X,Z),N,M,Y) -> c_3(f_2#(lt(N,M),N,M,Y,X,Z),lt#(N,M))
-->_2 lt#(s(X),s(Y)) -> c_9(lt#(X,Y)):4
3:S:f_3#(pair(Y,Z),N,X) -> c_6(append#(qsort(Y),add(X,qsort(Z))),qsort#(Y),qsort#(Z))
-->_3 qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X),split#(N,X)):5
-->_2 qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X),split#(N,X)):5
-->_1 append#(add(N,X),Y) -> c_1(append#(X,Y)):1
4:S:lt#(s(X),s(Y)) -> c_9(lt#(X,Y))
-->_1 lt#(s(X),s(Y)) -> c_9(lt#(X,Y)):4
5:S:qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X),split#(N,X))
-->_2 split#(N,add(M,Y)) -> c_12(f_1#(split(N,Y),N,M,Y),split#(N,Y)):6
-->_1 f_3#(pair(Y,Z),N,X) -> c_6(append#(qsort(Y),add(X,qsort(Z))),qsort#(Y),qsort#(Z)):3
6:S:split#(N,add(M,Y)) -> c_12(f_1#(split(N,Y),N,M,Y),split#(N,Y))
-->_2 split#(N,add(M,Y)) -> c_12(f_1#(split(N,Y),N,M,Y),split#(N,Y)):6
-->_1 f_1#(pair(X,Z),N,M,Y) -> c_3(f_2#(lt(N,M),N,M,Y,X,Z),lt#(N,M)):2
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
f_1#(pair(X,Z),N,M,Y) -> c_3(lt#(N,M))
* Step 5: DecomposeDG WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
append#(add(N,X),Y) -> c_1(append#(X,Y))
f_1#(pair(X,Z),N,M,Y) -> c_3(lt#(N,M))
f_3#(pair(Y,Z),N,X) -> c_6(append#(qsort(Y),add(X,qsort(Z))),qsort#(Y),qsort#(Z))
lt#(s(X),s(Y)) -> c_9(lt#(X,Y))
qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X),split#(N,X))
split#(N,add(M,Y)) -> c_12(f_1#(split(N,Y),N,M,Y),split#(N,Y))
- Weak TRS:
append(add(N,X),Y) -> add(N,append(X,Y))
append(nil(),Y) -> Y
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
f_3(pair(Y,Z),N,X) -> append(qsort(Y),add(X,qsort(Z)))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
qsort(add(N,X)) -> f_3(split(N,X),N,X)
qsort(nil()) -> nil()
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
- Signature:
{append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2,append#/2,f_1#/4,f_2#/6,f_3#/3,lt#/2,qsort#/1
,split#/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/3,c_7/0,c_8/0
,c_9/1,c_10/2,c_11/0,c_12/2,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,f_1#,f_2#,f_3#,lt#,qsort#
,split#} and constructors {0,add,false,nil,pair,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
f_3#(pair(Y,Z),N,X) -> c_6(append#(qsort(Y),add(X,qsort(Z))),qsort#(Y),qsort#(Z))
qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X),split#(N,X))
and a lower component
append#(add(N,X),Y) -> c_1(append#(X,Y))
f_1#(pair(X,Z),N,M,Y) -> c_3(lt#(N,M))
lt#(s(X),s(Y)) -> c_9(lt#(X,Y))
split#(N,add(M,Y)) -> c_12(f_1#(split(N,Y),N,M,Y),split#(N,Y))
Further, following extension rules are added to the lower component.
f_3#(pair(Y,Z),N,X) -> append#(qsort(Y),add(X,qsort(Z)))
f_3#(pair(Y,Z),N,X) -> qsort#(Y)
f_3#(pair(Y,Z),N,X) -> qsort#(Z)
qsort#(add(N,X)) -> f_3#(split(N,X),N,X)
qsort#(add(N,X)) -> split#(N,X)
** Step 5.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f_3#(pair(Y,Z),N,X) -> c_6(append#(qsort(Y),add(X,qsort(Z))),qsort#(Y),qsort#(Z))
qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X),split#(N,X))
- Weak TRS:
append(add(N,X),Y) -> add(N,append(X,Y))
append(nil(),Y) -> Y
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
f_3(pair(Y,Z),N,X) -> append(qsort(Y),add(X,qsort(Z)))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
qsort(add(N,X)) -> f_3(split(N,X),N,X)
qsort(nil()) -> nil()
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
- Signature:
{append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2,append#/2,f_1#/4,f_2#/6,f_3#/3,lt#/2,qsort#/1
,split#/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/3,c_7/0,c_8/0
,c_9/1,c_10/2,c_11/0,c_12/2,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,f_1#,f_2#,f_3#,lt#,qsort#
,split#} and constructors {0,add,false,nil,pair,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:f_3#(pair(Y,Z),N,X) -> c_6(append#(qsort(Y),add(X,qsort(Z))),qsort#(Y),qsort#(Z))
-->_3 qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X),split#(N,X)):2
-->_2 qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X),split#(N,X)):2
2:S:qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X),split#(N,X))
-->_1 f_3#(pair(Y,Z),N,X) -> c_6(append#(qsort(Y),add(X,qsort(Z))),qsort#(Y),qsort#(Z)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
