*** 1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
Strict TRS 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
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()
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {app,concat,less_leaves,minus,quot,reverse,shuffle}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
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.
*** 1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
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()
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
UsableRules
Proof:
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()
*** 1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
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()
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
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()
*** 1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
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 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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()
*** 1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
Strict DP Rules:
app#(add(n,x),y) -> c_1(app#(x,y))
Strict TRS Rules:
Weak DP Rules:
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 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Problem (S)
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
app#(add(n,x),y) -> c_1(app#(x,y))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
*** 1.1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
app#(add(n,x),y) -> c_1(app#(x,y))
Strict TRS Rules:
Weak DP Rules:
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 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
-->_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: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))
-->_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: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)):6
-->_1 app#(add(n,x),y) -> c_1(app#(x,y)):1
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))
*** 1.1.1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
app#(add(n,x),y) -> c_1(app#(x,y))
Strict TRS Rules:
Weak DP Rules:
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 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
UsableRules
Proof:
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))
*** 1.1.1.1.1.1.1.1 Progress [(?,O(n^3))] ***
Considered Problem:
Strict DP Rules:
app#(add(n,x),y) -> c_1(app#(x,y))
Strict TRS Rules:
Weak DP Rules:
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 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
Proof:
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))
*** 1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: shuffle#(add(n,x)) ->
c_14(shuffle#(reverse(x))
,reverse#(x))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{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 any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_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] x2 + [2]
p(app) = [1] x1 + [1] x2 + [0]
p(concat) = [2] x2 + [1]
p(cons) = [1]
p(false) = [1]
p(leaf) = [1]
p(less_leaves) = [1] x1 + [0]
p(minus) = [1] x1 + [2]
p(nil) = [0]
p(quot) = [2]
p(reverse) = [1] x1 + [0]
p(s) = [1] x1 + [1]
p(shuffle) = [1] x1 + [1]
p(true) = [1]
p(app#) = [2] x2 + [0]
p(concat#) = [1]
p(less_leaves#) = [8] x1 + [4]
p(minus#) = [0]
p(quot#) = [1] x1 + [0]
p(reverse#) = [1] x1 + [0]
p(shuffle#) = [2] x1 + [2]
p(c_1) = [2]
p(c_2) = [1]
p(c_3) = [1] x1 + [1]
p(c_4) = [1]
p(c_5) = [1]
p(c_6) = [2] x1 + [1] x2 + [1] x3 + [2]
p(c_7) = [2]
p(c_8) = [2]
p(c_9) = [2] x1 + [0]
p(c_10) = [1]
p(c_11) = [4] x1 + [1] x2 + [0]
p(c_12) = [8] x1 + [1]
p(c_13) = [2]
p(c_14) = [1] x1 + [0]
p(c_15) = [0]
Following rules are strictly oriented:
shuffle#(add(n,x)) = [2] x + [6]
> [2] x + [2]
= c_14(shuffle#(reverse(x))
,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()
*** 1.1.1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.1.1.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
app#(add(n,x),y) -> c_1(app#(x,y))
Strict TRS Rules:
Weak DP Rules:
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 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: app#(add(n,x),y) -> c_1(app#(x
,y))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1.1.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
app#(add(n,x),y) -> c_1(app#(x,y))
Strict TRS Rules:
Weak DP Rules:
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 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{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 any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_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) = 0
p(add) = 1 + x1 + x2
p(app) = x1 + x2
p(concat) = 2*x1 + x1*x2 + x1^2 + x2^2
p(cons) = x2
p(false) = 1
p(leaf) = 0
p(less_leaves) = 1 + x1 + x1*x2 + x2
p(minus) = 2 + 2*x1 + 4*x1^2 + 4*x2 + x2^2
p(nil) = 0
p(quot) = 1 + x1*x2 + 4*x2^2
p(reverse) = x1
p(s) = 0
p(shuffle) = 4 + x1 + 2*x1^2
p(true) = 0
p(app#) = 3 + x1 + 7*x1*x2
p(concat#) = x1 + x1*x2 + x1^2 + 4*x2^2
p(less_leaves#) = 2*x1 + x2
p(minus#) = 1 + 4*x1*x2 + 2*x2^2
p(quot#) = x1 + 2*x1*x2 + 2*x2
p(reverse#) = x1 + 4*x1^2
p(shuffle#) = 3 + 4*x1^2
p(c_1) = x1
p(c_2) = 0
p(c_3) = 1
p(c_4) = 1
p(c_5) = 1
p(c_6) = 0
p(c_7) = 1
p(c_8) = 0
p(c_9) = 1
p(c_10) = 0
p(c_11) = 1
p(c_12) = x1 + x2
p(c_13) = 0
p(c_14) = x1
p(c_15) = 1
Following rules are strictly oriented:
app#(add(n,x),y) = 4 + n + 7*n*y + x + 7*x*y + 7*y
> 3 + x + 7*x*y
= c_1(app#(x,y))
Following rules are (at-least) weakly oriented:
