*** 1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
Strict TRS Rules:
cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
fold#3(insert_ord(x2),Nil()) -> Nil()
fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2))
insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil())
insert_ord#2(leq(),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(x3) -> fold#3(insert_ord(leq()),x3)
Weak DP Rules:
Weak TRS Rules:
Signature:
{cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1,fold#3,insert_ord#2,leq#2,main}/{0,Cons,False,Nil,S,True,insert_ord,leq}
Applied Processor:
DependencyPairs {dpKind_ = DT}
Proof:
We add the following dependency tuples:
Strict DPs
cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2))
cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2()
fold#3#(insert_ord(x2),Nil()) -> c_3()
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2))
insert_ord#2#(leq(),x2,Nil()) -> c_5()
insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(0(),x8) -> c_7()
leq#2#(S(x12),0()) -> c_8()
leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
main#(x3) -> c_10(fold#3#(insert_ord(leq()),x3))
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(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2))
cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2()
fold#3#(insert_ord(x2),Nil()) -> c_3()
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2))
insert_ord#2#(leq(),x2,Nil()) -> c_5()
insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(0(),x8) -> c_7()
leq#2#(S(x12),0()) -> c_8()
leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
main#(x3) -> c_10(fold#3#(insert_ord(leq()),x3))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
fold#3(insert_ord(x2),Nil()) -> Nil()
fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2))
insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil())
insert_ord#2(leq(),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(x3) -> fold#3(insert_ord(leq()),x3)
Signature:
{cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}/{0,Cons,False,Nil,S,True,insert_ord,leq}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
fold#3(insert_ord(x2),Nil()) -> Nil()
fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2))
insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil())
insert_ord#2(leq(),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)
cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2))
cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2()
fold#3#(insert_ord(x2),Nil()) -> c_3()
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2))
insert_ord#2#(leq(),x2,Nil()) -> c_5()
insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(0(),x8) -> c_7()
leq#2#(S(x12),0()) -> c_8()
leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
main#(x3) -> c_10(fold#3#(insert_ord(leq()),x3))
*** 1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2))
cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2()
fold#3#(insert_ord(x2),Nil()) -> c_3()
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2))
insert_ord#2#(leq(),x2,Nil()) -> c_5()
insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(0(),x8) -> c_7()
leq#2#(S(x12),0()) -> c_8()
leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
main#(x3) -> c_10(fold#3#(insert_ord(leq()),x3))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
fold#3(insert_ord(x2),Nil()) -> Nil()
fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2))
insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil())
insert_ord#2(leq(),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)
Signature:
{cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}/{0,Cons,False,Nil,S,True,insert_ord,leq}
Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
Proof:
We estimate the number of application of
{2,3,5,7,8}
by application of
Pre({2,3,5,7,8}) = {1,4,6,9,10}.
Here rules are labelled as follows:
1: cond_insert_ord_x_ys_1#(False()
,x0
,x5
,x2) -> c_1(insert_ord#2#(leq()
,x0
,x2))
2: cond_insert_ord_x_ys_1#(True()
,x3
,x2
,x1) -> c_2()
3: fold#3#(insert_ord(x2),Nil()) ->
c_3()
4: fold#3#(insert_ord(x6)
,Cons(x4,x2)) ->
c_4(insert_ord#2#(x6
,x4
,fold#3(insert_ord(x6),x2))
,fold#3#(insert_ord(x6),x2))
5: insert_ord#2#(leq(),x2,Nil()) ->
c_5()