f_3#(pair(Y,Z),N,X) -> c_6(qsort#(Y),qsort#(Z))
qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X))
** Step 5.a:2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f_3#(pair(Y,Z),N,X) -> c_6(qsort#(Y),qsort#(Z))
qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X))
- Weak TRS:
append(add(N,X),Y) -> add(N,append(X,Y))
append(nil(),Y) -> Y
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
f_3(pair(Y,Z),N,X) -> append(qsort(Y),add(X,qsort(Z)))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
qsort(add(N,X)) -> f_3(split(N,X),N,X)
qsort(nil()) -> nil()
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
- Signature:
{append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2,append#/2,f_1#/4,f_2#/6,f_3#/3,lt#/2,qsort#/1
,split#/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0
,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,f_1#,f_2#,f_3#,lt#,qsort#
,split#} and constructors {0,add,false,nil,pair,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
f_3#(pair(Y,Z),N,X) -> c_6(qsort#(Y),qsort#(Z))
qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X))
** Step 5.a:3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f_3#(pair(Y,Z),N,X) -> c_6(qsort#(Y),qsort#(Z))
qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X))
- Weak TRS:
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
- Signature:
{append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2,append#/2,f_1#/4,f_2#/6,f_3#/3,lt#/2,qsort#/1
,split#/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0
,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,f_1#,f_2#,f_3#,lt#,qsort#
,split#} and constructors {0,add,false,nil,pair,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(f_1) = {1},
uargs(f_2) = {1},
uargs(f_3#) = {1},
uargs(c_6) = {1,2},
uargs(c_10) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(add) = [1] x1 + [1] x2 + [0]
p(append) = [0]
p(f_1) = [1] x1 + [0]
p(f_2) = [1] x1 + [1]
p(f_3) = [0]
p(false) = [1]
p(lt) = [1]
p(nil) = [0]
p(pair) = [2]
p(qsort) = [0]
p(s) = [0]
p(split) = [2]
p(true) = [1]
p(append#) = [1] x1 + [2]
p(f_1#) = [1] x2 + [1]
p(f_2#) = [1] x6 + [4]
p(f_3#) = [1] x1 + [0]
p(lt#) = [0]
p(qsort#) = [7]
p(split#) = [0]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [1] x2 + [0]
p(c_7) = [0]
p(c_8) = [0]
p(c_9) = [0]
p(c_10) = [1] x1 + [0]
p(c_11) = [0]
p(c_12) = [0]
p(c_13) = [0]
Following rules are strictly oriented:
qsort#(add(N,X)) = [7]
> [2]
= c_10(f_3#(split(N,X),N,X))
Following rules are (at-least) weakly oriented:
f_3#(pair(Y,Z),N,X) = [2]
>= [14]
= c_6(qsort#(Y),qsort#(Z))
f_1(pair(X,Z),N,M,Y) = [2]
>= [2]
= f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) = [2]
>= [2]
= pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) = [2]
>= [2]
= pair(X,add(M,Z))
lt(0(),s(X)) = [1]
>= [1]
= true()
lt(s(X),0()) = [1]
>= [1]
= false()
lt(s(X),s(Y)) = [1]
>= [1]
= lt(X,Y)
split(N,add(M,Y)) = [2]
>= [2]
= f_1(split(N,Y),N,M,Y)
split(N,nil()) = [2]
>= [2]
= pair(nil(),nil())
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 5.a:4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f_3#(pair(Y,Z),N,X) -> c_6(qsort#(Y),qsort#(Z))
- Weak DPs:
qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X))
- Weak TRS:
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
- Signature:
{append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2,append#/2,f_1#/4,f_2#/6,f_3#/3,lt#/2,qsort#/1
,split#/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0
,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,f_1#,f_2#,f_3#,lt#,qsort#
,split#} and constructors {0,add,false,nil,pair,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(f_1) = {1},
uargs(f_2) = {1},
uargs(f_3#) = {1},
uargs(c_6) = {1,2},
uargs(c_10) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(add) = [1] x2 + [1]
p(append) = [1] x2 + [4]
p(f_1) = [1] x1 + [1]
p(f_2) = [1] x1 + [1] x5 + [1] x6 + [0]
p(f_3) = [1] x1 + [1] x2 + [2] x3 + [2]
p(false) = [1]
p(lt) = [1]
p(nil) = [0]
p(pair) = [1] x1 + [1] x2 + [0]
p(qsort) = [1] x1 + [4]
p(s) = [1] x1 + [4]
p(split) = [1] x2 + [0]
p(true) = [1]
p(append#) = [2] x1 + [0]
p(f_1#) = [0]
