* Step 1: Sum WORST_CASE(Omega(n^1),O(n^2))
+ Considered Problem:
- Strict TRS:
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} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0
,insert_ord/1,leq/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1,fold#3,insert_ord#2,leq#2
,main} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
+ Applied Processor:
Sum {left = someStrategy, right = someStrategy}
+ Details:
()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
+ Considered Problem:
- Strict TRS:
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} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0
,insert_ord/1,leq/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1,fold#3,insert_ord#2,leq#2
,main} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
+ Applied Processor:
DecreasingLoops {bound = AnyLoop, narrow = 10}
+ Details:
The system has following decreasing Loops:
fold#3(insert_ord(x),z){z -> Cons(y,z)} =
fold#3(insert_ord(x),Cons(y,z)) ->^+ insert_ord#2(x,y,fold#3(insert_ord(x),z))
= C[fold#3(insert_ord(x),z) = fold#3(insert_ord(x),z){}]
** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
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} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0
,insert_ord/1,leq/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1,fold#3,insert_ord#2,leq#2
,main} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
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.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^2))
+ Considered Problem:
- 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 TRS:
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 runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
+ Applied Processor:
UsableRules
+ Details:
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))
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^2))
+ Considered Problem:
- 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 TRS:
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 runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
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))
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
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))
- Weak DPs:
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:
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 runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
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()
** Step 1.b:5: RemoveHeads WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
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))
- Weak TRS:
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 runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
+ Applied Processor:
RemoveHeads
+ Details:
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)))]
** Step 1.b:6: Decompose WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
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:
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 runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
+ Applied Processor:
Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
+ Details:
We analyse the complexity of following sub-problems (R) and (S).
Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
Problem (R)
- Strict DPs:
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 DPs:
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:
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 runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
Problem (S)
- Strict DPs:
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 DPs:
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:
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 runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
*** Step 1.b:6.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
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 DPs:
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:
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 runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {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}}
+ Details:
We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
4: leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
The strictly oriented rules are moved into the weak component.
**** Step 1.b:6.a:1.a:1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
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 DPs:
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:
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 runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {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 on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(c_1) = {1},
uargs(c_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) = 0
p(Cons) = 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(fold#3) = 2*x2
p(insert_ord) = 0
p(insert_ord#2) = 2*x2 + x3
p(leq) = 0
p(leq#2) = 0
p(main) = 2 + 2*x1^2
p(cond_insert_ord_x_ys_1#) = x2*x3 + 3*x2*x4
p(fold#3#) = 2 + 2*x1 + 3*x2^2
p(insert_ord#2#) = 3*x2*x3
p(leq#2#) = 2*x1*x2
p(main#) = 2 + 2*x1
p(c_1) = x1
p(c_2) = 0
p(c_3) = 1
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) = 0
Following rules are strictly oriented:
leq#2#(S(x4),S(x2)) = 2 + 2*x2 + 2*x2*x4 + 2*x4
> 2*x2*x4
= c_9(leq#2#(x4,x2))
Following rules are (at-least) weakly oriented:
cond_insert_ord_x_ys_1#(False(),x0,x5,x2) = 3*x0*x2 + x0*x5
>= 3*x0*x2
= c_1(insert_ord#2#(leq(),x0,x2))
fold#3#(insert_ord(x6),Cons(x4,x2)) = 2 + 6*x2*x4 + 3*x2^2 + 3*x4^2
>= 2 + 6*x2*x4 + 3*x2^2
= 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)) = 3*x2*x6 + 3*x4*x6
>= 3*x2*x6 + 3*x4*x6
= c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4))
cond_insert_ord_x_ys_1(False(),x0,x5,x2) = 2*x0 + x2 + x5
>= 2*x0 + x2 + x5
= Cons(x5,insert_ord#2(leq(),x0,x2))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) = x1 + x2 + 2*x3
>= x1 + x2 + x3
= Cons(x3,Cons(x2,x1))
fold#3(insert_ord(x2),Nil()) = 0
>= 0
= Nil()
fold#3(insert_ord(x6),Cons(x4,x2)) = 2*x2 + 2*x4
>= 2*x2 + 2*x4
= insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2))
insert_ord#2(leq(),x2,Nil()) = 2*x2
>= x2
= Cons(x2,Nil())
insert_ord#2(leq(),x6,Cons(x4,x2)) = x2 + x4 + 2*x6
>= x2 + x4 + 2*x6
= cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2)
**** Step 1.b:6.a:1.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
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))
- Weak DPs:
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))
leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
- Weak TRS:
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 runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 1.b:6.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
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))
- Weak DPs:
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))
leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
- Weak TRS:
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 runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
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)):2
2: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
3: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))
-->_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
-->_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)):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.
