* Step 1: DependencyPairs WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
less_leaves(x,leaf()) -> false()
less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z))
less_leaves(leaf(),cons(w,z)) -> true()
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
shuffle(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0
,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app,concat,less_leaves,minus,quot,reverse
,shuffle} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
app#(add(n,x),y) -> c_1(app#(x,y))
app#(nil(),y) -> c_2()
concat#(cons(u,v),y) -> c_3(concat#(v,y))
concat#(leaf(),y) -> c_4()
less_leaves#(x,leaf()) -> c_5()
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
less_leaves#(leaf(),cons(w,z)) -> c_7()
minus#(x,0()) -> c_8()
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(0(),s(y)) -> c_10()
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
reverse#(nil()) -> c_13()
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
shuffle#(nil()) -> c_15()
Weak DPs
and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
app#(nil(),y) -> c_2()
concat#(cons(u,v),y) -> c_3(concat#(v,y))
concat#(leaf(),y) -> c_4()
less_leaves#(x,leaf()) -> c_5()
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
less_leaves#(leaf(),cons(w,z)) -> c_7()
minus#(x,0()) -> c_8()
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(0(),s(y)) -> c_10()
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
reverse#(nil()) -> c_13()
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
shuffle#(nil()) -> c_15()
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
less_leaves(x,leaf()) -> false()
less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z))
less_leaves(leaf(),cons(w,z)) -> true()
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
quot(0(),s(y)) -> 0()
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
shuffle(add(n,x)) -> add(n,shuffle(reverse(x)))
shuffle(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
app#(add(n,x),y) -> c_1(app#(x,y))
app#(nil(),y) -> c_2()
concat#(cons(u,v),y) -> c_3(concat#(v,y))
concat#(leaf(),y) -> c_4()
less_leaves#(x,leaf()) -> c_5()
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
less_leaves#(leaf(),cons(w,z)) -> c_7()
minus#(x,0()) -> c_8()
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(0(),s(y)) -> c_10()
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
reverse#(nil()) -> c_13()
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
shuffle#(nil()) -> c_15()
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
app#(nil(),y) -> c_2()
concat#(cons(u,v),y) -> c_3(concat#(v,y))
concat#(leaf(),y) -> c_4()
less_leaves#(x,leaf()) -> c_5()
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
less_leaves#(leaf(),cons(w,z)) -> c_7()
minus#(x,0()) -> c_8()
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(0(),s(y)) -> c_10()
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
reverse#(nil()) -> c_13()
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
shuffle#(nil()) -> c_15()
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{2,4,5,7,8,10,13,15}
by application of
Pre({2,4,5,7,8,10,13,15}) = {1,3,6,9,11,12,14}.
Here rules are labelled as follows:
1: app#(add(n,x),y) -> c_1(app#(x,y))
2: app#(nil(),y) -> c_2()
3: concat#(cons(u,v),y) -> c_3(concat#(v,y))
4: concat#(leaf(),y) -> c_4()
5: less_leaves#(x,leaf()) -> c_5()
6: less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
7: less_leaves#(leaf(),cons(w,z)) -> c_7()
8: minus#(x,0()) -> c_8()
9: minus#(s(x),s(y)) -> c_9(minus#(x,y))
10: quot#(0(),s(y)) -> c_10()
11: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
12: reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
13: reverse#(nil()) -> c_13()
14: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
15: shuffle#(nil()) -> c_15()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
concat#(cons(u,v),y) -> c_3(concat#(v,y))
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak DPs:
app#(nil(),y) -> c_2()
concat#(leaf(),y) -> c_4()
less_leaves#(x,leaf()) -> c_5()
less_leaves#(leaf(),cons(w,z)) -> c_7()
minus#(x,0()) -> c_8()
quot#(0(),s(y)) -> c_10()
reverse#(nil()) -> c_13()
shuffle#(nil()) -> c_15()
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:app#(add(n,x),y) -> c_1(app#(x,y))
-->_1 app#(nil(),y) -> c_2():8
-->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1
2:S:concat#(cons(u,v),y) -> c_3(concat#(v,y))
-->_1 concat#(leaf(),y) -> c_4():9
-->_1 concat#(cons(u,v),y) -> c_3(concat#(v,y)):2
3:S:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
,concat#(u,v)
,concat#(w,z))
-->_1 less_leaves#(leaf(),cons(w,z)) -> c_7():11
-->_1 less_leaves#(x,leaf()) -> c_5():10
-->_3 concat#(leaf(),y) -> c_4():9
-->_2 concat#(leaf(),y) -> c_4():9
-->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
,concat#(u,v)
,concat#(w,z)):3
-->_3 concat#(cons(u,v),y) -> c_3(concat#(v,y)):2
-->_2 concat#(cons(u,v),y) -> c_3(concat#(v,y)):2
4:S:minus#(s(x),s(y)) -> c_9(minus#(x,y))
-->_1 minus#(x,0()) -> c_8():12
-->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):4
5:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(0(),s(y)) -> c_10():13
-->_2 minus#(x,0()) -> c_8():12
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):5
-->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):4
6:S:reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
-->_2 reverse#(nil()) -> c_13():14
-->_1 app#(nil(),y) -> c_2():8
-->_2 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):6
-->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1
7:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
-->_1 shuffle#(nil()) -> c_15():15
-->_2 reverse#(nil()) -> c_13():14
-->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):7
-->_2 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):6
8:W:app#(nil(),y) -> c_2()
9:W:concat#(leaf(),y) -> c_4()
10:W:less_leaves#(x,leaf()) -> c_5()
11:W:less_leaves#(leaf(),cons(w,z)) -> c_7()
12:W:minus#(x,0()) -> c_8()
13:W:quot#(0(),s(y)) -> c_10()
14:W:reverse#(nil()) -> c_13()
15:W:shuffle#(nil()) -> c_15()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
15: shuffle#(nil()) -> c_15()
14: reverse#(nil()) -> c_13()
13: quot#(0(),s(y)) -> c_10()
12: minus#(x,0()) -> c_8()
10: less_leaves#(x,leaf()) -> c_5()
11: less_leaves#(leaf(),cons(w,z)) -> c_7()
9: concat#(leaf(),y) -> c_4()
8: app#(nil(),y) -> c_2()
* Step 5: Decompose WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
concat#(cons(u,v),y) -> c_3(concat#(v,y))
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
- Strict DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
- Weak DPs:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
Problem (S)
- Strict DPs:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
** Step 5.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
- Weak DPs:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:app#(add(n,x),y) -> c_1(app#(x,y))
-->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1
2:W:concat#(cons(u,v),y) -> c_3(concat#(v,y))
-->_1 concat#(cons(u,v),y) -> c_3(concat#(v,y)):2
3:W:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
,concat#(u,v)
,concat#(w,z))
-->_3 concat#(cons(u,v),y) -> c_3(concat#(v,y)):2
-->_2 concat#(cons(u,v),y) -> c_3(concat#(v,y)):2
-->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
,concat#(u,v)
,concat#(w,z)):3
4:W:minus#(s(x),s(y)) -> c_9(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):4
5:W:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
-->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):4
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):5
6:W:reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
-->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1
-->_2 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):6
7:W:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
-->_2 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):6
-->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):7
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
4: minus#(s(x),s(y)) -> c_9(minus#(x,y))
3: less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
2: concat#(cons(u,v),y) -> c_3(concat#(v,y))
** Step 5.a:2: UsableRules WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
- Weak DPs:
reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
app#(add(n,x),y) -> c_1(app#(x,y))
reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
** Step 5.a:3: DecomposeDG WORST_CASE(?,O(n^3))
+ Considered Problem:
- Strict DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
- Weak DPs:
reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
and a lower component
app#(add(n,x),y) -> c_1(app#(x,y))
reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
Further, following extension rules are added to the lower component.
shuffle#(add(n,x)) -> reverse#(x)
shuffle#(add(n,x)) -> shuffle#(reverse(x))
*** Step 5.a:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
The strictly oriented rules are moved into the weak component.
**** Step 5.a:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_14) = {1}
Following symbols are considered usable:
{app,reverse,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}
TcT has computed the following interpretation:
p(0) = [0]
p(add) = [1] x1 + [1] x2 + [4]
p(app) = [1] x1 + [1] x2 + [0]
p(concat) = [4] x1 + [4] x2 + [0]
p(cons) = [1] x2 + [0]
p(false) = [1]
p(leaf) = [2]
p(less_leaves) = [1] x1 + [4] x2 + [1]
p(minus) = [1] x1 + [1] x2 + [1]
p(nil) = [0]
p(quot) = [1] x1 + [8]
p(reverse) = [1] x1 + [0]
p(s) = [1]
p(shuffle) = [1]
p(true) = [1]
p(app#) = [2] x1 + [2] x2 + [1]
p(concat#) = [1]
p(less_leaves#) = [2] x1 + [1] x2 + [0]
p(minus#) = [1] x1 + [1]
p(quot#) = [2]
p(reverse#) = [3]
p(shuffle#) = [4] x1 + [0]
p(c_1) = [1] x1 + [2]
p(c_2) = [1]
p(c_3) = [1] x1 + [0]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [1] x3 + [1]
p(c_7) = [0]
p(c_8) = [0]
p(c_9) = [4] x1 + [1]
p(c_10) = [2]
p(c_11) = [2] x2 + [1]
p(c_12) = [1] x1 + [0]
p(c_13) = [2]
p(c_14) = [1] x1 + [1] x2 + [0]
p(c_15) = [1]
Following rules are strictly oriented:
shuffle#(add(n,x)) = [4] n + [4] x + [16]
> [4] x + [3]
= c_14(shuffle#(reverse(x)),reverse#(x))
Following rules are (at-least) weakly oriented:
app(add(n,x),y) = [1] n + [1] x + [1] y + [4]
>= [1] n + [1] x + [1] y + [4]
= add(n,app(x,y))
app(nil(),y) = [1] y + [0]
>= [1] y + [0]
= y
reverse(add(n,x)) = [1] n + [1] x + [4]
>= [1] n + [1] x + [4]
= app(reverse(x),add(n,nil()))
reverse(nil()) = [0]
>= [0]
= nil()
**** Step 5.a:3.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 5.a:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
-->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
**** Step 5.a:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 5.a:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
- Weak DPs:
reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> reverse#(x)
shuffle#(add(n,x)) -> shuffle#(reverse(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: app#(add(n,x),y) -> c_1(app#(x,y))
The strictly oriented rules are moved into the weak component.