6: insert_ord#2#(leq()
,x6
,Cons(x4,x2)) ->
c_6(cond_insert_ord_x_ys_1#(leq#2(x6
,x4)
,x6
,x4
,x2)
,leq#2#(x6,x4))
7: leq#2#(0(),x8) -> c_7()
8: leq#2#(S(x12),0()) -> c_8()
9: leq#2#(S(x4),S(x2)) ->
c_9(leq#2#(x4,x2))
10: main#(x3) ->
c_10(fold#3#(insert_ord(leq())
,x3))
*** 1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2))
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2))
insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
main#(x3) -> c_10(fold#3#(insert_ord(leq()),x3))
Strict TRS Rules:
Weak DP Rules:
cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2()
fold#3#(insert_ord(x2),Nil()) -> c_3()
insert_ord#2#(leq(),x2,Nil()) -> c_5()
leq#2#(0(),x8) -> c_7()
leq#2#(S(x12),0()) -> c_8()
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
fold#3(insert_ord(x2),Nil()) -> Nil()
fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2))
insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil())
insert_ord#2(leq(),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)
Signature:
{cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}/{0,Cons,False,Nil,S,True,insert_ord,leq}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2))
-->_1 insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):3
-->_1 insert_ord#2#(leq(),x2,Nil()) -> c_5():8
2:S:fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2))
-->_1 insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):3
-->_1 insert_ord#2#(leq(),x2,Nil()) -> c_5():8
-->_2 fold#3#(insert_ord(x2),Nil()) -> c_3():7
-->_2 fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2)):2
3:S:insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(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_9(leq#2#(x4,x2)):4
-->_2 leq#2#(S(x12),0()) -> c_8():10
-->_2 leq#2#(0(),x8) -> c_7():9
-->_1 cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2():6
-->_1 cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)):1
4:S:leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
-->_1 leq#2#(S(x12),0()) -> c_8():10
-->_1 leq#2#(0(),x8) -> c_7():9
-->_1 leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)):4
5:S:main#(x3) -> c_10(fold#3#(insert_ord(leq()),x3))
-->_1 fold#3#(insert_ord(x2),Nil()) -> c_3():7
-->_1 fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2)):2
6:W:cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2()
7:W:fold#3#(insert_ord(x2),Nil()) -> c_3()
8:W:insert_ord#2#(leq(),x2,Nil()) -> c_5()
9:W:leq#2#(0(),x8) -> c_7()
10:W:leq#2#(S(x12),0()) -> c_8()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
7: fold#3#(insert_ord(x2),Nil()) ->
c_3()
8: insert_ord#2#(leq(),x2,Nil()) ->
c_5()
6: cond_insert_ord_x_ys_1#(True()
,x3
,x2
,x1) -> c_2()
9: leq#2#(0(),x8) -> c_7()
10: leq#2#(S(x12),0()) -> c_8()
*** 1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2))
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2))
insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
main#(x3) -> c_10(fold#3#(insert_ord(leq()),x3))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
fold#3(insert_ord(x2),Nil()) -> Nil()
fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2))
insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil())
insert_ord#2(leq(),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)
Signature:
{cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}/{0,Cons,False,Nil,S,True,insert_ord,leq}
Applied Processor:
RemoveHeads
Proof:
Consider the dependency graph
1:S:cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2))
-->_1 insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):3
2:S:fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2))
-->_1 insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):3
-->_2 fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2)):2
3:S:insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(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_9(leq#2#(x4,x2)):4
-->_1 cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)):1
4:S:leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
-->_1 leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)):4
5:S:main#(x3) -> c_10(fold#3#(insert_ord(leq()),x3))
-->_1 fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2)):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).