p(f_2#) = [2] x4 + [1] x5 + [0]
p(f_3#) = [1] x1 + [1]
p(lt#) = [4] x1 + [2]
p(qsort#) = [1] x1 + [0]
p(split#) = [2] x1 + [1] x2 + [0]
p(c_1) = [1] x1 + [0]
p(c_2) = [2]
p(c_3) = [2]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [1] x2 + [0]
p(c_7) = [0]
p(c_8) = [2]
p(c_9) = [0]
p(c_10) = [1] x1 + [0]
p(c_11) = [2]
p(c_12) = [1] x1 + [4] x2 + [1]
p(c_13) = [0]
Following rules are strictly oriented:
f_3#(pair(Y,Z),N,X) = [1] Y + [1] Z + [1]
> [1] Y + [1] Z + [0]
= c_6(qsort#(Y),qsort#(Z))
Following rules are (at-least) weakly oriented:
qsort#(add(N,X)) = [1] X + [1]
>= [1] X + [1]
= c_10(f_3#(split(N,X),N,X))
f_1(pair(X,Z),N,M,Y) = [1] X + [1] Z + [1]
>= [1] X + [1] Z + [1]
= f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) = [1] X + [1] Z + [1]
>= [1] X + [1] Z + [1]
= pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) = [1] X + [1] Z + [1]
>= [1] X + [1] Z + [1]
= pair(X,add(M,Z))
lt(0(),s(X)) = [1]
>= [1]
= true()
lt(s(X),0()) = [1]
>= [1]
= false()
lt(s(X),s(Y)) = [1]
>= [1]
= lt(X,Y)
split(N,add(M,Y)) = [1] Y + [1]
>= [1] Y + [1]
= f_1(split(N,Y),N,M,Y)
split(N,nil()) = [0]
>= [0]
= pair(nil(),nil())
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 5.a:5: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f_3#(pair(Y,Z),N,X) -> c_6(qsort#(Y),qsort#(Z))
qsort#(add(N,X)) -> c_10(f_3#(split(N,X),N,X))
- Weak TRS:
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
- Signature:
{append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2,append#/2,f_1#/4,f_2#/6,f_3#/3,lt#/2,qsort#/1
,split#/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/2,c_7/0,c_8/0
,c_9/1,c_10/1,c_11/0,c_12/2,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,f_1#,f_2#,f_3#,lt#,qsort#
,split#} and constructors {0,add,false,nil,pair,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
** Step 5.b:1: DecomposeDG WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
append#(add(N,X),Y) -> c_1(append#(X,Y))
f_1#(pair(X,Z),N,M,Y) -> c_3(lt#(N,M))
lt#(s(X),s(Y)) -> c_9(lt#(X,Y))
split#(N,add(M,Y)) -> c_12(f_1#(split(N,Y),N,M,Y),split#(N,Y))
- Weak DPs:
f_3#(pair(Y,Z),N,X) -> append#(qsort(Y),add(X,qsort(Z)))
f_3#(pair(Y,Z),N,X) -> qsort#(Y)
f_3#(pair(Y,Z),N,X) -> qsort#(Z)
qsort#(add(N,X)) -> f_3#(split(N,X),N,X)
qsort#(add(N,X)) -> split#(N,X)
- Weak TRS:
append(add(N,X),Y) -> add(N,append(X,Y))
append(nil(),Y) -> Y
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
f_3(pair(Y,Z),N,X) -> append(qsort(Y),add(X,qsort(Z)))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
qsort(add(N,X)) -> f_3(split(N,X),N,X)
qsort(nil()) -> nil()
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
- Signature:
{append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2,append#/2,f_1#/4,f_2#/6,f_3#/3,lt#/2,qsort#/1
,split#/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/3,c_7/0,c_8/0
,c_9/1,c_10/2,c_11/0,c_12/2,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,f_1#,f_2#,f_3#,lt#,qsort#
,split#} and constructors {0,add,false,nil,pair,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
append#(add(N,X),Y) -> c_1(append#(X,Y))
f_3#(pair(Y,Z),N,X) -> append#(qsort(Y),add(X,qsort(Z)))
f_3#(pair(Y,Z),N,X) -> qsort#(Y)
f_3#(pair(Y,Z),N,X) -> qsort#(Z)
qsort#(add(N,X)) -> f_3#(split(N,X),N,X)
qsort#(add(N,X)) -> split#(N,X)
split#(N,add(M,Y)) -> c_12(f_1#(split(N,Y),N,M,Y),split#(N,Y))
and a lower component
f_1#(pair(X,Z),N,M,Y) -> c_3(lt#(N,M))
lt#(s(X),s(Y)) -> c_9(lt#(X,Y))
Further, following extension rules are added to the lower component.
append#(add(N,X),Y) -> append#(X,Y)
f_3#(pair(Y,Z),N,X) -> append#(qsort(Y),add(X,qsort(Z)))
f_3#(pair(Y,Z),N,X) -> qsort#(Y)
f_3#(pair(Y,Z),N,X) -> qsort#(Z)
qsort#(add(N,X)) -> f_3#(split(N,X),N,X)
qsort#(add(N,X)) -> split#(N,X)
split#(N,add(M,Y)) -> f_1#(split(N,Y),N,M,Y)
split#(N,add(M,Y)) -> split#(N,Y)
*** Step 5.b:1.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
append#(add(N,X),Y) -> c_1(append#(X,Y))
split#(N,add(M,Y)) -> c_12(f_1#(split(N,Y),N,M,Y),split#(N,Y))
- Weak DPs:
f_3#(pair(Y,Z),N,X) -> append#(qsort(Y),add(X,qsort(Z)))
f_3#(pair(Y,Z),N,X) -> qsort#(Y)
f_3#(pair(Y,Z),N,X) -> qsort#(Z)
qsort#(add(N,X)) -> f_3#(split(N,X),N,X)