**** Step 5.a:3.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
- Weak DPs:
reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> reverse#(x)
shuffle#(add(n,x)) -> shuffle#(reverse(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_1) = {1},
uargs(c_12) = {1,2}
Following symbols are considered usable:
{app,reverse,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}
TcT has computed the following interpretation:
p(0) = 1
p(add) = 1 + x2
p(app) = x1 + x2
p(concat) = x1 + 4*x1^2
p(cons) = x2
p(false) = 1
p(leaf) = 0
p(less_leaves) = 2 + 2*x1 + x2^2
p(minus) = 1
p(nil) = 0
p(quot) = 1 + x1 + 4*x1^2 + x2 + 2*x2^2
p(reverse) = x1
p(s) = 0
p(shuffle) = 2*x1 + x1^2
p(true) = 0
p(app#) = 1 + x1 + x2 + x2^2
p(concat#) = x1 + 4*x1*x2 + x2^2
p(less_leaves#) = 2*x1*x2 + x1^2 + x2
p(minus#) = x1*x2 + 4*x1^2 + x2 + x2^2
p(quot#) = 1
p(reverse#) = 5 + 6*x1 + x1^2
p(shuffle#) = 2 + 2*x1 + 2*x1^2
p(c_1) = x1
p(c_2) = 0
p(c_3) = x1
p(c_4) = 0
p(c_5) = 0
p(c_6) = x2
p(c_7) = 0
p(c_8) = 1
p(c_9) = 0
p(c_10) = 0
p(c_11) = x2
p(c_12) = x1 + x2
p(c_13) = 1
p(c_14) = x1 + x2
p(c_15) = 0
Following rules are strictly oriented:
app#(add(n,x),y) = 2 + x + y + y^2
> 1 + x + y + y^2
= c_1(app#(x,y))
Following rules are (at-least) weakly oriented:
reverse#(add(n,x)) = 12 + 8*x + x^2
>= 8 + 7*x + x^2
= c_12(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) = 6 + 6*x + 2*x^2
>= 5 + 6*x + x^2
= reverse#(x)
shuffle#(add(n,x)) = 6 + 6*x + 2*x^2
>= 2 + 2*x + 2*x^2
= shuffle#(reverse(x))
app(add(n,x),y) = 1 + x + y
>= 1 + x + y
= add(n,app(x,y))
app(nil(),y) = y
>= y
= y
reverse(add(n,x)) = 1 + x
>= 1 + x
= app(reverse(x),add(n,nil()))
reverse(nil()) = 0
>= 0
= nil()
**** Step 5.a:3.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> reverse#(x)
shuffle#(add(n,x)) -> shuffle#(reverse(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 5.a:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> reverse#(x)
shuffle#(add(n,x)) -> shuffle#(reverse(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:app#(add(n,x),y) -> c_1(app#(x,y))
-->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1
2:W:reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
-->_2 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):2
-->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1
3:W:shuffle#(add(n,x)) -> reverse#(x)
-->_1 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):2
4:W:shuffle#(add(n,x)) -> shuffle#(reverse(x))
-->_1 shuffle#(add(n,x)) -> shuffle#(reverse(x)):4
-->_1 shuffle#(add(n,x)) -> reverse#(x):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: shuffle#(add(n,x)) -> shuffle#(reverse(x))
3: shuffle#(add(n,x)) -> reverse#(x)
2: reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
1: app#(add(n,x),y) -> c_1(app#(x,y))
**** Step 5.a:3.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak DPs:
app#(add(n,x),y) -> c_1(app#(x,y))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:concat#(cons(u,v),y) -> c_3(concat#(v,y))
-->_1 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
2:S:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
,concat#(u,v)
,concat#(w,z))
-->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
,concat#(u,v)
,concat#(w,z)):2
-->_3 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
-->_2 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
3:S:minus#(s(x),s(y)) -> c_9(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):3
4:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):4
-->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):3
5:S:reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
-->_1 app#(add(n,x),y) -> c_1(app#(x,y)):7
-->_2 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):5
6:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
-->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):6
-->_2 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):5
7:W:app#(add(n,x),y) -> c_1(app#(x,y))
-->_1 app#(add(n,x),y) -> c_1(app#(x,y)):7
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
7: app#(add(n,x),y) -> c_1(app#(x,y))
** Step 5.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/2,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:concat#(cons(u,v),y) -> c_3(concat#(v,y))
-->_1 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
2:S:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
,concat#(u,v)
,concat#(w,z))
-->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
,concat#(u,v)
,concat#(w,z)):2
-->_3 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
-->_2 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
3:S:minus#(s(x),s(y)) -> c_9(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):3
4:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):4
-->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):3
5:S:reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x))
-->_2 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):5
6:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
-->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):6
-->_2 reverse#(add(n,x)) -> c_12(app#(reverse(x),add(n,nil())),reverse#(x)):5
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
reverse#(add(n,x)) -> c_12(reverse#(x))
** Step 5.b:3: Decompose WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
- Strict DPs:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
- Weak DPs:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
Problem (S)
- Strict DPs:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak DPs:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
*** Step 5.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
- Weak DPs:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:concat#(cons(u,v),y) -> c_3(concat#(v,y))
-->_1 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
2:W:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
,concat#(u,v)
,concat#(w,z))
-->_3 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
-->_2 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
-->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
,concat#(u,v)
,concat#(w,z)):2
3:W:minus#(s(x),s(y)) -> c_9(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):3
4:W:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
-->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):3
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):4
5:W:reverse#(add(n,x)) -> c_12(reverse#(x))
-->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):5
6:W:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
-->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):5
-->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):6
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
5: reverse#(add(n,x)) -> c_12(reverse#(x))
4: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
3: minus#(s(x),s(y)) -> c_9(minus#(x,y))
*** Step 5.b:3.a:2: UsableRules WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
- Weak DPs:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
concat#(cons(u,v),y) -> c_3(concat#(v,y))
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
*** Step 5.b:3.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
- Weak DPs:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
- Weak TRS:
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: concat#(cons(u,v),y) -> c_3(concat#(v,y))
The strictly oriented rules are moved into the weak component.