[(5,main#(x3) -> c_10(fold#3#(insert_ord(leq()),x3)))]
*** 1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2))
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2))
insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
fold#3(insert_ord(x2),Nil()) -> Nil()
fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2))
insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil())
insert_ord#2(leq(),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)
Signature:
{cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}/{0,Cons,False,Nil,S,True,insert_ord,leq}
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(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2))
insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
Strict TRS Rules:
Weak DP Rules:
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2))
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
fold#3(insert_ord(x2),Nil()) -> Nil()
fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2))
insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil())
insert_ord#2(leq(),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)
Signature:
{cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}/{0,Cons,False,Nil,S,True,insert_ord,leq}
Problem (S)
Strict DP Rules:
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2))
Strict TRS Rules:
Weak DP Rules:
cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2))
insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
fold#3(insert_ord(x2),Nil()) -> Nil()
fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2))
insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil())
insert_ord#2(leq(),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)
Signature:
{cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}/{0,Cons,False,Nil,S,True,insert_ord,leq}
*** 1.1.1.1.1.1.1 Progress [(?,O(n^2))] ***
Considered Problem:
Strict DP Rules:
cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2))
insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
Strict TRS Rules:
Weak DP Rules:
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2))
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
fold#3(insert_ord(x2),Nil()) -> Nil()
fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2))
insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil())
insert_ord#2(leq(),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)
Signature:
{cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}/{0,Cons,False,Nil,S,True,insert_ord,leq}
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:
3: insert_ord#2#(leq()
,x6
,Cons(x4,x2)) ->
c_6(cond_insert_ord_x_ys_1#(leq#2(x6
,x4)
,x6
,x4
,x2)
,leq#2#(x6,x4))
4: leq#2#(S(x4),S(x2)) ->
c_9(leq#2#(x4,x2))
Consider the set of all dependency pairs
1: cond_insert_ord_x_ys_1#(False()
,x0
,x5
,x2) -> c_1(insert_ord#2#(leq()
,x0
,x2))
2: fold#3#(insert_ord(x6)
,Cons(x4,x2)) ->
c_4(insert_ord#2#(x6
,x4
,fold#3(insert_ord(x6),x2))
,fold#3#(insert_ord(x6),x2))
3: insert_ord#2#(leq()
,x6
,Cons(x4,x2)) ->
c_6(cond_insert_ord_x_ys_1#(leq#2(x6
,x4)
,x6
,x4
,x2)
,leq#2#(x6,x4))
4: leq#2#(S(x4),S(x2)) ->
c_9(leq#2#(x4,x2))
Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}induces the complexity certificateTIME (?,O(n^2))
SPACE(?,?)on application of the dependency pairs
{3,4}
These cover all (indirect) predecessors of dependency pairs
{1,3,4}
their number of applications is equally bounded.
The dependency pairs are shifted 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(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2))
insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
Strict TRS Rules:
Weak DP Rules:
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2))
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
fold#3(insert_ord(x2),Nil()) -> Nil()
fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2))
insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil())
insert_ord#2(leq(),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)
Signature:
{cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}/{0,Cons,False,Nil,S,True,insert_ord,leq}
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_6) = {1,2},
uargs(c_9) = {1}
Following symbols are considered usable:
{cond_insert_ord_x_ys_1,fold#3,insert_ord#2,cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}
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) = 3 + x2 + x3 + x4
p(fold#3) = x1*x2 + x1^2 + x2
p(insert_ord) = 1
p(insert_ord#2) = 2 + x2 + x3
p(leq) = 0
p(leq#2) = 0
p(main) = 2*x1^2
p(cond_insert_ord_x_ys_1#) = x2*x3 + x2*x4 + x2^2 + 2*x4
p(fold#3#) = 1 + x1 + 3*x1^2 + 2*x2^2
p(insert_ord#2#) = x2*x3 + x2^2 + 2*x3
p(leq#2#) = x1
p(main#) = 2
p(c_1) = x1
p(c_2) = 1
p(c_3) = 0
p(c_4) = x1 + x2
p(c_5) = 1
p(c_6) = x1 + x2
p(c_7) = 1
p(c_8) = 0
p(c_9) = x1
p(c_10) = x1
Following rules are strictly oriented:
insert_ord#2#(leq() = 2 + 2*x2 + x2*x6 + 2*x4 + x4*x6 + x6 + x6^2
,x6
,Cons(x4,x2))
> 2*x2 + x2*x6 + x4*x6 + x6 + x6^2
= c_6(cond_insert_ord_x_ys_1#(leq#2(x6
,x4)
,x6
,x4
,x2)
,leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) = 1 + x4
> x4
= c_9(leq#2#(x4,x2))
Following rules are (at-least) weakly oriented:
cond_insert_ord_x_ys_1#(False() = x0*x2 + x0*x5 + x0^2 + 2*x2
,x0
,x5
,x2)
>= x0*x2 + x0^2 + 2*x2
= c_1(insert_ord#2#(leq(),x0,x2))
fold#3#(insert_ord(x6) = 7 + 4*x2 + 4*x2*x4 + 2*x2^2 + 4*x4 + 2*x4^2
,Cons(x4,x2))
>= 7 + 4*x2 + 2*x2*x4 + 2*x2^2 + x4 + x4^2
= c_4(insert_ord#2#(x6
,x4
,fold#3(insert_ord(x6),x2))
,fold#3#(insert_ord(x6),x2))
cond_insert_ord_x_ys_1(False() = 3 + x0 + x2 + x5
,x0
,x5
,x2)
>= 3 + x0 + x2 + x5
= Cons(x5
,insert_ord#2(leq(),x0,x2))
cond_insert_ord_x_ys_1(True() = 3 + x1 + x2 + x3
,x3
,x2
,x1)
>= 2 + x1 + x2 + x3
= Cons(x3,Cons(x2,x1))
fold#3(insert_ord(x2),Nil()) = 1
>= 0
= Nil()
fold#3(insert_ord(x6) = 3 + 2*x2 + 2*x4
,Cons(x4,x2))
>= 3 + 2*x2 + x4
= insert_ord#2(x6
,x4
,fold#3(insert_ord(x6),x2))
insert_ord#2(leq(),x2,Nil()) = 2 + x2
>= 1 + x2
= Cons(x2,Nil())
insert_ord#2(leq() = 3 + x2 + x4 + x6
,x6
,Cons(x4,x2))
>= 3 + x2 + x4 + x6
= cond_insert_ord_x_ys_1(leq#2(x6
,x4)
,x6
,x4
,x2)
*** 1.1.1.1.1.1.1.1.1 Progress [(?,O(1))] ***
Considered Problem:
Strict DP Rules:
cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2))
Strict TRS Rules:
Weak DP Rules:
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2))
insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
fold#3(insert_ord(x2),Nil()) -> Nil()
fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2))
insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil())
insert_ord#2(leq(),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)
Signature:
{cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}/{0,Cons,False,Nil,S,True,insert_ord,leq}
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(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2))
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2))
insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
fold#3(insert_ord(x2),Nil()) -> Nil()
fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2))
insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil())
insert_ord#2(leq(),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)
Signature:
{cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}/{0,Cons,False,Nil,S,True,insert_ord,leq}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2))
-->_1 insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):3
2:W:fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2))
-->_1 insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):3
-->_2 fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2)):2
3:W:insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(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_9(leq#2#(x4,x2)):4
-->_1 cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)):1
4:W:leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
-->_1 leq#2#(S(x4),S(x2)) -> c_9(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.
2: fold#3#(insert_ord(x6)
,Cons(x4,x2)) ->
c_4(insert_ord#2#(x6
,x4
,fold#3(insert_ord(x6),x2))
,fold#3#(insert_ord(x6),x2))
1: cond_insert_ord_x_ys_1#(False()
,x0
,x5
,x2) -> c_1(insert_ord#2#(leq()
,x0
,x2))
3: insert_ord#2#(leq()
,x6
,Cons(x4,x2)) ->
c_6(cond_insert_ord_x_ys_1#(leq#2(x6
,x4)
,x6
,x4
,x2)
,leq#2#(x6,x4))
4: leq#2#(S(x4),S(x2)) ->
c_9(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(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
fold#3(insert_ord(x2),Nil()) -> Nil()
fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2))
insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil())
insert_ord#2(leq(),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)
Signature:
{cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}/{0,Cons,False,Nil,S,True,insert_ord,leq}
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:
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2))
Strict TRS Rules:
Weak DP Rules:
cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2))
insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
fold#3(insert_ord(x2),Nil()) -> Nil()
fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2))
insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil())
insert_ord#2(leq(),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)
Signature:
{cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}/{0,Cons,False,Nil,S,True,insert_ord,leq}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:S:fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2))
-->_1 insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):3
-->_2 fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2)):1
2:W:cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2))
-->_1 insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)):3