qsort#(add(N,X)) -> split#(N,X)
- Weak TRS:
append(add(N,X),Y) -> add(N,append(X,Y))
append(nil(),Y) -> Y
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
f_3(pair(Y,Z),N,X) -> append(qsort(Y),add(X,qsort(Z)))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
qsort(add(N,X)) -> f_3(split(N,X),N,X)
qsort(nil()) -> nil()
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
- Signature:
{append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2,append#/2,f_1#/4,f_2#/6,f_3#/3,lt#/2,qsort#/1
,split#/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/3,c_7/0,c_8/0
,c_9/1,c_10/2,c_11/0,c_12/2,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,f_1#,f_2#,f_3#,lt#,qsort#
,split#} and constructors {0,add,false,nil,pair,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:append#(add(N,X),Y) -> c_1(append#(X,Y))
-->_1 append#(add(N,X),Y) -> c_1(append#(X,Y)):1
2:S:split#(N,add(M,Y)) -> c_12(f_1#(split(N,Y),N,M,Y),split#(N,Y))
-->_2 split#(N,add(M,Y)) -> c_12(f_1#(split(N,Y),N,M,Y),split#(N,Y)):2
3:W:f_3#(pair(Y,Z),N,X) -> append#(qsort(Y),add(X,qsort(Z)))
-->_1 append#(add(N,X),Y) -> c_1(append#(X,Y)):1
4:W:f_3#(pair(Y,Z),N,X) -> qsort#(Y)
-->_1 qsort#(add(N,X)) -> split#(N,X):7
-->_1 qsort#(add(N,X)) -> f_3#(split(N,X),N,X):6
5:W:f_3#(pair(Y,Z),N,X) -> qsort#(Z)
-->_1 qsort#(add(N,X)) -> split#(N,X):7
-->_1 qsort#(add(N,X)) -> f_3#(split(N,X),N,X):6
6:W:qsort#(add(N,X)) -> f_3#(split(N,X),N,X)
-->_1 f_3#(pair(Y,Z),N,X) -> qsort#(Z):5
-->_1 f_3#(pair(Y,Z),N,X) -> qsort#(Y):4
-->_1 f_3#(pair(Y,Z),N,X) -> append#(qsort(Y),add(X,qsort(Z))):3
7:W:qsort#(add(N,X)) -> split#(N,X)
-->_1 split#(N,add(M,Y)) -> c_12(f_1#(split(N,Y),N,M,Y),split#(N,Y)):2
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
split#(N,add(M,Y)) -> c_12(split#(N,Y))
*** Step 5.b:1.a:2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
append#(add(N,X),Y) -> c_1(append#(X,Y))
split#(N,add(M,Y)) -> c_12(split#(N,Y))
- Weak DPs:
f_3#(pair(Y,Z),N,X) -> append#(qsort(Y),add(X,qsort(Z)))
f_3#(pair(Y,Z),N,X) -> qsort#(Y)
f_3#(pair(Y,Z),N,X) -> qsort#(Z)
qsort#(add(N,X)) -> f_3#(split(N,X),N,X)
qsort#(add(N,X)) -> split#(N,X)
- Weak TRS:
append(add(N,X),Y) -> add(N,append(X,Y))
append(nil(),Y) -> Y
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
f_3(pair(Y,Z),N,X) -> append(qsort(Y),add(X,qsort(Z)))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
qsort(add(N,X)) -> f_3(split(N,X),N,X)
qsort(nil()) -> nil()
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
- Signature:
{append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2,append#/2,f_1#/4,f_2#/6,f_3#/3,lt#/2,qsort#/1
,split#/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/3,c_7/0,c_8/0
,c_9/1,c_10/2,c_11/0,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,f_1#,f_2#,f_3#,lt#,qsort#
,split#} and constructors {0,add,false,nil,pair,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(add) = {2},
uargs(append) = {1,2},
uargs(f_1) = {1},
uargs(f_2) = {1},
uargs(f_3) = {1},
uargs(append#) = {1,2},
uargs(f_3#) = {1},
uargs(c_1) = {1},
uargs(c_12) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(add) = [1] x2 + [2]
p(append) = [1] x1 + [1] x2 + [0]
p(f_1) = [1] x1 + [2]
p(f_2) = [1] x1 + [1] x5 + [1] x6 + [0]
p(f_3) = [1] x1 + [2]
p(false) = [2]
p(lt) = [2]
p(nil) = [0]
p(pair) = [1] x1 + [1] x2 + [0]
p(qsort) = [1] x1 + [0]
p(s) = [1]
p(split) = [1] x2 + [0]
p(true) = [2]
p(append#) = [1] x1 + [1] x2 + [0]
p(f_1#) = [1] x1 + [1] x4 + [1]
p(f_2#) = [1] x1 + [2] x4 + [1] x5 + [2] x6 + [0]
p(f_3#) = [1] x1 + [3]
p(lt#) = [1] x2 + [1]
p(qsort#) = [1] x1 + [1]
p(split#) = [0]
p(c_1) = [1] x1 + [0]
p(c_2) = [0]
p(c_3) = [1]
p(c_4) = [1]
p(c_5) = [2]
p(c_6) = [1] x1 + [1] x3 + [0]
p(c_7) = [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [4]
p(c_10) = [2] x1 + [1] x2 + [0]
p(c_11) = [2]
p(c_12) = [1] x1 + [0]
p(c_13) = [1]
Following rules are strictly oriented:
append#(add(N,X),Y) = [1] X + [1] Y + [2]
> [1] X + [1] Y + [0]
= c_1(append#(X,Y))
Following rules are (at-least) weakly oriented:
f_3#(pair(Y,Z),N,X) = [1] Y + [1] Z + [3]
>= [1] Y + [1] Z + [2]
= append#(qsort(Y),add(X,qsort(Z)))
f_3#(pair(Y,Z),N,X) = [1] Y + [1] Z + [3]