**** Step 5.b:3.a:3.a:1: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
- Weak DPs:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
- Weak TRS:
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_3) = {1},
uargs(c_6) = {1,2,3}
Following symbols are considered usable:
{concat,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}
TcT has computed the following interpretation:
p(0) = [0]
[2]
p(add) = [0 1] x1 + [1]
[0 0] [1]
p(app) = [1 0] x1 + [0]
[4 4] [0]
p(concat) = [1 0] x1 + [1 0] x2 + [1]
[0 4] [0 1] [0]
p(cons) = [1 4] x1 + [1 0] x2 + [0]
[0 1] [0 1] [2]
p(false) = [1]
[1]
p(leaf) = [2]
[3]
p(less_leaves) = [4 0] x1 + [0 0] x2 + [1]
[1 0] [4 1] [1]
p(minus) = [2 1] x1 + [0 0] x2 + [0]
[1 0] [0 1] [4]
p(nil) = [0]
[1]
p(quot) = [1 4] x2 + [0]
[1 4] [0]
p(reverse) = [0]
[0]
p(s) = [0 1] x1 + [1]
[0 0] [4]
p(shuffle) = [0]
[1]
p(true) = [0]
[0]
p(app#) = [0 1] x2 + [1]
[0 2] [0]
p(concat#) = [0 1] x1 + [0]
[0 0] [1]
p(less_leaves#) = [1 1] x1 + [2 2] x2 + [6]
[0 0] [1 0] [1]
p(minus#) = [0 0] x1 + [1 1] x2 + [0]
[0 1] [0 1] [2]
p(quot#) = [1 4] x2 + [0]
[0 1] [1]
p(reverse#) = [2 0] x1 + [0]
[1 1] [0]
p(shuffle#) = [1]
[0]
p(c_1) = [1]
[1]
p(c_2) = [1]
[0]
p(c_3) = [1 1] x1 + [0]
[0 1] [0]
p(c_4) = [1]
[1]
p(c_5) = [0]
[1]
p(c_6) = [1 0] x1 + [1 0] x2 + [2 2] x3 + [1]
[0 0] [0 1] [0 0] [0]
p(c_7) = [1]
[1]
p(c_8) = [1]
[1]
p(c_9) = [1]
[1]
p(c_10) = [0]
[0]
p(c_11) = [4 0] x2 + [0]
[2 1] [1]
p(c_12) = [1 2] x1 + [0]
[1 0] [1]
p(c_13) = [4]
[1]
p(c_14) = [0 0] x1 + [1 2] x2 + [0]
[0 1] [4 1] [0]
p(c_15) = [1]
[1]
Following rules are strictly oriented:
concat#(cons(u,v),y) = [0 1] u + [0 1] v + [2]
[0 0] [0 0] [1]
> [0 1] v + [1]
[0 0] [1]
= c_3(concat#(v,y))
Following rules are (at-least) weakly oriented:
less_leaves#(cons(u,v),cons(w,z)) = [1 5] u + [1 1] v + [2 10] w + [2 2] z + [12]
[0 0] [0 0] [1 4] [1 0] [1]
>= [1 5] u + [1 1] v + [2 10] w + [2 2] z + [12]
[0 0] [0 0] [0 0] [0 0] [1]
= c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
concat(cons(u,v),y) = [1 4] u + [1 0] v + [1 0] y + [1]
[0 4] [0 4] [0 1] [8]
>= [1 4] u + [1 0] v + [1 0] y + [1]
[0 1] [0 4] [0 1] [2]
= cons(u,concat(v,y))
concat(leaf(),y) = [1 0] y + [3]
[0 1] [12]
>= [1 0] y + [0]
[0 1] [0]
= y
**** Step 5.b:3.a:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
- Weak TRS:
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 5.b:3.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
- Weak TRS:
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:concat#(cons(u,v),y) -> c_3(concat#(v,y))
-->_1 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
2:W:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
,concat#(u,v)
,concat#(w,z))
-->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
,concat#(u,v)
,concat#(w,z)):2
-->_3 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
-->_2 concat#(cons(u,v),y) -> c_3(concat#(v,y)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
,concat#(u,v)
,concat#(w,z))
1: concat#(cons(u,v),y) -> c_3(concat#(v,y))
**** Step 5.b:3.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 5.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak DPs:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
,concat#(u,v)
,concat#(w,z))
-->_3 concat#(cons(u,v),y) -> c_3(concat#(v,y)):6
-->_2 concat#(cons(u,v),y) -> c_3(concat#(v,y)):6
-->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
,concat#(u,v)
,concat#(w,z)):1
2:S:minus#(s(x),s(y)) -> c_9(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):2
3:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):3
-->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):2
4:S:reverse#(add(n,x)) -> c_12(reverse#(x))
-->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):4
5:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
-->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):5
-->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):4
6:W:concat#(cons(u,v),y) -> c_3(concat#(v,y))
-->_1 concat#(cons(u,v),y) -> c_3(concat#(v,y)):6
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: concat#(cons(u,v),y) -> c_3(concat#(v,y))
*** Step 5.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)),concat#(u,v),concat#(w,z))
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,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_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
,concat#(u,v)
,concat#(w,z))
-->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))
,concat#(u,v)
,concat#(w,z)):1
2:S:minus#(s(x),s(y)) -> c_9(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):2
3:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):3
-->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):2
4:S:reverse#(add(n,x)) -> c_12(reverse#(x))
-->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):4
5:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
-->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):5
-->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):4
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
*** Step 5.b:3.b:3: Decompose WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
- Strict DPs:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
- Weak DPs:
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1
,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
Problem (S)
- Strict DPs:
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak DPs:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1
,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
**** Step 5.b:3.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
- Weak DPs:
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
-->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))):1
2:W:minus#(s(x),s(y)) -> c_9(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):2
3:W:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
-->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):2
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):3
4:W:reverse#(add(n,x)) -> c_12(reverse#(x))
-->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):4
5:W:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
-->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):4
-->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):5
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
4: reverse#(add(n,x)) -> c_12(reverse#(x))
3: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
2: minus#(s(x),s(y)) -> c_9(minus#(x,y))
**** Step 5.b:3.b:3.a:2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
**** Step 5.b:3.b:3.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
- Weak TRS:
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
The strictly oriented rules are moved into the weak component.
