WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
cond_insTree_t_xs_1(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) ->
Cons(cond_link_t1_t2_2(leq#2(x8,x4)
,Node(x54,Elem(x8),x38)
,Node(x30,Elem(x4),x14)
,x54
,Elem(x8)
,x38
,Elem(x4)
,x14)
,Nil())
cond_insTree_t_xs_1(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> Cons(Node(x5,x6,x7),Cons(Node(x2,x3,x4),x1))
cond_link_t1_t2_2(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) ->
Node(S(x5),Elem(x2),Cons(Node(x11,Elem(x10),x9),x1))
cond_link_t1_t2_2(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) ->
Node(S(x5),Elem(x4),Cons(Node(x8,Elem(x7),x6),x3))
leq#2(0(),x20) -> True()
leq#2(S(x24),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
lt#2(0(),0()) -> False()
lt#2(0(),S(x20)) -> True()
lt#2(S(x20),0()) -> False()
lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
main(x4,Nil()) -> Cons(x4,Nil())
main(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> cond_insTree_t_xs_1(lt#2(x100,x48)
,Node(x100,x132,x164)
,Node(x48,x64,x80)
,x28)
- Signature:
{cond_insTree_t_xs_1/4,cond_link_t1_t2_2/8,leq#2/2,lt#2/2,main/2} / {0/0,Cons/2,Elem/1,False/0,Nil/0,Node/3
,S/1,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_insTree_t_xs_1,cond_link_t1_t2_2,leq#2,lt#2
,main} and constructors {0,Cons,Elem,False,Nil,Node,S,True}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:
Strict DPs
cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) ->
c_1(cond_link_t1_t2_2#(leq#2(x8,x4)
,Node(x54,Elem(x8),x38)
,Node(x30,Elem(x4),x14)
,x54
,Elem(x8)
,x38
,Elem(x4)
,x14)
,leq#2#(x8,x4))
cond_insTree_t_xs_1#(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> c_2()
cond_link_t1_t2_2#(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_3()
cond_link_t1_t2_2#(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_4()
leq#2#(0(),x20) -> c_5()
leq#2#(S(x24),0()) -> c_6()
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
lt#2#(0(),0()) -> c_8()
lt#2#(0(),S(x20)) -> c_9()
lt#2#(S(x20),0()) -> c_10()
lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2))
main#(x4,Nil()) -> c_12()
main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48)
,Node(x100,x132,x164)
,Node(x48,x64,x80)
,x28)
,lt#2#(x100,x48))
Weak DPs
and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) ->
c_1(cond_link_t1_t2_2#(leq#2(x8,x4)
,Node(x54,Elem(x8),x38)
,Node(x30,Elem(x4),x14)
,x54
,Elem(x8)
,x38
,Elem(x4)
,x14)
,leq#2#(x8,x4))
cond_insTree_t_xs_1#(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> c_2()
cond_link_t1_t2_2#(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_3()
cond_link_t1_t2_2#(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_4()
leq#2#(0(),x20) -> c_5()
leq#2#(S(x24),0()) -> c_6()
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
lt#2#(0(),0()) -> c_8()
lt#2#(0(),S(x20)) -> c_9()
lt#2#(S(x20),0()) -> c_10()
lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2))
main#(x4,Nil()) -> c_12()
main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48)
,Node(x100,x132,x164)
,Node(x48,x64,x80)
,x28)
,lt#2#(x100,x48))
- Weak TRS:
cond_insTree_t_xs_1(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) ->
Cons(cond_link_t1_t2_2(leq#2(x8,x4)
,Node(x54,Elem(x8),x38)
,Node(x30,Elem(x4),x14)
,x54
,Elem(x8)
,x38
,Elem(x4)
,x14)
,Nil())
cond_insTree_t_xs_1(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> Cons(Node(x5,x6,x7),Cons(Node(x2,x3,x4),x1))
cond_link_t1_t2_2(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) ->
Node(S(x5),Elem(x2),Cons(Node(x11,Elem(x10),x9),x1))
cond_link_t1_t2_2(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) ->
Node(S(x5),Elem(x4),Cons(Node(x8,Elem(x7),x6),x3))
leq#2(0(),x20) -> True()
leq#2(S(x24),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
lt#2(0(),0()) -> False()
lt#2(0(),S(x20)) -> True()
lt#2(S(x20),0()) -> False()
lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
main(x4,Nil()) -> Cons(x4,Nil())
main(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> cond_insTree_t_xs_1(lt#2(x100,x48)
,Node(x100,x132,x164)
,Node(x48,x64,x80)
,x28)
- Signature:
{cond_insTree_t_xs_1/4,cond_link_t1_t2_2/8,leq#2/2,lt#2/2,main/2,cond_insTree_t_xs_1#/4,cond_link_t1_t2_2#/8
,leq#2#/2,lt#2#/2,main#/2} / {0/0,Cons/2,Elem/1,False/0,Nil/0,Node/3,S/1,True/0,c_1/2,c_2/0,c_3/0,c_4/0
,c_5/0,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_insTree_t_xs_1#,cond_link_t1_t2_2#,leq#2#,lt#2#
,main#} and constructors {0,Cons,Elem,False,Nil,Node,S,True}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{2,3,4,5,6,8,9,10,12}
by application of
Pre({2,3,4,5,6,8,9,10,12}) = {1,7,11,13}.