>= [1] Y + [1]
= qsort#(Y)
f_3#(pair(Y,Z),N,X) = [1] Y + [1] Z + [3]
>= [1] Z + [1]
= qsort#(Z)
qsort#(add(N,X)) = [1] X + [3]
>= [1] X + [3]
= f_3#(split(N,X),N,X)
qsort#(add(N,X)) = [1] X + [3]
>= [0]
= split#(N,X)
split#(N,add(M,Y)) = [0]
>= [0]
= c_12(split#(N,Y))
append(add(N,X),Y) = [1] X + [1] Y + [2]
>= [1] X + [1] Y + [2]
= add(N,append(X,Y))
append(nil(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
f_1(pair(X,Z),N,M,Y) = [1] X + [1] Z + [2]
>= [1] X + [1] Z + [2]
= f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) = [1] X + [1] Z + [2]
>= [1] X + [1] Z + [2]
= pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) = [1] X + [1] Z + [2]
>= [1] X + [1] Z + [2]
= pair(X,add(M,Z))
f_3(pair(Y,Z),N,X) = [1] Y + [1] Z + [2]
>= [1] Y + [1] Z + [2]
= append(qsort(Y),add(X,qsort(Z)))
lt(0(),s(X)) = [2]
>= [2]
= true()
lt(s(X),0()) = [2]
>= [2]
= false()
lt(s(X),s(Y)) = [2]
>= [2]
= lt(X,Y)
qsort(add(N,X)) = [1] X + [2]
>= [1] X + [2]
= f_3(split(N,X),N,X)
qsort(nil()) = [0]
>= [0]
= nil()
split(N,add(M,Y)) = [1] Y + [2]
>= [1] Y + [2]
= f_1(split(N,Y),N,M,Y)
split(N,nil()) = [0]
>= [0]
= pair(nil(),nil())
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 5.b:1.a:3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
split#(N,add(M,Y)) -> c_12(split#(N,Y))
- Weak DPs:
append#(add(N,X),Y) -> c_1(append#(X,Y))
f_3#(pair(Y,Z),N,X) -> append#(qsort(Y),add(X,qsort(Z)))
f_3#(pair(Y,Z),N,X) -> qsort#(Y)
f_3#(pair(Y,Z),N,X) -> qsort#(Z)
qsort#(add(N,X)) -> f_3#(split(N,X),N,X)
qsort#(add(N,X)) -> split#(N,X)
- Weak TRS:
append(add(N,X),Y) -> add(N,append(X,Y))
append(nil(),Y) -> Y
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
f_3(pair(Y,Z),N,X) -> append(qsort(Y),add(X,qsort(Z)))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
qsort(add(N,X)) -> f_3(split(N,X),N,X)
qsort(nil()) -> nil()
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
- Signature:
{append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2,append#/2,f_1#/4,f_2#/6,f_3#/3,lt#/2,qsort#/1
,split#/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/3,c_7/0,c_8/0
,c_9/1,c_10/2,c_11/0,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,f_1#,f_2#,f_3#,lt#,qsort#
,split#} and constructors {0,add,false,nil,pair,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(add) = {2},
uargs(append) = {1,2},
uargs(f_1) = {1},
uargs(f_2) = {1},
uargs(f_3) = {1},
uargs(append#) = {1,2},
uargs(f_3#) = {1},
uargs(c_1) = {1},
uargs(c_12) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(add) = [1] x2 + [2]
p(append) = [1] x1 + [1] x2 + [0]
p(f_1) = [1] x1 + [2]
p(f_2) = [1] x1 + [1] x5 + [1] x6 + [2]
p(f_3) = [1] x1 + [2]
p(false) = [0]
p(lt) = [0]
p(nil) = [0]
p(pair) = [1] x1 + [1] x2 + [0]
p(qsort) = [1] x1 + [0]
p(s) = [1] x1 + [0]
p(split) = [1] x2 + [0]
p(true) = [0]
p(append#) = [1] x1 + [1] x2 + [0]
p(f_1#) = [1] x2 + [2] x3 + [2] x4 + [2]
p(f_2#) = [2] x3 + [2] x4 + [1] x5 + [2]
p(f_3#) = [1] x1 + [2]
p(lt#) = [1] x1 + [2] x2 + [0]
p(qsort#) = [1] x1 + [2]
p(split#) = [1] x2 + [4]
p(c_1) = [1] x1 + [2]
p(c_2) = [0]
p(c_3) = [0]
p(c_4) = [1]
p(c_5) = [0]
p(c_6) = [1] x1 + [1] x3 + [1]
p(c_7) = [2]
p(c_8) = [4]
p(c_9) = [1] x1 + [1]
p(c_10) = [1] x1 + [0]
p(c_11) = [0]
p(c_12) = [1] x1 + [0]
p(c_13) = [1]
Following rules are strictly oriented:
split#(N,add(M,Y)) = [1] Y + [6]
> [1] Y + [4]
= c_12(split#(N,Y))
Following rules are (at-least) weakly oriented:
append#(add(N,X),Y) = [1] X + [1] Y + [2]
>= [1] X + [1] Y + [2]
= c_1(append#(X,Y))
f_3#(pair(Y,Z),N,X) = [1] Y + [1] Z + [2]
>= [1] Y + [1] Z + [2]
= append#(qsort(Y),add(X,qsort(Z)))
f_3#(pair(Y,Z),N,X) = [1] Y + [1] Z + [2]
>= [1] Y + [2]
= qsort#(Y)
f_3#(pair(Y,Z),N,X) = [1] Y + [1] Z + [2]
>= [1] Z + [2]
= qsort#(Z)
qsort#(add(N,X)) = [1] X + [4]
>= [1] X + [2]
= f_3#(split(N,X),N,X)
qsort#(add(N,X)) = [1] X + [4]
>= [1] X + [4]
= split#(N,X)
append(add(N,X),Y) = [1] X + [1] Y + [2]
>= [1] X + [1] Y + [2]
= add(N,append(X,Y))
append(nil(),Y) = [1] Y + [0]
>= [1] Y + [0]
= Y
f_1(pair(X,Z),N,M,Y) = [1] X + [1] Z + [2]
>= [1] X + [1] Z + [2]
= f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) = [1] X + [1] Z + [2]