reverse#(add(n,x)) = 5 + 9*n + 8*n*x + 4*n^2 + 9*x + 4*x^2
>= 3 + 7*n*x + 9*x + 4*x^2
= c_12(app#(reverse(x)
,add(n,nil()))
,reverse#(x))
shuffle#(add(n,x)) = 7 + 8*n + 8*n*x + 4*n^2 + 8*x + 4*x^2
>= x + 4*x^2
= reverse#(x)
shuffle#(add(n,x)) = 7 + 8*n + 8*n*x + 4*n^2 + 8*x + 4*x^2
>= 3 + 4*x^2
= shuffle#(reverse(x))
app(add(n,x),y) = 1 + n + x + y
>= 1 + n + x + y
= add(n,app(x,y))
app(nil(),y) = y
>= y
= y
reverse(add(n,x)) = 1 + n + x
>= 1 + n + x
= app(reverse(x),add(n,nil()))
reverse(nil()) = 0
>= 0
= nil()
*** 1.1.1.1.1.1.1.1.2.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
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 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.1.2.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
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 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.1.1.1.2.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
app#(add(n,x),y) -> c_1(app#(x,y))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
SimplifyRHS
Proof:
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))
*** 1.1.1.1.1.2.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
Strict DP Rules:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
Strict TRS Rules:
Weak DP Rules:
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 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Problem (S)
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
*** 1.1.1.1.1.2.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
Strict TRS Rules:
Weak DP Rules:
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 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
-->_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: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))
*** 1.1.1.1.1.2.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
Strict TRS Rules:
Weak DP Rules:
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 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
UsableRules
Proof:
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))
*** 1.1.1.1.1.2.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
Strict TRS Rules:
Weak DP Rules:
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 Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: concat#(cons(u,v),y) ->
c_3(concat#(v,y))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.2.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
Strict TRS Rules:
Weak DP Rules:
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 Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{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 any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_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) = 1
p(add) = 1
p(app) = 1 + 2*x1*x2 + x1^2 + x2 + x2^2
p(concat) = x1 + x2
p(cons) = 1 + x1 + x2
p(false) = 0
p(leaf) = 1
p(less_leaves) = 2 + x1 + x1^2 + 4*x2^2
p(minus) = 1 + 2*x1 + 2*x1*x2 + x2
p(nil) = 1
p(quot) = 2 + x1^2
p(reverse) = x1 + 4*x1^2
p(s) = 0
p(shuffle) = 4 + x1
p(true) = 1
p(app#) = x1*x2
p(concat#) = x1
p(less_leaves#) = 4*x1 + 4*x1^2 + 4*x2 + 2*x2^2
p(minus#) = x1 + x1*x2 + x1^2 + 2*x2
p(quot#) = 4 + 2*x1*x2 + x2 + 4*x2^2
p(reverse#) = 1 + 2*x1 + x1^2
p(shuffle#) = 1 + x1
p(c_1) = 0
p(c_2) = 1
p(c_3) = x1
p(c_4) = 1
p(c_5) = 1
p(c_6) = x1 + x2 + x3
p(c_7) = 1
p(c_8) = 1
p(c_9) = x1
p(c_10) = 1
p(c_11) = 1 + x1 + x2
p(c_12) = 0
p(c_13) = 1
p(c_14) = x2
p(c_15) = 1
Following rules are strictly oriented:
concat#(cons(u,v),y) = 1 + u + v
> v
= c_3(concat#(v,y))
Following rules are (at-least) weakly oriented:
less_leaves#(cons(u,v) = 14 + 12*u + 8*u*v + 4*u^2 + 12*v + 4*v^2 + 8*w + 4*w*z + 2*w^2 + 8*z + 2*z^2
,cons(w,z))
>= 5*u + 8*u*v + 4*u^2 + 4*v + 4*v^2 + 5*w + 4*w*z + 2*w^2 + 4*z + 2*z^2
= c_6(less_leaves#(concat(u,v)
,concat(w,z))
,concat#(u,v)
,concat#(w,z))
concat(cons(u,v),y) = 1 + u + v + y
>= 1 + u + v + y
= cons(u,concat(v,y))
concat(leaf(),y) = 1 + y
>= y
= y
*** 1.1.1.1.1.2.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
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 Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.2.1.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
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 Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.2.1.1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.2.1.1.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
concat#(cons(u,v),y) -> c_3(concat#(v,y))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.2.1.1.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
SimplifyRHS
Proof:
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)))
*** 1.1.1.1.1.2.1.1.2.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
Strict DP Rules:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
Strict TRS Rules:
Weak DP Rules:
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 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Problem (S)
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
*** 1.1.1.1.1.2.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
Strict TRS Rules:
Weak DP Rules:
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 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
-->_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: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))
*** 1.1.1.1.1.2.1.1.2.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
UsableRules
Proof:
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)))
*** 1.1.1.1.1.2.1.1.2.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: 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.