***** Step 5.b:3.b:3.a:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
- Weak TRS:
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_6) = {1}
Following symbols are considered usable:
{concat,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}
TcT has computed the following interpretation:
p(0) = [0]
p(add) = [1]
p(app) = [8] x1 + [1] x2 + [0]
p(concat) = [1] x1 + [1] x2 + [0]
p(cons) = [1] x1 + [1] x2 + [1]
p(false) = [2]
p(leaf) = [1]
p(less_leaves) = [1] x1 + [1] x2 + [0]
p(minus) = [1] x2 + [0]
p(nil) = [2]
p(quot) = [1] x1 + [1]
p(reverse) = [1] x1 + [4]
p(s) = [1]
p(shuffle) = [1]
p(true) = [1]
p(app#) = [1] x1 + [1] x2 + [0]
p(concat#) = [4] x2 + [4]
p(less_leaves#) = [12] x1 + [8] x2 + [9]
p(minus#) = [1] x2 + [2]
p(quot#) = [2] x1 + [1] x2 + [1]
p(reverse#) = [1] x1 + [0]
p(shuffle#) = [2] x1 + [8]
p(c_1) = [1] x1 + [0]
p(c_2) = [0]
p(c_3) = [1] x1 + [2]
p(c_4) = [1]
p(c_5) = [0]
p(c_6) = [1] x1 + [1]
p(c_7) = [4]
p(c_8) = [0]
p(c_9) = [1] x1 + [4]
p(c_10) = [2]
p(c_11) = [2] x1 + [8]
p(c_12) = [1]
p(c_13) = [0]
p(c_14) = [1] x1 + [1] x2 + [1]
p(c_15) = [1]
Following rules are strictly oriented:
less_leaves#(cons(u,v),cons(w,z)) = [12] u + [12] v + [8] w + [8] z + [29]
> [12] u + [12] v + [8] w + [8] z + [10]
= c_6(less_leaves#(concat(u,v),concat(w,z)))
Following rules are (at-least) weakly oriented:
concat(cons(u,v),y) = [1] u + [1] v + [1] y + [1]
>= [1] u + [1] v + [1] y + [1]
= cons(u,concat(v,y))
concat(leaf(),y) = [1] y + [1]
>= [1] y + [0]
= y
***** Step 5.b:3.b:3.a:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
- Weak TRS:
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 5.b:3.b:3.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
- Weak TRS:
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
-->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
***** Step 5.b:3.b:3.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
**** Step 5.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak DPs:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:minus#(s(x),s(y)) -> c_9(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):1
2:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):2
-->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):1
3:S:reverse#(add(n,x)) -> c_12(reverse#(x))
-->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):3
4:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
-->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):4
-->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):3
5:W:less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
-->_1 less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z))):5
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
**** Step 5.b:3.b:3.b:2: Decompose WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
- Strict DPs:
minus#(s(x),s(y)) -> c_9(minus#(x,y))
- Weak DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1
,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
Problem (S)
- Strict DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak DPs:
minus#(s(x),s(y)) -> c_9(minus#(x,y))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1
,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
***** Step 5.b:3.b:3.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
minus#(s(x),s(y)) -> c_9(minus#(x,y))
- Weak DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:minus#(s(x),s(y)) -> c_9(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):1
2:W:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
-->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):1
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):2
3:W:reverse#(add(n,x)) -> c_12(reverse#(x))
-->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):3
4:W:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
-->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):3
-->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):4
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
3: reverse#(add(n,x)) -> c_12(reverse#(x))
***** Step 5.b:3.b:3.b:2.a:2: UsableRules WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
minus#(s(x),s(y)) -> c_9(minus#(x,y))
- Weak DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
***** Step 5.b:3.b:3.b:2.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
minus#(s(x),s(y)) -> c_9(minus#(x,y))
- Weak DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: minus#(s(x),s(y)) -> c_9(minus#(x,y))
The strictly oriented rules are moved into the weak component.
