*** 1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
insert#3(x2,Nil()) -> Cons(x2,Nil())
insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2)
leq#2(0(),x8) -> True()
leq#2(S(x12),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
main(x1) -> sort#2(x1)
sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2))
sort#2(Nil()) -> Nil()
Weak DP Rules:
Weak TRS Rules:
Signature:
{cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1,insert#3,leq#2,main,sort#2}/{0,Cons,False,Nil,S,True}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
We add the following dependency tuples:
Strict DPs
cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1))
cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2()
insert#3#(x2,Nil()) -> c_3()
insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(0(),x8) -> c_5()
leq#2#(S(x12),0()) -> c_6()
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
main#(x1) -> c_8(sort#2#(x1))
sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2))
sort#2#(Nil()) -> c_10()
Weak DPs
and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1))
cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2()
insert#3#(x2,Nil()) -> c_3()
insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(0(),x8) -> c_5()
leq#2#(S(x12),0()) -> c_6()
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
main#(x1) -> c_8(sort#2#(x1))
sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2))
sort#2#(Nil()) -> c_10()
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
insert#3(x2,Nil()) -> Cons(x2,Nil())
insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2)
leq#2(0(),x8) -> True()
leq#2(S(x12),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
main(x1) -> sort#2(x1)
sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2))
sort#2(Nil()) -> Nil()
Signature:
{cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}/{0,Cons,False,Nil,S,True}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
insert#3(x2,Nil()) -> Cons(x2,Nil())
insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2)
leq#2(0(),x8) -> True()
leq#2(S(x12),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2))
sort#2(Nil()) -> Nil()
cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1))
cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2()
insert#3#(x2,Nil()) -> c_3()
insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(0(),x8) -> c_5()
leq#2#(S(x12),0()) -> c_6()
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
main#(x1) -> c_8(sort#2#(x1))
sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2))
sort#2#(Nil()) -> c_10()
*** 1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1))
cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2()
insert#3#(x2,Nil()) -> c_3()
insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(0(),x8) -> c_5()
leq#2#(S(x12),0()) -> c_6()
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
main#(x1) -> c_8(sort#2#(x1))
sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2))
sort#2#(Nil()) -> c_10()
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
insert#3(x2,Nil()) -> Cons(x2,Nil())
insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2)
leq#2(0(),x8) -> True()
leq#2(S(x12),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2))
sort#2(Nil()) -> Nil()
Signature:
{cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}/{0,Cons,False,Nil,S,True}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{2,3,5,6,10}
by application of
Pre({2,3,5,6,10}) = {1,4,7,8,9}.
Here rules are labelled as follows:
1: cond_insert_ord_x_ys_1#(False()
,x3
,x2
,x1) -> c_1(insert#3#(x3,x1))
2: cond_insert_ord_x_ys_1#(True()
,x3
,x2
,x1) -> c_2()
3: insert#3#(x2,Nil()) -> c_3()
4: insert#3#(x6,Cons(x4,x2)) ->
c_4(cond_insert_ord_x_ys_1#(leq#2(x6
,x4)
,x6
,x4
,x2)
,leq#2#(x6,x4))
5: leq#2#(0(),x8) -> c_5()
6: leq#2#(S(x12),0()) -> c_6()
7: leq#2#(S(x4),S(x2)) ->
c_7(leq#2#(x4,x2))
8: main#(x1) -> c_8(sort#2#(x1))
9: sort#2#(Cons(x4,x2)) ->
c_9(insert#3#(x4,sort#2(x2))
,sort#2#(x2))
10: sort#2#(Nil()) -> c_10()
*** 1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1))
insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
main#(x1) -> c_8(sort#2#(x1))
sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2))
Strict TRS Rules:
Weak DP Rules:
cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2()
insert#3#(x2,Nil()) -> c_3()
leq#2#(0(),x8) -> c_5()
leq#2#(S(x12),0()) -> c_6()
sort#2#(Nil()) -> c_10()
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
insert#3(x2,Nil()) -> Cons(x2,Nil())
insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2)
leq#2(0(),x8) -> True()
leq#2(S(x12),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2))
sort#2(Nil()) -> Nil()
Signature:
{cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}/{0,Cons,False,Nil,S,True}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1))
-->_1 insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):2
-->_1 insert#3#(x2,Nil()) -> c_3():7
2:S:insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