4: leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2))
**** Step 1.b:6.a:1.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
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))
- Weak DPs:
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:
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 runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
+ Applied Processor:
SimplifyRHS
+ Details:
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)):2
2: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))
-->_1 cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)):1
3: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))
-->_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
-->_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)):2
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2))
**** Step 1.b:6.a:1.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
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))
- Weak DPs:
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:
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_6/1,c_7/0,c_8/0,c_9/1,c_10/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {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 = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
2: insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,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: insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2))
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))
Processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2))
SPACE(?,?)on application of the dependency pairs
{2}
These cover all (indirect) predecessors of dependency pairs
{1,2}
their number of applications is equally bounded.
The dependency pairs are shifted into the weak component.
***** Step 1.b:6.a:1.b:3.a:1: NaturalMI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict DPs:
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))
- Weak DPs:
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:
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_6/1,c_7/0,c_8/0,c_9/1,c_10/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
+ Applied Processor:
NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
The following argument positions are considered usable:
uargs(c_1) = {1},
uargs(c_4) = {1,2},
uargs(c_6) = {1}
Following symbols are considered usable:
{cond_insert_ord_x_ys_1,fold#3,insert_ord#2,leq#2,cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#}
TcT has computed the following interpretation:
p(0) = [0]
[0]
[0]
p(Cons) = [0 1 0] [1 1 0] [0]
[0 0 0] x1 + [0 0 1] x2 + [0]
[0 0 0] [0 0 1] [1]
p(False) = [1]
[0]
[0]
p(Nil) = [0]
[0]
[1]
p(S) = [1 1 0] [1]
[0 0 0] x1 + [0]
[0 0 0] [0]
p(True) = [1]
[0]
[0]
p(cond_insert_ord_x_ys_1) = [0 0 0] [0 1 0] [0 1 0] [1 1 1] [0]
[0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0 0 1] x4 + [1]
[1 0 0] [0 0 0] [0 0 0] [0 0 1] [1]
p(fold#3) = [1 0 0] [0]
[0 1 0] x2 + [0]
[0 0 1] [0]
p(insert_ord) = [1 0 1] [0]
[0 0 1] x1 + [0]
[0 0 1] [0]
p(insert_ord#2) = [0 1 0] [1 1 0] [0]
[0 0 0] x2 + [0 0 1] x3 + [0]
[0 0 0] [0 0 1] [1]
p(leq) = [0]
[0]
[0]
p(leq#2) = [0 0 0] [1]
[1 0 0] x2 + [0]
[0 0 0] [1]
p(main) = [0]
[0]
[0]
p(cond_insert_ord_x_ys_1#) = [0 0 0] [0 0 1] [0]
[0 1 0] x2 + [1 0 0] x4 + [0]
[0 0 0] [0 1 1] [0]
p(fold#3#) = [0 0 1] [1 1 0] [0]
[0 1 0] x1 + [1 0 1] x2 + [1]
[1 0 0] [1 1 1] [0]
p(insert_ord#2#) = [0 0 0] [0 0 0] [0 0 1] [0]
[1 0 1] x1 + [1 1 1] x2 + [1 1 1] x3 + [0]