****** Step 5.b:3.b:3.b:2.a:3.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
minus#(s(x),s(y)) -> c_9(minus#(x,y))
- Weak DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_9) = {1},
uargs(c_11) = {1,2}
Following symbols are considered usable:
{minus,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}
TcT has computed the following interpretation:
p(0) = 1
p(add) = 1 + x1 + x2
p(app) = 0
p(concat) = 0
p(cons) = 1 + x1 + x2
p(false) = 1
p(leaf) = 1
p(less_leaves) = x1 + x1*x2 + x1^2 + 2*x2
p(minus) = x1
p(nil) = 0
p(quot) = 2 + x1
p(reverse) = 1
p(s) = 1 + x1
p(shuffle) = 4
p(true) = 0
p(app#) = 2 + x1 + 2*x1^2 + 4*x2^2
p(concat#) = 2 + x1 + 4*x1^2
p(less_leaves#) = x1^2 + x2 + x2^2
p(minus#) = 5 + 3*x2
p(quot#) = 4*x1 + 4*x1*x2 + x2 + x2^2
p(reverse#) = 1 + x1 + 2*x1^2
p(shuffle#) = x1 + x1^2
p(c_1) = 1
p(c_2) = 1
p(c_3) = x1
p(c_4) = 1
p(c_5) = 1
p(c_6) = 0
p(c_7) = 0
p(c_8) = 0
p(c_9) = x1
p(c_10) = 1
p(c_11) = x1 + x2
p(c_12) = 1
p(c_13) = 0
p(c_14) = x1
p(c_15) = 1
Following rules are strictly oriented:
minus#(s(x),s(y)) = 8 + 3*y
> 5 + 3*y
= c_9(minus#(x,y))
Following rules are (at-least) weakly oriented:
quot#(s(x),s(y)) = 10 + 8*x + 4*x*y + 7*y + y^2
>= 7 + 8*x + 4*x*y + 6*y + y^2
= c_11(quot#(minus(x,y),s(y)),minus#(x,y))
minus(x,0()) = x
>= x
= x
minus(s(x),s(y)) = 1 + x
>= x
= minus(x,y)
****** Step 5.b:3.b:3.b:2.a:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
****** Step 5.b:3.b:3.b:2.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
minus#(s(x),s(y)) -> c_9(minus#(x,y))
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:minus#(s(x),s(y)) -> c_9(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):1
2:W:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):2
-->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
1: minus#(s(x),s(y)) -> c_9(minus#(x,y))
****** Step 5.b:3.b:3.b:2.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
***** Step 5.b:3.b:3.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak DPs:
minus#(s(x),s(y)) -> c_9(minus#(x,y))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
-->_2 minus#(s(x),s(y)) -> c_9(minus#(x,y)):4
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):1
2:S:reverse#(add(n,x)) -> c_12(reverse#(x))
-->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):2
3:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
-->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):3
-->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):2
4:W:minus#(s(x),s(y)) -> c_9(minus#(x,y))
-->_1 minus#(s(x),s(y)) -> c_9(minus#(x,y)):4
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: minus#(s(x),s(y)) -> c_9(minus#(x,y))
***** Step 5.b:3.b:3.b:2.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/2,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):1
2:S:reverse#(add(n,x)) -> c_12(reverse#(x))
-->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):2
3:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
-->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):3
-->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):2
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
***** Step 5.b:3.b:3.b:2.b:3: UsableRules WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
concat(cons(u,v),y) -> cons(u,concat(v,y))
concat(leaf(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
***** Step 5.b:3.b:3.b:2.b:4: Decompose WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
- Strict DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
- Weak DPs:
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1
,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
Problem (S)
- Strict DPs:
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1
,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
****** Step 5.b:3.b:3.b:2.b:4.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
- Weak DPs:
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y))):1
2:W:reverse#(add(n,x)) -> c_12(reverse#(x))
-->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):2
3:W:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
-->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):2
-->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
2: reverse#(add(n,x)) -> c_12(reverse#(x))
****** Step 5.b:3.b:3.b:2.b:4.a:2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
****** Step 5.b:3.b:3.b:2.b:4.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
The strictly oriented rules are moved into the weak component.
******* Step 5.b:3.b:3.b:2.b:4.a:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_11) = {1}
Following symbols are considered usable:
{minus,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}
TcT has computed the following interpretation:
p(0) = [2]
p(add) = [1]
p(app) = [8] x1 + [4]
p(concat) = [2] x1 + [1] x2 + [1]
p(cons) = [2]
p(false) = [1]
p(leaf) = [8]
p(less_leaves) = [0]
p(minus) = [1] x1 + [0]
p(nil) = [0]
p(quot) = [1] x1 + [0]
p(reverse) = [0]
p(s) = [1] x1 + [1]
p(shuffle) = [0]
p(true) = [1]
p(app#) = [1] x1 + [1] x2 + [4]
p(concat#) = [1] x2 + [1]
p(less_leaves#) = [1]
p(minus#) = [1] x1 + [1]
p(quot#) = [4] x1 + [0]
p(reverse#) = [1] x1 + [1]
p(shuffle#) = [2]
p(c_1) = [8] x1 + [2]
p(c_2) = [4]
p(c_3) = [0]
p(c_4) = [1]
p(c_5) = [1]
p(c_6) = [1]
p(c_7) = [1]
p(c_8) = [1]
p(c_9) = [1] x1 + [1]
p(c_10) = [0]
p(c_11) = [1] x1 + [2]
p(c_12) = [8] x1 + [1]
p(c_13) = [1]
p(c_14) = [1] x1 + [8]
p(c_15) = [0]
Following rules are strictly oriented:
quot#(s(x),s(y)) = [4] x + [4]
> [4] x + [2]
= c_11(quot#(minus(x,y),s(y)))
Following rules are (at-least) weakly oriented:
minus(x,0()) = [1] x + [0]
>= [1] x + [0]
= x
minus(s(x),s(y)) = [1] x + [1]
>= [1] x + [0]
= minus(x,y)
******* Step 5.b:3.b:3.b:2.b:4.a:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
******* Step 5.b:3.b:3.b:2.b:4.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
******* Step 5.b:3.b:3.b:2.b:4.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
****** Step 5.b:3.b:3.b:2.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak DPs:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:reverse#(add(n,x)) -> c_12(reverse#(x))
-->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):1
2:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
-->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):2
-->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):1
3:W:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
-->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y))):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
****** Step 5.b:3.b:3.b:2.b:4.b:2: Decompose WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
- Strict DPs:
reverse#(add(n,x)) -> c_12(reverse#(x))
- Weak DPs:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1
,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
Problem (S)
- Strict DPs:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak DPs:
reverse#(add(n,x)) -> c_12(reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1
,c_4/0,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
******* Step 5.b:3.b:3.b:2.b:4.b:2.a:1: UsableRules WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
reverse#(add(n,x)) -> c_12(reverse#(x))
- Weak DPs:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
******* Step 5.b:3.b:3.b:2.b:4.b:2.a:2: DecomposeDG WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
reverse#(add(n,x)) -> c_12(reverse#(x))
- Weak DPs:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
+ Details:
We decompose the input problem according to the dependency graph into the upper component
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
and a lower component
reverse#(add(n,x)) -> c_12(reverse#(x))
Further, following extension rules are added to the lower component.