*** 1.1.1.1.1.2.1.1.2.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{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 any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_6) = {1}
Following symbols are considered usable:
{concat,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}
TcT has computed the following interpretation:
p(0) = [1]
p(add) = [2]
p(app) = [2] x1 + [1] x2 + [2]
p(concat) = [1] x1 + [1] x2 + [0]
p(cons) = [1] x1 + [1] x2 + [2]
p(false) = [1]
p(leaf) = [1]
p(less_leaves) = [1] x1 + [0]
p(minus) = [1] x1 + [4]
p(nil) = [0]
p(quot) = [1] x1 + [1]
p(reverse) = [1] x1 + [2]
p(s) = [0]
p(shuffle) = [2]
p(true) = [2]
p(app#) = [1]
p(concat#) = [1] x1 + [0]
p(less_leaves#) = [8] x1 + [0]
p(minus#) = [8] x2 + [1]
p(quot#) = [1] x1 + [1]
p(reverse#) = [0]
p(shuffle#) = [8] x1 + [2]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [8] x1 + [0]
p(c_4) = [1]
p(c_5) = [0]
p(c_6) = [1] x1 + [14]
p(c_7) = [4]
p(c_8) = [2]
p(c_9) = [1] x1 + [0]
p(c_10) = [1]
p(c_11) = [2] x1 + [1] x2 + [1]
p(c_12) = [4]
p(c_13) = [0]
p(c_14) = [1] x2 + [0]
p(c_15) = [2]
Following rules are strictly oriented:
less_leaves#(cons(u,v) = [8] u + [8] v + [16]
,cons(w,z))
> [8] u + [8] v + [14]
= 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 + [2]
>= [1] u + [1] v + [1] y + [2]
= cons(u,concat(v,y))
concat(leaf(),y) = [1] y + [1]
>= [1] y + [0]
= y
*** 1.1.1.1.1.2.1.1.2.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
Weak TRS Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.2.1.1.2.1.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
Weak TRS Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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)))
*** 1.1.1.1.1.2.1.1.2.1.1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.2.1.1.2.1.1.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
less_leaves#(cons(u,v),cons(w,z)) -> c_6(less_leaves#(concat(u,v),concat(w,z)))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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)))
*** 1.1.1.1.1.2.1.1.2.1.1.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
Strict DP Rules:
minus#(s(x),s(y)) -> c_9(minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
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 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Problem (S)
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
minus#(s(x),s(y)) -> c_9(minus#(x,y))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
minus#(s(x),s(y)) -> c_9(minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
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 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
-->_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: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))
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
minus#(s(x),s(y)) -> c_9(minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
UsableRules
Proof:
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))
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
minus#(s(x),s(y)) -> c_9(minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
Weak TRS Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: minus#(s(x),s(y)) ->
c_9(minus#(x,y))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
minus#(s(x),s(y)) -> c_9(minus#(x,y))
Strict TRS Rules:
Weak DP Rules:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y))
Weak TRS Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{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 any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_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) = 0
p(add) = x1
p(app) = 4*x1*x2 + x1^2 + 2*x2^2
p(concat) = 1 + x1^2 + x2^2
p(cons) = x2
p(false) = 0
p(leaf) = 0
p(less_leaves) = 4 + x1 + 4*x1^2 + 4*x2^2
p(minus) = x1
p(nil) = 0
p(quot) = 2*x1 + x2
p(reverse) = 2*x1 + x1^2
p(s) = 1 + x1
p(shuffle) = 2*x1 + 2*x1^2
p(true) = 1
p(app#) = 1 + 4*x1 + x1*x2 + 2*x2^2
p(concat#) = x1
p(less_leaves#) = 2 + 4*x1 + 2*x1*x2 + 2*x1^2 + x2 + x2^2
p(minus#) = 4*x1
p(quot#) = 1 + 2*x1^2
p(reverse#) = 2 + x1 + 2*x1^2
p(shuffle#) = x1 + x1^2
p(c_1) = 1
p(c_2) = 1
p(c_3) = 0
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) = 1 + x1 + x2
p(c_12) = 1
p(c_13) = 0
p(c_14) = x1 + x2
p(c_15) = 0
Following rules are strictly oriented:
minus#(s(x),s(y)) = 4 + 4*x
> 4*x
= c_9(minus#(x,y))
Following rules are (at-least) weakly oriented:
quot#(s(x),s(y)) = 3 + 4*x + 2*x^2
>= 2 + 4*x + 2*x^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)