[0 0 0] [0 1 0] [0 1 1] [0]
p(leq#2#) = [0]
[0]
[0]
p(main#) = [0]
[0]
[0]
p(c_1) = [1 0 0] [0]
[0 0 0] x1 + [0]
[0 0 0] [0]
p(c_2) = [0]
[0]
[0]
p(c_3) = [0]
[0]
[0]
p(c_4) = [1 0 0] [1 0 0] [0]
[0 0 0] x1 + [0 0 0] x2 + [0]
[0 0 1] [0 1 0] [0]
p(c_5) = [0]
[0]
[0]
p(c_6) = [1 0 0] [0]
[0 1 1] x1 + [1]
[0 0 0] [0]
p(c_7) = [0]
[0]
[0]
p(c_8) = [0]
[0]
[0]
p(c_9) = [0]
[0]
[0]
p(c_10) = [0]
[0]
[0]
Following rules are strictly oriented:
insert_ord#2#(leq(),x6,Cons(x4,x2)) = [0 0 1] [0 0 0] [0 0 0] [1]
[1 1 2] x2 + [0 1 0] x4 + [1 1 1] x6 + [1]
[0 0 2] [0 0 0] [0 1 0] [1]
> [0 0 1] [0 0 0] [0]
[1 1 1] x2 + [0 1 0] x6 + [1]
[0 0 0] [0 0 0] [0]
= c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2))
Following rules are (at-least) weakly oriented:
cond_insert_ord_x_ys_1#(False(),x0,x5,x2) = [0 0 0] [0 0 1] [0]
[0 1 0] x0 + [1 0 0] x2 + [0]
[0 0 0] [0 1 1] [0]
>= [0 0 1] [0]
[0 0 0] x2 + [0]
[0 0 0] [0]
= c_1(insert_ord#2#(leq(),x0,x2))
fold#3#(insert_ord(x6),Cons(x4,x2)) = [1 1 1] [0 1 0] [0 0 1] [0]
[1 1 1] x2 + [0 1 0] x4 + [0 0 1] x6 + [2]
[1 1 2] [0 1 0] [1 0 1] [1]
>= [1 1 1] [0 0 0] [0 0 1] [0]
[0 0 0] x2 + [0 0 0] x4 + [0 0 0] x6 + [0]
[1 1 2] [0 1 0] [0 0 1] [1]
= 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(),x0,x5,x2) = [0 1 0] [1 1 1] [0 1 0] [0]
[0 0 0] x0 + [0 0 1] x2 + [0 0 0] x5 + [1]
[0 0 0] [0 0 1] [0 0 0] [2]
>= [0 1 0] [1 1 1] [0 1 0] [0]
[0 0 0] x0 + [0 0 1] x2 + [0 0 0] x5 + [1]
[0 0 0] [0 0 1] [0 0 0] [2]
= Cons(x5,insert_ord#2(leq(),x0,x2))
cond_insert_ord_x_ys_1(True(),x3,x2,x1) = [1 1 1] [0 1 0] [0 1 0] [0]
[0 0 1] x1 + [0 0 0] x2 + [0 0 0] x3 + [1]
[0 0 1] [0 0 0] [0 0 0] [2]
>= [1 1 1] [0 1 0] [0 1 0] [0]
[0 0 1] x1 + [0 0 0] x2 + [0 0 0] x3 + [1]
[0 0 1] [0 0 0] [0 0 0] [2]
= Cons(x3,Cons(x2,x1))
fold#3(insert_ord(x2),Nil()) = [0]
[0]
[1]
>= [0]
[0]
[1]
= Nil()
fold#3(insert_ord(x6),Cons(x4,x2)) = [1 1 0] [0 1 0] [0]
[0 0 1] x2 + [0 0 0] x4 + [0]
[0 0 1] [0 0 0] [1]
>= [1 1 0] [0 1 0] [0]
[0 0 1] x2 + [0 0 0] x4 + [0]
[0 0 1] [0 0 0] [1]
= insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2))
insert_ord#2(leq(),x2,Nil()) = [0 1 0] [0]
[0 0 0] x2 + [1]
[0 0 0] [2]
>= [0 1 0] [0]
[0 0 0] x2 + [1]
[0 0 0] [2]
= Cons(x2,Nil())
insert_ord#2(leq(),x6,Cons(x4,x2)) = [1 1 1] [0 1 0] [0 1 0] [0]
[0 0 1] x2 + [0 0 0] x4 + [0 0 0] x6 + [1]
[0 0 1] [0 0 0] [0 0 0] [2]
>= [1 1 1] [0 1 0] [0 1 0] [0]
[0 0 1] x2 + [0 0 0] x4 + [0 0 0] x6 + [1]
[0 0 1] [0 0 0] [0 0 0] [2]
= cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2)
leq#2(0(),x8) = [0 0 0] [1]
[1 0 0] x8 + [0]
[0 0 0] [1]
>= [1]
[0]
[0]
= True()
leq#2(S(x12),0()) = [1]
[0]
[1]
>= [1]
[0]
[0]
= False()
leq#2(S(x4),S(x2)) = [0 0 0] [1]
[1 1 0] x2 + [1]
[0 0 0] [1]
>= [0 0 0] [1]
[1 0 0] x2 + [0]
[0 0 0] [1]
= leq#2(x4,x2)
***** Step 1.b:6.a:1.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2))
- Weak DPs:
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))
- Weak TRS:
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_6/1,c_7/0,c_8/0,c_9/1,c_10/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
***** Step 1.b:6.a:1.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
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))
- Weak TRS:
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_6/1,c_7/0,c_8/0,c_9/1,c_10/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
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)):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)):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))