>= [1] X + [1] Z + [2]
= pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) = [1] X + [1] Z + [2]
>= [1] X + [1] Z + [2]
= pair(X,add(M,Z))
f_3(pair(Y,Z),N,X) = [1] Y + [1] Z + [2]
>= [1] Y + [1] Z + [2]
= append(qsort(Y),add(X,qsort(Z)))
lt(0(),s(X)) = [0]
>= [0]
= true()
lt(s(X),0()) = [0]
>= [0]
= false()
lt(s(X),s(Y)) = [0]
>= [0]
= lt(X,Y)
qsort(add(N,X)) = [1] X + [2]
>= [1] X + [2]
= f_3(split(N,X),N,X)
qsort(nil()) = [0]
>= [0]
= nil()
split(N,add(M,Y)) = [1] Y + [2]
>= [1] Y + [2]
= f_1(split(N,Y),N,M,Y)
split(N,nil()) = [0]
>= [0]
= pair(nil(),nil())
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 5.b:1.a:4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
append#(add(N,X),Y) -> c_1(append#(X,Y))
f_3#(pair(Y,Z),N,X) -> append#(qsort(Y),add(X,qsort(Z)))
f_3#(pair(Y,Z),N,X) -> qsort#(Y)
f_3#(pair(Y,Z),N,X) -> qsort#(Z)
qsort#(add(N,X)) -> f_3#(split(N,X),N,X)
qsort#(add(N,X)) -> split#(N,X)
split#(N,add(M,Y)) -> c_12(split#(N,Y))
- Weak TRS:
append(add(N,X),Y) -> add(N,append(X,Y))
append(nil(),Y) -> Y
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
f_3(pair(Y,Z),N,X) -> append(qsort(Y),add(X,qsort(Z)))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
qsort(add(N,X)) -> f_3(split(N,X),N,X)
qsort(nil()) -> nil()
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
- Signature:
{append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2,append#/2,f_1#/4,f_2#/6,f_3#/3,lt#/2,qsort#/1
,split#/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/3,c_7/0,c_8/0
,c_9/1,c_10/2,c_11/0,c_12/1,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,f_1#,f_2#,f_3#,lt#,qsort#
,split#} and constructors {0,add,false,nil,pair,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 5.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f_1#(pair(X,Z),N,M,Y) -> c_3(lt#(N,M))
lt#(s(X),s(Y)) -> c_9(lt#(X,Y))
- Weak DPs:
append#(add(N,X),Y) -> append#(X,Y)
f_3#(pair(Y,Z),N,X) -> append#(qsort(Y),add(X,qsort(Z)))
f_3#(pair(Y,Z),N,X) -> qsort#(Y)
f_3#(pair(Y,Z),N,X) -> qsort#(Z)
qsort#(add(N,X)) -> f_3#(split(N,X),N,X)
qsort#(add(N,X)) -> split#(N,X)
split#(N,add(M,Y)) -> f_1#(split(N,Y),N,M,Y)
split#(N,add(M,Y)) -> split#(N,Y)
- Weak TRS:
append(add(N,X),Y) -> add(N,append(X,Y))
append(nil(),Y) -> Y
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
f_3(pair(Y,Z),N,X) -> append(qsort(Y),add(X,qsort(Z)))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
qsort(add(N,X)) -> f_3(split(N,X),N,X)
qsort(nil()) -> nil()
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
- Signature:
{append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2,append#/2,f_1#/4,f_2#/6,f_3#/3,lt#/2,qsort#/1
,split#/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/3,c_7/0,c_8/0
,c_9/1,c_10/2,c_11/0,c_12/2,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,f_1#,f_2#,f_3#,lt#,qsort#
,split#} and constructors {0,add,false,nil,pair,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:f_1#(pair(X,Z),N,M,Y) -> c_3(lt#(N,M))
-->_1 lt#(s(X),s(Y)) -> c_9(lt#(X,Y)):2
2:S:lt#(s(X),s(Y)) -> c_9(lt#(X,Y))
-->_1 lt#(s(X),s(Y)) -> c_9(lt#(X,Y)):2
3:W:append#(add(N,X),Y) -> append#(X,Y)
-->_1 append#(add(N,X),Y) -> append#(X,Y):3
4:W:f_3#(pair(Y,Z),N,X) -> append#(qsort(Y),add(X,qsort(Z)))
-->_1 append#(add(N,X),Y) -> append#(X,Y):3
5:W:f_3#(pair(Y,Z),N,X) -> qsort#(Y)
-->_1 qsort#(add(N,X)) -> split#(N,X):8
-->_1 qsort#(add(N,X)) -> f_3#(split(N,X),N,X):7
6:W:f_3#(pair(Y,Z),N,X) -> qsort#(Z)
-->_1 qsort#(add(N,X)) -> split#(N,X):8
-->_1 qsort#(add(N,X)) -> f_3#(split(N,X),N,X):7
7:W:qsort#(add(N,X)) -> f_3#(split(N,X),N,X)
-->_1 f_3#(pair(Y,Z),N,X) -> qsort#(Z):6
-->_1 f_3#(pair(Y,Z),N,X) -> qsort#(Y):5
-->_1 f_3#(pair(Y,Z),N,X) -> append#(qsort(Y),add(X,qsort(Z))):4
8:W:qsort#(add(N,X)) -> split#(N,X)
-->_1 split#(N,add(M,Y)) -> split#(N,Y):10
-->_1 split#(N,add(M,Y)) -> f_1#(split(N,Y),N,M,Y):9
9:W:split#(N,add(M,Y)) -> f_1#(split(N,Y),N,M,Y)
-->_1 f_1#(pair(X,Z),N,M,Y) -> c_3(lt#(N,M)):1
10:W:split#(N,add(M,Y)) -> split#(N,Y)
-->_1 split#(N,add(M,Y)) -> split#(N,Y):10
-->_1 split#(N,add(M,Y)) -> f_1#(split(N,Y),N,M,Y):9
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: f_3#(pair(Y,Z),N,X) -> append#(qsort(Y),add(X,qsort(Z)))