-->_2 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):3
-->_2 leq#2#(S(x12),0()) -> c_6():9
-->_2 leq#2#(0(),x8) -> c_5():8
-->_1 cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2():6
-->_1 cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1)):1
3:S:leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
-->_1 leq#2#(S(x12),0()) -> c_6():9
-->_1 leq#2#(0(),x8) -> c_5():8
-->_1 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):3
4:S:main#(x1) -> c_8(sort#2#(x1))
-->_1 sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)):5
-->_1 sort#2#(Nil()) -> c_10():10
5:S:sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2))
-->_2 sort#2#(Nil()) -> c_10():10
-->_1 insert#3#(x2,Nil()) -> c_3():7
-->_2 sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)):5
-->_1 insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):2
6:W:cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2()
7:W:insert#3#(x2,Nil()) -> c_3()
8:W:leq#2#(0(),x8) -> c_5()
9:W:leq#2#(S(x12),0()) -> c_6()
10:W:sort#2#(Nil()) -> c_10()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
10: sort#2#(Nil()) -> c_10()
7: insert#3#(x2,Nil()) -> c_3()
6: cond_insert_ord_x_ys_1#(True()
,x3
,x2
,x1) -> c_2()
8: leq#2#(0(),x8) -> c_5()
9: leq#2#(S(x12),0()) -> c_6()
*** 1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1))
insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
main#(x1) -> c_8(sort#2#(x1))
sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
insert#3(x2,Nil()) -> Cons(x2,Nil())
insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2)
leq#2(0(),x8) -> True()
leq#2(S(x12),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2))
sort#2(Nil()) -> Nil()
Signature:
{cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}/{0,Cons,False,Nil,S,True}
Applied Processor:
RemoveHeads
Proof:
Consider the dependency graph
1:S:cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1))
-->_1 insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):2
2:S:insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
-->_2 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):3
-->_1 cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1)):1
3:S:leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
-->_1 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):3
4:S:main#(x1) -> c_8(sort#2#(x1))
-->_1 sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)):5
5:S:sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2))
-->_2 sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)):5
-->_1 insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):2
Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
[(4,main#(x1) -> c_8(sort#2#(x1)))]
*** 1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1))
insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
insert#3(x2,Nil()) -> Cons(x2,Nil())
insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2)
leq#2(0(),x8) -> True()
leq#2(S(x12),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2))
sort#2(Nil()) -> Nil()
Signature:
{cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}/{0,Cons,False,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:
cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1))
insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
Strict TRS Rules:
Weak DP Rules:
sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2))
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
insert#3(x2,Nil()) -> Cons(x2,Nil())
insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2)
leq#2(0(),x8) -> True()
leq#2(S(x12),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2))
sort#2(Nil()) -> Nil()
Signature:
{cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}/{0,Cons,False,Nil,S,True}
Problem (S)
Strict DP Rules:
sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2))
Strict TRS Rules:
Weak DP Rules:
cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1))
insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
insert#3(x2,Nil()) -> Cons(x2,Nil())
insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2)
leq#2(0(),x8) -> True()
leq#2(S(x12),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2))
sort#2(Nil()) -> Nil()
Signature:
{cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}/{0,Cons,False,Nil,S,True}
*** 1.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1))
insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
Strict TRS Rules:
Weak DP Rules:
sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2))
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
insert#3(x2,Nil()) -> Cons(x2,Nil())
insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2)
leq#2(0(),x8) -> True()
leq#2(S(x12),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2))
sort#2(Nil()) -> Nil()
Signature:
{cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}/{0,Cons,False,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: cond_insert_ord_x_ys_1#(False()
,x3
,x2
,x1) -> c_1(insert#3#(x3,x1))
2: insert#3#(x6,Cons(x4,x2)) ->