shuffle#(add(n,x)) -> reverse#(x)
shuffle#(add(n,x)) -> shuffle#(reverse(x))
******** Step 5.b:3.b:3.b:2.b:4.b:2.a:2.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
The strictly oriented rules are moved into the weak component.
********* Step 5.b:3.b:3.b:2.b:4.b:2.a:2.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_14) = {1}
Following symbols are considered usable:
{app,reverse,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}
TcT has computed the following interpretation:
p(0) = [1]
p(add) = [1] x1 + [1] x2 + [2]
p(app) = [1] x1 + [1] x2 + [0]
p(concat) = [2] x1 + [2]
p(cons) = [0]
p(false) = [1]
p(leaf) = [0]
p(less_leaves) = [1] x1 + [0]
p(minus) = [1] x1 + [1] x2 + [0]
p(nil) = [0]
p(quot) = [4]
p(reverse) = [1] x1 + [1]
p(s) = [1]
p(shuffle) = [2] x1 + [1]
p(true) = [0]
p(app#) = [2] x2 + [0]
p(concat#) = [1] x1 + [1]
p(less_leaves#) = [1] x1 + [2]
p(minus#) = [1] x2 + [2]
p(quot#) = [0]
p(reverse#) = [1]
p(shuffle#) = [12] x1 + [0]
p(c_1) = [0]
p(c_2) = [2]
p(c_3) = [0]
p(c_4) = [0]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1]
p(c_8) = [2]
p(c_9) = [1] x1 + [1]
p(c_10) = [0]
p(c_11) = [1]
p(c_12) = [1] x1 + [4]
p(c_13) = [2]
p(c_14) = [1] x1 + [9] x2 + [0]
p(c_15) = [4]
Following rules are strictly oriented:
shuffle#(add(n,x)) = [12] n + [12] x + [24]
> [12] x + [21]
= c_14(shuffle#(reverse(x)),reverse#(x))
Following rules are (at-least) weakly oriented:
app(add(n,x),y) = [1] n + [1] x + [1] y + [2]
>= [1] n + [1] x + [1] y + [2]
= add(n,app(x,y))
app(nil(),y) = [1] y + [0]
>= [1] y + [0]
= y
reverse(add(n,x)) = [1] n + [1] x + [3]
>= [1] n + [1] x + [3]
= app(reverse(x),add(n,nil()))
reverse(nil()) = [1]
>= [0]
= nil()
********* Step 5.b:3.b:3.b:2.b:4.b:2.a:2.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
********* Step 5.b:3.b:3.b:2.b:4.b:2.a:2.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
-->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
********* Step 5.b:3.b:3.b:2.b:4.b:2.a:2.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
******** Step 5.b:3.b:3.b:2.b:4.b:2.a:2.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
reverse#(add(n,x)) -> c_12(reverse#(x))
- Weak DPs:
shuffle#(add(n,x)) -> reverse#(x)
shuffle#(add(n,x)) -> shuffle#(reverse(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: reverse#(add(n,x)) -> c_12(reverse#(x))
The strictly oriented rules are moved into the weak component.
********* Step 5.b:3.b:3.b:2.b:4.b:2.a:2.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
reverse#(add(n,x)) -> c_12(reverse#(x))
- Weak DPs:
shuffle#(add(n,x)) -> reverse#(x)
shuffle#(add(n,x)) -> shuffle#(reverse(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_12) = {1}
Following symbols are considered usable:
{app,reverse,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}
TcT has computed the following interpretation:
p(0) = [0]
p(add) = [1] x2 + [1]
p(app) = [1] x1 + [1] x2 + [0]
p(concat) = [1] x1 + [0]
p(cons) = [1] x1 + [1] x2 + [1]
p(false) = [4]
p(leaf) = [1]
p(less_leaves) = [1] x1 + [1] x2 + [1]
p(minus) = [8] x1 + [8]
p(nil) = [0]
p(quot) = [1] x1 + [2]
p(reverse) = [1] x1 + [0]
p(s) = [2]
p(shuffle) = [1] x1 + [0]
p(true) = [1]
p(app#) = [0]
p(concat#) = [1] x1 + [1] x2 + [1]
p(less_leaves#) = [0]
p(minus#) = [2] x1 + [2] x2 + [1]
p(quot#) = [1] x1 + [1] x2 + [0]
p(reverse#) = [1] x1 + [0]
p(shuffle#) = [1] x1 + [0]
p(c_1) = [1]
p(c_2) = [0]
p(c_3) = [1] x1 + [1]
p(c_4) = [1]
p(c_5) = [0]
p(c_6) = [1] x1 + [0]
p(c_7) = [1]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [1]
p(c_11) = [1] x1 + [8]
p(c_12) = [1] x1 + [0]
p(c_13) = [0]
p(c_14) = [1] x1 + [1]
p(c_15) = [1]
Following rules are strictly oriented:
reverse#(add(n,x)) = [1] x + [1]
> [1] x + [0]
= c_12(reverse#(x))
Following rules are (at-least) weakly oriented:
shuffle#(add(n,x)) = [1] x + [1]
>= [1] x + [0]
= reverse#(x)
shuffle#(add(n,x)) = [1] x + [1]
>= [1] x + [0]
= shuffle#(reverse(x))
app(add(n,x),y) = [1] x + [1] y + [1]
>= [1] x + [1] y + [1]
= add(n,app(x,y))
app(nil(),y) = [1] y + [0]
>= [1] y + [0]
= y
reverse(add(n,x)) = [1] x + [1]
>= [1] x + [1]
= app(reverse(x),add(n,nil()))
reverse(nil()) = [0]
>= [0]
= nil()
********* Step 5.b:3.b:3.b:2.b:4.b:2.a:2.b:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> reverse#(x)
shuffle#(add(n,x)) -> shuffle#(reverse(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