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
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 Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
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 Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
minus#(s(x),s(y)) -> c_9(minus#(x,y))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
SimplifyRHS
Proof:
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)))
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
UsableRules
Proof:
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))
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
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))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
Strict DP Rules:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
Strict TRS Rules:
Weak DP Rules:
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Problem (S)
Strict DP Rules:
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
Strict TRS Rules:
Weak DP Rules:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
Strict TRS Rules:
Weak DP Rules:
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
UsableRules
Proof:
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)))
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: quot#(s(x),s(y)) ->
c_11(quot#(minus(x,y),s(y)))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{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 any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_11) = {1}
Following symbols are considered usable:
{minus,app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}
TcT has computed the following interpretation:
p(0) = [0]
p(add) = [1] x1 + [1] x2 + [0]
p(app) = [0]
p(concat) = [0]
p(cons) = [1] x1 + [1] x2 + [0]
p(false) = [0]
p(leaf) = [0]
p(less_leaves) = [0]
p(minus) = [1] x1 + [0]
p(nil) = [0]
p(quot) = [0]
p(reverse) = [0]
p(s) = [1] x1 + [1]
p(shuffle) = [0]
p(true) = [0]
p(app#) = [0]
p(concat#) = [0]
p(less_leaves#) = [0]
p(minus#) = [2] x2 + [0]
p(quot#) = [8] x1 + [9] x2 + [7]
p(reverse#) = [1]
p(shuffle#) = [2]
p(c_1) = [0]
p(c_2) = [0]
p(c_3) = [1] x1 + [2]
p(c_4) = [1]
p(c_5) = [1]
p(c_6) = [8] x1 + [0]
p(c_7) = [1]
p(c_8) = [0]
p(c_9) = [8] x1 + [1]
p(c_10) = [0]
p(c_11) = [1] x1 + [0]
p(c_12) = [1]
p(c_13) = [1]
p(c_14) = [2] x2 + [1]
p(c_15) = [1]
Following rules are strictly oriented:
quot#(s(x),s(y)) = [8] x + [9] y + [24]
> [8] x + [9] y + [16]
= 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)
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
Weak TRS Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
Weak TRS Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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)))
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.2 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
Strict TRS Rules:
Weak DP Rules:
quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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)))
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.2.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
Proof:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
Strict DP Rules:
reverse#(add(n,x)) -> c_12(reverse#(x))
Strict TRS Rules:
Weak DP Rules:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Problem (S)
Strict DP Rules:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
Strict TRS Rules:
Weak DP Rules:
reverse#(add(n,x)) -> c_12(reverse#(x))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.2.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
reverse#(add(n,x)) -> c_12(reverse#(x))
Strict TRS Rules:
Weak DP Rules:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
UsableRules
Proof:
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))
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.2.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
reverse#(add(n,x)) -> c_12(reverse#(x))
Strict TRS Rules:
Weak DP Rules:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: reverse#(add(n,x)) ->
c_12(reverse#(x))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.2.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
reverse#(add(n,x)) -> c_12(reverse#(x))
Strict TRS Rules:
Weak DP Rules:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{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 any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_12) = {1},
uargs(c_14) = {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) = 0
p(add) = 1 + x2
p(app) = x1 + x2
p(concat) = 1 + x1*x2 + x1^2 + 4*x2 + x2^2
p(cons) = 1