-->_1 cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)):1
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))
***** Step 1.b:6.a:1.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
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_6/1,c_7/0,c_8/0,c_9/1,c_10/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 1.b:6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
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 DPs:
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:
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 runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
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))
*** Step 1.b:6.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
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:
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 runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
+ Applied Processor:
SimplifyRHS
+ Details:
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))
*** Step 1.b:6.b:3: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(fold#3#(insert_ord(x6),x2))
- Weak TRS:
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 runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(fold#3#(insert_ord(x6),x2))
*** Step 1.b:6.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(fold#3#(insert_ord(x6),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 runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {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}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: 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.
**** Step 1.b:6.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(fold#3#(insert_ord(x6),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 runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {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 on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_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 + [2]
p(False) = [0]
p(Nil) = [0]
p(S) = [1] x1 + [0]
p(True) = [0]
p(cond_insert_ord_x_ys_1) = [0]
p(fold#3) = [0]
p(insert_ord) = [1] x1 + [0]
p(insert_ord#2) = [1] x1 + [2] x2 + [0]
p(leq) = [4]
p(leq#2) = [8] x1 + [1]
p(main) = [1] x1 + [0]
p(cond_insert_ord_x_ys_1#) = [1] x3 + [1] x4 + [1]
p(fold#3#) = [8] x2 + [0]
p(insert_ord#2#) = [1] x2 + [0]
p(leq#2#) = [2] x1 + [1]
p(main#) = [1] x1 + [1]
p(c_1) = [1]
p(c_2) = [1]
p(c_3) = [4]
p(c_4) = [1] x1 + [15]
p(c_5) = [1]
p(c_6) = [1] x1 + [1] x2 + [1]
p(c_7) = [0]
p(c_8) = [2]
p(c_9) = [2] x1 + [1]
p(c_10) = [8] x1 + [8]
Following rules are strictly oriented:
fold#3#(insert_ord(x6),Cons(x4,x2)) = [8] x2 + [8] x4 + [16]
> [8] x2 + [15]
= c_4(fold#3#(insert_ord(x6),x2))
Following rules are (at-least) weakly oriented:
**** Step 1.b:6.b:4.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(fold#3#(insert_ord(x6),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 runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 1.b:6.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(fold#3#(insert_ord(x6),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 runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
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))
**** Step 1.b:6.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- 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 runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2#
,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(Omega(n^1),O(n^2))