Here rules are labelled as follows:
1: cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) ->
c_1(cond_link_t1_t2_2#(leq#2(x8,x4)
,Node(x54,Elem(x8),x38)
,Node(x30,Elem(x4),x14)
,x54
,Elem(x8)
,x38
,Elem(x4)
,x14)
,leq#2#(x8,x4))
2: cond_insTree_t_xs_1#(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> c_2()
3: cond_link_t1_t2_2#(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) ->
c_3()
4: cond_link_t1_t2_2#(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) ->
c_4()
5: leq#2#(0(),x20) -> c_5()
6: leq#2#(S(x24),0()) -> c_6()
7: leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
8: lt#2#(0(),0()) -> c_8()
9: lt#2#(0(),S(x20)) -> c_9()
10: lt#2#(S(x20),0()) -> c_10()
11: lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2))
12: main#(x4,Nil()) -> c_12()
13: main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48)
,Node(x100,x132,x164)
,Node(x48,x64,x80)
,x28)
,lt#2#(x100,x48))
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) ->
c_1(cond_link_t1_t2_2#(leq#2(x8,x4)
,Node(x54,Elem(x8),x38)
,Node(x30,Elem(x4),x14)
,x54
,Elem(x8)
,x38
,Elem(x4)
,x14)
,leq#2#(x8,x4))
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2))
main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48)
,Node(x100,x132,x164)
,Node(x48,x64,x80)
,x28)
,lt#2#(x100,x48))
- Weak DPs:
cond_insTree_t_xs_1#(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> c_2()
cond_link_t1_t2_2#(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_3()
cond_link_t1_t2_2#(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) -> c_4()
leq#2#(0(),x20) -> c_5()
leq#2#(S(x24),0()) -> c_6()
lt#2#(0(),0()) -> c_8()
lt#2#(0(),S(x20)) -> c_9()
lt#2#(S(x20),0()) -> c_10()
main#(x4,Nil()) -> c_12()
- Weak TRS:
cond_insTree_t_xs_1(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) ->
Cons(cond_link_t1_t2_2(leq#2(x8,x4)
,Node(x54,Elem(x8),x38)
,Node(x30,Elem(x4),x14)
,x54
,Elem(x8)
,x38
,Elem(x4)
,x14)
,Nil())
cond_insTree_t_xs_1(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> Cons(Node(x5,x6,x7),Cons(Node(x2,x3,x4),x1))
cond_link_t1_t2_2(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) ->
Node(S(x5),Elem(x2),Cons(Node(x11,Elem(x10),x9),x1))
cond_link_t1_t2_2(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) ->
Node(S(x5),Elem(x4),Cons(Node(x8,Elem(x7),x6),x3))
leq#2(0(),x20) -> True()
leq#2(S(x24),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
lt#2(0(),0()) -> False()
lt#2(0(),S(x20)) -> True()
lt#2(S(x20),0()) -> False()
lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
main(x4,Nil()) -> Cons(x4,Nil())
main(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> cond_insTree_t_xs_1(lt#2(x100,x48)
,Node(x100,x132,x164)
,Node(x48,x64,x80)
,x28)
- Signature:
{cond_insTree_t_xs_1/4,cond_link_t1_t2_2/8,leq#2/2,lt#2/2,main/2,cond_insTree_t_xs_1#/4,cond_link_t1_t2_2#/8
,leq#2#/2,lt#2#/2,main#/2} / {0/0,Cons/2,Elem/1,False/0,Nil/0,Node/3,S/1,True/0,c_1/2,c_2/0,c_3/0,c_4/0
,c_5/0,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_insTree_t_xs_1#,cond_link_t1_t2_2#,leq#2#,lt#2#
,main#} and constructors {0,Cons,Elem,False,Nil,Node,S,True}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) ->
c_1(cond_link_t1_t2_2#(leq#2(x8,x4)
,Node(x54,Elem(x8),x38)
,Node(x30,Elem(x4),x14)
,x54
,Elem(x8)
,x38
,Elem(x4)
,x14)
,leq#2#(x8,x4))
-->_2 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):2
-->_2 leq#2#(S(x24),0()) -> c_6():9
-->_2 leq#2#(0(),x20) -> c_5():8
-->_1 cond_link_t1_t2_2#(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) ->
c_4():7