p(false) = 1
p(leaf) = 1
p(less_leaves) = 4*x1 + 4*x1^2 + 4*x2 + x2^2
p(minus) = 1 + 2*x1^2
p(nil) = 0
p(quot) = 4*x1*x2 + x1^2
p(reverse) = x1
p(s) = 0
p(shuffle) = 0
p(true) = 0
p(app#) = 2*x1*x2 + 2*x1^2 + x2^2
p(concat#) = 1 + x1 + x1*x2 + x1^2 + 4*x2 + 4*x2^2
p(less_leaves#) = x1^2 + x2
p(minus#) = 1 + x2 + x2^2
p(quot#) = 2 + x1
p(reverse#) = 1 + 4*x1
p(shuffle#) = 6 + 4*x1 + 4*x1^2
p(c_1) = 1 + x1
p(c_2) = 1
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) = 0
p(c_10) = 1
p(c_11) = 1
p(c_12) = x1
p(c_13) = 0
p(c_14) = 1 + x1 + x2
p(c_15) = 0
Following rules are strictly oriented:
reverse#(add(n,x)) = 5 + 4*x
> 1 + 4*x
= c_12(reverse#(x))
Following rules are (at-least) weakly oriented:
shuffle#(add(n,x)) = 14 + 12*x + 4*x^2
>= 8 + 8*x + 4*x^2
= c_14(shuffle#(reverse(x))
,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()
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.2.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.2.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
reverse#(add(n,x)) -> c_12(reverse#(x))
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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)) -> 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
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: shuffle#(add(n,x)) ->
c_14(shuffle#(reverse(x))
,reverse#(x))
1: reverse#(add(n,x)) ->
c_12(reverse#(x))
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.2.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.2.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
Strict TRS Rules:
Weak DP Rules:
reverse#(add(n,x)) -> c_12(reverse#(x))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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))
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.2.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)),reverse#(x))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
SimplifyRHS
Proof:
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)))
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.2.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
UsableRules
Proof:
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)))
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.2.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{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, greedy = NoGreedy}}
Proof:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
1: shuffle#(add(n,x)) ->
c_14(shuffle#(reverse(x)))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.2.1.2.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
Strict TRS Rules:
Weak DP Rules:
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{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 any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
Proof:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_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) = [4]
p(add) = [1] x2 + [1]
p(app) = [1] x1 + [1] x2 + [0]
p(concat) = [4] x2 + [1]
p(cons) = [1] x2 + [2]
p(false) = [4]
p(leaf) = [0]
p(less_leaves) = [1] x1 + [1] x2 + [2]
p(minus) = [1]
p(nil) = [0]
p(quot) = [1]
p(reverse) = [1] x1 + [0]
p(s) = [4]
p(shuffle) = [4]
p(true) = [2]
p(app#) = [2] x2 + [1]
p(concat#) = [2] x1 + [1]
p(less_leaves#) = [1] x2 + [0]
p(minus#) = [1]
p(quot#) = [1] x1 + [1] x2 + [0]
p(reverse#) = [4] x1 + [1]
p(shuffle#) = [1] x1 + [0]
p(c_1) = [1]
p(c_2) = [1]
p(c_3) = [1]
p(c_4) = [1]
p(c_5) = [2]
p(c_6) = [1] x1 + [0]
p(c_7) = [0]
p(c_8) = [1]
p(c_9) = [0]
p(c_10) = [0]
p(c_11) = [1]
p(c_12) = [0]
p(c_13) = [1]
p(c_14) = [1] x1 + [0]
p(c_15) = [0]
Following rules are strictly oriented:
shuffle#(add(n,x)) = [1] x + [1]
> [1] x + [0]
= c_14(shuffle#(reverse(x)))
Following rules are (at-least) weakly oriented:
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()
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.2.1.2.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.2.1.2.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
shuffle#(add(n,x)) -> c_14(shuffle#(reverse(x)))
Weak TRS 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()
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
RemoveWeakSuffixes
Proof:
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)))
*** 1.1.1.1.1.2.1.1.2.1.1.2.1.2.1.1.1.2.1.2.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
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
basic terms: {app#,concat#,less_leaves#,minus#,quot#,reverse#,shuffle#}/{0,add,cons,false,leaf,nil,s,true}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).