3: append#(add(N,X),Y) -> append#(X,Y)
*** Step 5.b:1.b:2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f_1#(pair(X,Z),N,M,Y) -> c_3(lt#(N,M))
lt#(s(X),s(Y)) -> c_9(lt#(X,Y))
- Weak DPs:
f_3#(pair(Y,Z),N,X) -> qsort#(Y)
f_3#(pair(Y,Z),N,X) -> qsort#(Z)
qsort#(add(N,X)) -> f_3#(split(N,X),N,X)
qsort#(add(N,X)) -> split#(N,X)
split#(N,add(M,Y)) -> f_1#(split(N,Y),N,M,Y)
split#(N,add(M,Y)) -> split#(N,Y)
- Weak TRS:
append(add(N,X),Y) -> add(N,append(X,Y))
append(nil(),Y) -> Y
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
f_3(pair(Y,Z),N,X) -> append(qsort(Y),add(X,qsort(Z)))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
qsort(add(N,X)) -> f_3(split(N,X),N,X)
qsort(nil()) -> nil()
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
- Signature:
{append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2,append#/2,f_1#/4,f_2#/6,f_3#/3,lt#/2,qsort#/1
,split#/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/3,c_7/0,c_8/0
,c_9/1,c_10/2,c_11/0,c_12/2,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,f_1#,f_2#,f_3#,lt#,qsort#
,split#} and constructors {0,add,false,nil,pair,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
f_1#(pair(X,Z),N,M,Y) -> c_3(lt#(N,M))
f_3#(pair(Y,Z),N,X) -> qsort#(Y)
f_3#(pair(Y,Z),N,X) -> qsort#(Z)
lt#(s(X),s(Y)) -> c_9(lt#(X,Y))
qsort#(add(N,X)) -> f_3#(split(N,X),N,X)
qsort#(add(N,X)) -> split#(N,X)
split#(N,add(M,Y)) -> f_1#(split(N,Y),N,M,Y)
split#(N,add(M,Y)) -> split#(N,Y)
*** Step 5.b:1.b:3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
f_1#(pair(X,Z),N,M,Y) -> c_3(lt#(N,M))
lt#(s(X),s(Y)) -> c_9(lt#(X,Y))
- Weak DPs:
f_3#(pair(Y,Z),N,X) -> qsort#(Y)
f_3#(pair(Y,Z),N,X) -> qsort#(Z)
qsort#(add(N,X)) -> f_3#(split(N,X),N,X)
qsort#(add(N,X)) -> split#(N,X)
split#(N,add(M,Y)) -> f_1#(split(N,Y),N,M,Y)
split#(N,add(M,Y)) -> split#(N,Y)
- Weak TRS:
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
- Signature:
{append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2,append#/2,f_1#/4,f_2#/6,f_3#/3,lt#/2,qsort#/1
,split#/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/3,c_7/0,c_8/0
,c_9/1,c_10/2,c_11/0,c_12/2,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,f_1#,f_2#,f_3#,lt#,qsort#
,split#} and constructors {0,add,false,nil,pair,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(f_1) = {1},
uargs(f_2) = {1},
uargs(f_1#) = {1},
uargs(f_3#) = {1},
uargs(c_3) = {1},
uargs(c_9) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(add) = [1] x1 + [1] x2 + [0]
p(append) = [2] x2 + [0]
p(f_1) = [1] x1 + [0]
p(f_2) = [1] x1 + [0]
p(f_3) = [4] x1 + [2] x2 + [2] x3 + [0]
p(false) = [0]
p(lt) = [0]
p(nil) = [0]
p(pair) = [0]
p(qsort) = [1]
p(s) = [1] x1 + [4]
p(split) = [0]
p(true) = [0]
p(append#) = [4] x1 + [2]
p(f_1#) = [1] x1 + [1]
p(f_2#) = [2] x2 + [4] x6 + [4]
p(f_3#) = [1] x1 + [1]
p(lt#) = [0]
p(qsort#) = [1]
p(split#) = [1]
p(c_1) = [1] x1 + [0]
p(c_2) = [0]
p(c_3) = [1] x1 + [0]
p(c_4) = [2]
p(c_5) = [2]
p(c_6) = [1] x3 + [1]
p(c_7) = [2]
p(c_8) = [1]
p(c_9) = [1] x1 + [0]
p(c_10) = [1] x1 + [0]
p(c_11) = [0]
p(c_12) = [4] x2 + [0]
p(c_13) = [1]
Following rules are strictly oriented:
f_1#(pair(X,Z),N,M,Y) = [1]
> [0]
= c_3(lt#(N,M))
Following rules are (at-least) weakly oriented:
f_3#(pair(Y,Z),N,X) = [1]
>= [1]
= qsort#(Y)
f_3#(pair(Y,Z),N,X) = [1]
>= [1]
= qsort#(Z)
lt#(s(X),s(Y)) = [0]
>= [0]
= c_9(lt#(X,Y))
qsort#(add(N,X)) = [1]
>= [1]
= f_3#(split(N,X),N,X)
qsort#(add(N,X)) = [1]
>= [1]
= split#(N,X)
split#(N,add(M,Y)) = [1]
>= [1]
= f_1#(split(N,Y),N,M,Y)
split#(N,add(M,Y)) = [1]
>= [1]
= split#(N,Y)
f_1(pair(X,Z),N,M,Y) = [0]
>= [0]
= f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) = [0]
>= [0]
= pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) = [0]
>= [0]
= pair(X,add(M,Z))
lt(0(),s(X)) = [0]
>= [0]
= true()
lt(s(X),0()) = [0]
>= [0]
= false()
lt(s(X),s(Y)) = [0]
>= [0]
= lt(X,Y)
split(N,add(M,Y)) = [0]
>= [0]
= f_1(split(N,Y),N,M,Y)
split(N,nil()) = [0]
>= [0]
= pair(nil(),nil())
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 5.b:1.b:4: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
lt#(s(X),s(Y)) -> c_9(lt#(X,Y))