c_4(cond_insert_ord_x_ys_1#(leq#2(x6
,x4)
,x6
,x4
,x2)
,leq#2#(x6,x4))
3: leq#2#(S(x4),S(x2)) ->
c_7(leq#2#(x4,x2))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1))
insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
Strict TRS Rules:
Weak DP Rules:
sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2))
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
insert#3(x2,Nil()) -> Cons(x2,Nil())
insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2)
leq#2(0(),x8) -> True()
leq#2(S(x12),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2))
sort#2(Nil()) -> Nil()
Signature:
{cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}/{0,Cons,False,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_4) = {1,2},
uargs(c_7) = {1},
uargs(c_9) = {1,2}
Following symbols are considered usable:
{cond_insert_ord_x_ys_1,insert#3,sort#2,cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}
TcT has computed the following interpretation:
p(0) = 1
p(Cons) = 1 + x1 + x2
p(False) = 0
p(Nil) = 0
p(S) = 1 + x1
p(True) = 0
p(cond_insert_ord_x_ys_1) = 2 + x2 + x3 + x4
p(insert#3) = 1 + x1 + x2
p(leq#2) = 2*x1*x2 + 2*x2
p(main) = 2
p(sort#2) = x1
p(cond_insert_ord_x_ys_1#) = 5 + 6*x2 + 2*x2*x3 + 4*x2*x4 + x3 + 4*x4
p(insert#3#) = 2 + 5*x1 + 4*x1*x2 + 4*x2
p(leq#2#) = 2*x1
p(main#) = 2 + x1
p(sort#2#) = 3*x1 + 3*x1^2
p(c_1) = x1
p(c_2) = 1
p(c_3) = 0
p(c_4) = x1 + x2
p(c_5) = 0
p(c_6) = 1
p(c_7) = x1
p(c_8) = 0
p(c_9) = 1 + x1 + x2
p(c_10) = 1
Following rules are strictly oriented:
cond_insert_ord_x_ys_1#(False() = 5 + 4*x1 + 4*x1*x3 + x2 + 2*x2*x3 + 6*x3
,x3
,x2
,x1)
> 2 + 4*x1 + 4*x1*x3 + 5*x3
= c_1(insert#3#(x3,x1))
insert#3#(x6,Cons(x4,x2)) = 6 + 4*x2 + 4*x2*x6 + 4*x4 + 4*x4*x6 + 9*x6
> 5 + 4*x2 + 4*x2*x6 + x4 + 2*x4*x6 + 8*x6
= c_4(cond_insert_ord_x_ys_1#(leq#2(x6
,x4)
,x6
,x4
,x2)
,leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) = 2 + 2*x4
> 2*x4
= c_7(leq#2#(x4,x2))
Following rules are (at-least) weakly oriented:
sort#2#(Cons(x4,x2)) = 6 + 9*x2 + 6*x2*x4 + 3*x2^2 + 9*x4 + 3*x4^2
>= 3 + 7*x2 + 4*x2*x4 + 3*x2^2 + 5*x4
= c_9(insert#3#(x4,sort#2(x2))
,sort#2#(x2))
cond_insert_ord_x_ys_1(False() = 2 + x1 + x2 + x3
,x3
,x2
,x1)
>= 2 + x1 + x2 + x3
= Cons(x2,insert#3(x3,x1))
cond_insert_ord_x_ys_1(True() = 2 + x1 + x2 + x3
,x3
,x2
,x1)
>= 2 + x1 + x2 + x3
= Cons(x3,Cons(x2,x1))
insert#3(x2,Nil()) = 1 + x2
>= 1 + x2
= Cons(x2,Nil())
insert#3(x6,Cons(x4,x2)) = 2 + x2 + x4 + x6
>= 2 + x2 + x4 + x6
= cond_insert_ord_x_ys_1(leq#2(x6
,x4)
,x6
,x4
,x2)
sort#2(Cons(x4,x2)) = 1 + x2 + x4
>= 1 + x2 + x4
= insert#3(x4,sort#2(x2))
sort#2(Nil()) = 0
>= 0
= Nil()
*** 1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1))
insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2))
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
insert#3(x2,Nil()) -> Cons(x2,Nil())
insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2)
leq#2(0(),x8) -> True()
leq#2(S(x12),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2))
sort#2(Nil()) -> Nil()
Signature:
{cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}/{0,Cons,False,Nil,S,True}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1))
insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2))
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
insert#3(x2,Nil()) -> Cons(x2,Nil())
insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2)
leq#2(0(),x8) -> True()
leq#2(S(x12),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2))
sort#2(Nil()) -> Nil()
Signature:
{cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}/{0,Cons,False,Nil,S,True}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1))
-->_1 insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):2
2:W:insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
-->_2 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):3
-->_1 cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1)):1
3:W:leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
-->_1 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):3
4:W:sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2))
-->_2 sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)):4
-->_1 insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: sort#2#(Cons(x4,x2)) ->
c_9(insert#3#(x4,sort#2(x2))
,sort#2#(x2))
1: cond_insert_ord_x_ys_1#(False()
,x3
,x2
,x1) -> c_1(insert#3#(x3,x1))
2: insert#3#(x6,Cons(x4,x2)) ->
c_4(cond_insert_ord_x_ys_1#(leq#2(x6
,x4)
,x6
,x4
,x2)
,leq#2#(x6,x4))
3: leq#2#(S(x4),S(x2)) ->
c_7(leq#2#(x4,x2))
*** 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:
cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
insert#3(x2,Nil()) -> Cons(x2,Nil())
insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2)
leq#2(0(),x8) -> True()
leq#2(S(x12),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2))
sort#2(Nil()) -> Nil()
Signature:
{cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}/{0,Cons,False,Nil,S,True}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).