********* Step 5.b:3.b:3.b:2.b:4.b:2.a:2.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> reverse#(x)
shuffle#(add(n,x)) -> shuffle#(reverse(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:reverse#(add(n,x)) -> c_12(reverse#(x))
-->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):1
2:W:shuffle#(add(n,x)) -> reverse#(x)
-->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):1
3:W:shuffle#(add(n,x)) -> shuffle#(reverse(x))
-->_1 shuffle#(add(n,x)) -> shuffle#(reverse(x)):3
-->_1 shuffle#(add(n,x)) -> reverse#(x):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: shuffle#(add(n,x)) -> shuffle#(reverse(x))
2: shuffle#(add(n,x)) -> reverse#(x)
1: reverse#(add(n,x)) -> c_12(reverse#(x))
********* Step 5.b:3.b:3.b:2.b:4.b:2.a:2.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
******* Step 5.b:3.b:3.b:2.b:4.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak DPs:
reverse#(add(n,x)) -> c_12(reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
-->_2 reverse#(add(n,x)) -> c_12(reverse#(x)):2
-->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):1
2:W:reverse#(add(n,x)) -> c_12(reverse#(x))
-->_1 reverse#(add(n,x)) -> c_12(reverse#(x)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: reverse#(add(n,x)) -> c_12(reverse#(x))
******* Step 5.b:3.b:3.b:2.b:4.b:2.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/2,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
-->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
******* Step 5.b:3.b:3.b:2.b:4.b:2.b:3: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
minus(x,0()) -> x
minus(s(x),s(y)) -> minus(x,y)
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/1,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
******* Step 5.b:3.b:3.b:2.b:4.b:2.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/1,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
The strictly oriented rules are moved into the weak component.
******** Step 5.b:3.b:3.b:2.b:4.b:2.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/1,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_14) = {1}
Following symbols are considered usable:
{app,reverse,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}
TcT has computed the following interpretation:
p(0) = [1]
p(add) = [1] x2 + [2]
p(app) = [1] x1 + [1] x2 + [0]
p(concat) = [1]
p(cons) = [1] x2 + [0]
p(false) = [2]
p(leaf) = [0]
p(less_leaves) = [1] x1 + [0]
p(minus) = [1] x1 + [2] x2 + [1]
p(nil) = [0]
p(quot) = [2] x1 + [1] x2 + [1]
p(reverse) = [1] x1 + [0]
p(s) = [0]
p(shuffle) = [2]
p(true) = [1]
p(app#) = [1]
p(concat#) = [4] x2 + [1]
p(less_leaves#) = [1] x1 + [4] x2 + [1]
p(minus#) = [0]
p(quot#) = [1] x1 + [1]
p(reverse#) = [0]
p(shuffle#) = [1] x1 + [0]
p(c_1) = [1]
p(c_2) = [2]
p(c_3) = [8] x1 + [0]
p(c_4) = [2]
p(c_5) = [2]
p(c_6) = [1] x1 + [1]
p(c_7) = [2]
p(c_8) = [0]
p(c_9) = [1]
p(c_10) = [1]
p(c_11) = [8] x1 + [0]
p(c_12) = [1] x1 + [4]
p(c_13) = [4]
p(c_14) = [1] x1 + [0]
p(c_15) = [0]
Following rules are strictly oriented:
shuffle#(add(n,x)) = [1] x + [2]
> [1] x + [0]
= c_14(shuffle#(reverse(x)))
Following rules are (at-least) weakly oriented:
app(add(n,x),y) = [1] x + [1] y + [2]
>= [1] x + [1] y + [2]
= add(n,app(x,y))
app(nil(),y) = [1] y + [0]
>= [1] y + [0]
= y
reverse(add(n,x)) = [1] x + [2]
>= [1] x + [2]
= app(reverse(x),add(n,nil()))
reverse(nil()) = [0]
>= [0]
= nil()
******** Step 5.b:3.b:3.b:2.b:4.b:2.b:4.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/1,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
******** Step 5.b:3.b:3.b:2.b:4.b:2.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/1,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
-->_1 shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x))):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
******** Step 5.b:3.b:3.b:2.b:4.b:2.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
app(add(n,x),y) -> add(n,app(x,y))
app(nil(),y) -> y
reverse(add(n,x)) -> app(reverse(x),add(n,nil()))
reverse(nil()) -> nil()
- Signature:
{app/2,concat/2,less_leaves/2,minus/2,quot/2,reverse/1,shuffle/1,app#/2,concat#/2,less_leaves#/2,minus#/2
,quot#/2,reverse#/1,shuffle#/1} / {0/0,add/2,cons/2,false/0,leaf/0,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/1,c_4/0
,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0,c_14/1,c_15/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {app#,concat#,less_leaves#,minus#,quot#,reverse#
,shuffle#} and constructors {0,add,cons,false,leaf,nil,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^3))