- Weak DPs:
f_1#(pair(X,Z),N,M,Y) -> c_3(lt#(N,M))
f_3#(pair(Y,Z),N,X) -> qsort#(Y)
f_3#(pair(Y,Z),N,X) -> qsort#(Z)
qsort#(add(N,X)) -> f_3#(split(N,X),N,X)
qsort#(add(N,X)) -> split#(N,X)
split#(N,add(M,Y)) -> f_1#(split(N,Y),N,M,Y)
split#(N,add(M,Y)) -> split#(N,Y)
- Weak TRS:
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
- Signature:
{append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2,append#/2,f_1#/4,f_2#/6,f_3#/3,lt#/2,qsort#/1
,split#/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/3,c_7/0,c_8/0
,c_9/1,c_10/2,c_11/0,c_12/2,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,f_1#,f_2#,f_3#,lt#,qsort#
,split#} and constructors {0,add,false,nil,pair,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(f_1) = {1},
uargs(f_2) = {1},
uargs(f_1#) = {1},
uargs(f_3#) = {1},
uargs(c_3) = {1},
uargs(c_9) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [0]
p(add) = [1] x1 + [1] x2 + [4]
p(append) = [1] x1 + [1] x2 + [1]
p(f_1) = [1] x1 + [1] x3 + [4]
p(f_2) = [1] x1 + [1] x3 + [1] x5 + [1] x6 + [4]
p(f_3) = [2] x2 + [1]
p(false) = [0]
p(lt) = [0]
p(nil) = [0]
p(pair) = [1] x1 + [1] x2 + [0]
p(qsort) = [1]
p(s) = [1] x1 + [6]
p(split) = [1] x2 + [0]
p(true) = [0]
p(append#) = [4] x1 + [1] x2 + [2]
p(f_1#) = [1] x1 + [1] x2 + [1] x3 + [3]
p(f_2#) = [1] x1 + [1] x2 + [4] x3 + [1] x4 + [4] x5 + [1]
p(f_3#) = [1] x1 + [0]
p(lt#) = [1] x1 + [0]
p(qsort#) = [1] x1 + [0]
p(split#) = [1] x1 + [1] x2 + [1]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [1] x1 + [3]
p(c_4) = [2]
p(c_5) = [0]
p(c_6) = [2] x3 + [0]
p(c_7) = [1]
p(c_8) = [0]
p(c_9) = [1] x1 + [1]
p(c_10) = [1] x1 + [2] x2 + [0]
p(c_11) = [0]
p(c_12) = [1] x1 + [0]
p(c_13) = [0]
Following rules are strictly oriented:
lt#(s(X),s(Y)) = [1] X + [6]
> [1] X + [1]
= c_9(lt#(X,Y))
Following rules are (at-least) weakly oriented:
f_1#(pair(X,Z),N,M,Y) = [1] M + [1] N + [1] X + [1] Z + [3]
>= [1] N + [3]
= c_3(lt#(N,M))
f_3#(pair(Y,Z),N,X) = [1] Y + [1] Z + [0]
>= [1] Y + [0]
= qsort#(Y)
f_3#(pair(Y,Z),N,X) = [1] Y + [1] Z + [0]
>= [1] Z + [0]
= qsort#(Z)
qsort#(add(N,X)) = [1] N + [1] X + [4]
>= [1] X + [0]
= f_3#(split(N,X),N,X)
qsort#(add(N,X)) = [1] N + [1] X + [4]
>= [1] N + [1] X + [1]
= split#(N,X)
split#(N,add(M,Y)) = [1] M + [1] N + [1] Y + [5]
>= [1] M + [1] N + [1] Y + [3]
= f_1#(split(N,Y),N,M,Y)
split#(N,add(M,Y)) = [1] M + [1] N + [1] Y + [5]
>= [1] N + [1] Y + [1]
= split#(N,Y)
f_1(pair(X,Z),N,M,Y) = [1] M + [1] X + [1] Z + [4]
>= [1] M + [1] X + [1] Z + [4]
= f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) = [1] M + [1] X + [1] Z + [4]
>= [1] M + [1] X + [1] Z + [4]
= pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) = [1] M + [1] X + [1] Z + [4]
>= [1] M + [1] X + [1] Z + [4]
= pair(X,add(M,Z))
lt(0(),s(X)) = [0]
>= [0]
= true()
lt(s(X),0()) = [0]
>= [0]
= false()
lt(s(X),s(Y)) = [0]
>= [0]
= lt(X,Y)
split(N,add(M,Y)) = [1] M + [1] Y + [4]
>= [1] M + [1] Y + [4]
= f_1(split(N,Y),N,M,Y)
split(N,nil()) = [0]
>= [0]
= pair(nil(),nil())
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 5.b:1.b:5: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
f_1#(pair(X,Z),N,M,Y) -> c_3(lt#(N,M))
f_3#(pair(Y,Z),N,X) -> qsort#(Y)
f_3#(pair(Y,Z),N,X) -> qsort#(Z)
lt#(s(X),s(Y)) -> c_9(lt#(X,Y))
qsort#(add(N,X)) -> f_3#(split(N,X),N,X)
qsort#(add(N,X)) -> split#(N,X)
split#(N,add(M,Y)) -> f_1#(split(N,Y),N,M,Y)
split#(N,add(M,Y)) -> split#(N,Y)
- Weak TRS:
f_1(pair(X,Z),N,M,Y) -> f_2(lt(N,M),N,M,Y,X,Z)
f_2(false(),N,M,Y,X,Z) -> pair(add(M,X),Z)
f_2(true(),N,M,Y,X,Z) -> pair(X,add(M,Z))
lt(0(),s(X)) -> true()
lt(s(X),0()) -> false()
lt(s(X),s(Y)) -> lt(X,Y)
split(N,add(M,Y)) -> f_1(split(N,Y),N,M,Y)
split(N,nil()) -> pair(nil(),nil())
- Signature:
{append/2,f_1/4,f_2/6,f_3/3,lt/2,qsort/1,split/2,append#/2,f_1#/4,f_2#/6,f_3#/3,lt#/2,qsort#/1
,split#/2} / {0/0,add/2,false/0,nil/0,pair/2,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0,c_5/0,c_6/3,c_7/0,c_8/0
,c_9/1,c_10/2,c_11/0,c_12/2,c_13/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {append#,f_1#,f_2#,f_3#,lt#,qsort#
,split#} and constructors {0,add,false,nil,pair,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^3))