3:W:insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(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_9(leq#2#(x4,x2)):4
-->_1 cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)):2
4:W:leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
-->_1 leq#2#(S(x4),S(x2)) -> c_9(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_ord#2#(leq()
,x6
,Cons(x4,x2)) ->
c_6(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()
,x0
,x5
,x2) -> c_1(insert_ord#2#(leq()
,x0
,x2))
4: leq#2#(S(x4),S(x2)) ->
c_9(leq#2#(x4,x2))
*** 1.1.1.1.1.1.2.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
fold#3(insert_ord(x2),Nil()) -> Nil()
fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2))
insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil())
insert_ord#2(leq(),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)
Signature:
{cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}/{0,Cons,False,Nil,S,True,insert_ord,leq}
Applied Processor:
SimplifyRHS
Proof:
Consider the dependency graph
1:S:fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2))
-->_2 fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)),fold#3#(insert_ord(x6),x2)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(fold#3#(insert_ord(x6),x2))
*** 1.1.1.1.1.1.2.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(fold#3#(insert_ord(x6),x2))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1))
fold#3(insert_ord(x2),Nil()) -> Nil()
fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2))
insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil())
insert_ord#2(leq(),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)
Signature:
{cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}/{0,Cons,False,Nil,S,True,insert_ord,leq}
Applied Processor:
UsableRules
Proof:
We replace rewrite rules by usable rules:
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(fold#3#(insert_ord(x6),x2))
*** 1.1.1.1.1.1.2.1.1.1 Progress [(?,O(n^1))] ***
Considered Problem:
Strict DP Rules:
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(fold#3#(insert_ord(x6),x2))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}/{0,Cons,False,Nil,S,True,insert_ord,leq}
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: fold#3#(insert_ord(x6)
,Cons(x4,x2)) ->
c_4(fold#3#(insert_ord(x6),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:
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(fold#3#(insert_ord(x6),x2))
Strict TRS Rules:
Weak DP Rules:
Weak TRS Rules:
Signature:
{cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}/{0,Cons,False,Nil,S,True,insert_ord,leq}
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_4) = {1}
Following symbols are considered usable:
{cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}
TcT has computed the following interpretation:
p(0) = [0]
p(Cons) = [1] x1 + [1] x2 + [4]
p(False) = [8]
p(Nil) = [1]
p(S) = [0]
p(True) = [1]
p(cond_insert_ord_x_ys_1) = [1] x1 + [2]
p(fold#3) = [1] x2 + [1]
p(insert_ord) = [1] x1 + [0]
p(insert_ord#2) = [2] x2 + [2] x3 + [0]
p(leq) = [2]
p(leq#2) = [4] x1 + [1] x2 + [1]
p(main) = [1]
p(cond_insert_ord_x_ys_1#) = [2] x4 + [1]
p(fold#3#) = [1] x1 + [4] x2 + [0]
p(insert_ord#2#) = [1] x2 + [1]
p(leq#2#) = [2] x2 + [0]
p(main#) = [1]
p(c_1) = [1] x1 + [1]
p(c_2) = [1]
p(c_3) = [8]
p(c_4) = [1] x1 + [12]
p(c_5) = [1]
p(c_6) = [1] x2 + [1]
p(c_7) = [1]
p(c_8) = [0]
p(c_9) = [1]
p(c_10) = [1]
Following rules are strictly oriented:
fold#3#(insert_ord(x6) = [4] x2 + [4] x4 + [1] x6 + [16]
,Cons(x4,x2))
> [4] x2 + [1] x6 + [12]
= c_4(fold#3#(insert_ord(x6),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:
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(fold#3#(insert_ord(x6),x2))
Weak TRS Rules:
Signature:
{cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}/{0,Cons,False,Nil,S,True,insert_ord,leq}
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:
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(fold#3#(insert_ord(x6),x2))
Weak TRS Rules:
Signature:
{cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}/{0,Cons,False,Nil,S,True,insert_ord,leq}
Applied Processor:
RemoveWeakSuffixes
Proof:
Consider the dependency graph
1:W:fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(fold#3#(insert_ord(x6),x2))
-->_1 fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(fold#3#(insert_ord(x6),x2)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: fold#3#(insert_ord(x6)
,Cons(x4,x2)) ->
c_4(fold#3#(insert_ord(x6),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,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1}
Obligation:
Innermost
basic terms: {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#,main#}/{0,Cons,False,Nil,S,True,insert_ord,leq}
Applied Processor:
EmptyProcessor
Proof:
The problem is already closed. The intended complexity is O(1).