*** 1.1.1.1.1.1.2 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2))
Strict TRS Rules:
Weak DP Rules:
cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1))
insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
insert#3(x2,Nil()) -> Cons(x2,Nil())
insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2)
leq#2(0(),x8) -> True()
leq#2(S(x12),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2))
sort#2(Nil()) -> Nil()
Signature:
{cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}/{0,Cons,False,Nil,S,True}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2))
-->_1 insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):3
-->_2 sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)):1
2:W:cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1))
-->_1 insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):3
3:W:insert#3#(x6,Cons(x4,x2)) -> c_4(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
-->_2 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):4
-->_1 cond_insert_ord_x_ys_1#(False(),x3,x2,x1) -> c_1(insert#3#(x3,x1)):2
4:W:leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
-->_1 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):4
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: insert#3#(x6,Cons(x4,x2)) ->
c_4(cond_insert_ord_x_ys_1#(leq#2(x6
,x4)
,x6
,x4
,x2)
,leq#2#(x6,x4))
2: cond_insert_ord_x_ys_1#(False()
,x3
,x2
,x1) -> c_1(insert#3#(x3,x1))
4: leq#2#(S(x4),S(x2)) ->
c_7(leq#2#(x4,x2))
*** 1.1.1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
insert#3(x2,Nil()) -> Cons(x2,Nil())
insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2)
leq#2(0(),x8) -> True()
leq#2(S(x12),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2))
sort#2(Nil()) -> Nil()
Signature:
{cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/2,c_10/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}/{0,Cons,False,Nil,S,True}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2))
-->_2 sort#2#(Cons(x4,x2)) -> c_9(insert#3#(x4,sort#2(x2)),sort#2#(x2)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
sort#2#(Cons(x4,x2)) -> c_9(sort#2#(x2))
*** 1.1.1.1.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sort#2#(Cons(x4,x2)) -> c_9(sort#2#(x2))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
insert#3(x2,Nil()) -> Cons(x2,Nil())
insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2)
leq#2(0(),x8) -> True()
leq#2(S(x12),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2))
sort#2(Nil()) -> Nil()
Signature:
{cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/1,c_10/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}/{0,Cons,False,Nil,S,True}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
sort#2#(Cons(x4,x2)) -> c_9(sort#2#(x2))
*** 1.1.1.1.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sort#2#(Cons(x4,x2)) -> c_9(sort#2#(x2))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/1,c_10/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}/{0,Cons,False,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: sort#2#(Cons(x4,x2)) ->
c_9(sort#2#(x2))
The strictly oriented rules are moved into the weak component.
*** 1.1.1.1.1.1.2.1.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
sort#2#(Cons(x4,x2)) -> c_9(sort#2#(x2))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/1,c_10/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}/{0,Cons,False,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_9) = {1}
Following symbols are considered usable:
{cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}
TcT has computed the following interpretation:
p(0) = [0]
p(Cons) = [1] x1 + [1] x2 + [8]
p(False) = [0]
p(Nil) = [0]
p(S) = [1] x1 + [0]
p(True) = [0]
p(cond_insert_ord_x_ys_1) = [0]
p(insert#3) = [0]
p(leq#2) = [0]
p(main) = [0]
p(sort#2) = [0]
p(cond_insert_ord_x_ys_1#) = [8] x1 + [2] x3 + [1] x4 + [0]
p(insert#3#) = [8] x1 + [2]
p(leq#2#) = [1]
p(main#) = [8]
p(sort#2#) = [2] x1 + [1]
p(c_1) = [1] x1 + [4]
p(c_2) = [1]
p(c_3) = [1]
p(c_4) = [4] x2 + [8]
p(c_5) = [1]
p(c_6) = [1]
p(c_7) = [1] x1 + [1]
p(c_8) = [0]
p(c_9) = [1] x1 + [0]
p(c_10) = [8]
Following rules are strictly oriented:
sort#2#(Cons(x4,x2)) = [2] x2 + [2] x4 + [17]
> [2] x2 + [1]
= c_9(sort#2#(x2))
Following rules are (at-least) weakly oriented:
*** 1.1.1.1.1.1.2.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
sort#2#(Cons(x4,x2)) -> c_9(sort#2#(x2))
Weak TRS Rules:
Signature:
{cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/1,c_10/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}/{0,Cons,False,Nil,S,True}
Applied Processor:
Assumption
Proof:
()
*** 1.1.1.1.1.1.2.1.1.1.2 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
sort#2#(Cons(x4,x2)) -> c_9(sort#2#(x2))
Weak TRS Rules:
Signature:
{cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/1,c_10/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}/{0,Cons,False,Nil,S,True}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:sort#2#(Cons(x4,x2)) -> c_9(sort#2#(x2))
-->_1 sort#2#(Cons(x4,x2)) -> c_9(sort#2#(x2)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: sort#2#(Cons(x4,x2)) ->
c_9(sort#2#(x2))
*** 1.1.1.1.1.1.2.1.1.1.2.1 Progress [(O(1),O(1))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1,cond_insert_ord_x_ys_1#/4,insert#3#/2,leq#2#/2,main#/1,sort#2#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/1,c_9/1,c_10/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,insert#3#,leq#2#,main#,sort#2#}/{0,Cons,False,Nil,S,True}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).