-->_1 cond_link_t1_t2_2#(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) ->
c_3():6
2:S:leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
-->_1 leq#2#(S(x24),0()) -> c_6():9
-->_1 leq#2#(0(),x20) -> c_5():8
-->_1 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):2
3:S:lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2))
-->_1 lt#2#(S(x20),0()) -> c_10():12
-->_1 lt#2#(0(),S(x20)) -> c_9():11
-->_1 lt#2#(0(),0()) -> c_8():10
-->_1 lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)):3
4:S:main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48)
,Node(x100,x132,x164)
,Node(x48,x64,x80)
,x28)
,lt#2#(x100,x48))
-->_2 lt#2#(S(x20),0()) -> c_10():12
-->_2 lt#2#(0(),S(x20)) -> c_9():11
-->_2 lt#2#(0(),0()) -> c_8():10
-->_1 cond_insTree_t_xs_1#(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> c_2():5
-->_2 lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)):3
-->_1 cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) ->
c_1(cond_link_t1_t2_2#(leq#2(x8,x4)
,Node(x54,Elem(x8),x38)
,Node(x30,Elem(x4),x14)
,x54
,Elem(x8)
,x38
,Elem(x4)
,x14)
,leq#2#(x8,x4)):1
5:W:cond_insTree_t_xs_1#(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> c_2()
6:W:cond_link_t1_t2_2#(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) ->
c_3()
7:W:cond_link_t1_t2_2#(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) ->
c_4()
8:W:leq#2#(0(),x20) -> c_5()
9:W:leq#2#(S(x24),0()) -> c_6()
10:W:lt#2#(0(),0()) -> c_8()
11:W:lt#2#(0(),S(x20)) -> c_9()
12:W:lt#2#(S(x20),0()) -> c_10()
13:W:main#(x4,Nil()) -> c_12()
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
13: main#(x4,Nil()) -> c_12()
5: cond_insTree_t_xs_1#(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> c_2()
10: lt#2#(0(),0()) -> c_8()
11: lt#2#(0(),S(x20)) -> c_9()
12: lt#2#(S(x20),0()) -> c_10()
6: cond_link_t1_t2_2#(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) ->
c_3()
7: cond_link_t1_t2_2#(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) ->
c_4()
8: leq#2#(0(),x20) -> c_5()
9: leq#2#(S(x24),0()) -> c_6()
* Step 4: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) ->
c_1(cond_link_t1_t2_2#(leq#2(x8,x4)
,Node(x54,Elem(x8),x38)
,Node(x30,Elem(x4),x14)
,x54
,Elem(x8)
,x38
,Elem(x4)
,x14)
,leq#2#(x8,x4))
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2))
main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48)
,Node(x100,x132,x164)
,Node(x48,x64,x80)
,x28)
,lt#2#(x100,x48))
- Weak TRS:
cond_insTree_t_xs_1(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) ->
Cons(cond_link_t1_t2_2(leq#2(x8,x4)
,Node(x54,Elem(x8),x38)
,Node(x30,Elem(x4),x14)
,x54
,Elem(x8)
,x38
,Elem(x4)
,x14)
,Nil())
cond_insTree_t_xs_1(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> Cons(Node(x5,x6,x7),Cons(Node(x2,x3,x4),x1))
cond_link_t1_t2_2(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) ->
Node(S(x5),Elem(x2),Cons(Node(x11,Elem(x10),x9),x1))
cond_link_t1_t2_2(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) ->
Node(S(x5),Elem(x4),Cons(Node(x8,Elem(x7),x6),x3))
leq#2(0(),x20) -> True()
leq#2(S(x24),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
lt#2(0(),0()) -> False()
lt#2(0(),S(x20)) -> True()
lt#2(S(x20),0()) -> False()
lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
main(x4,Nil()) -> Cons(x4,Nil())
main(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> cond_insTree_t_xs_1(lt#2(x100,x48)
,Node(x100,x132,x164)
,Node(x48,x64,x80)
,x28)
- Signature:
{cond_insTree_t_xs_1/4,cond_link_t1_t2_2/8,leq#2/2,lt#2/2,main/2,cond_insTree_t_xs_1#/4,cond_link_t1_t2_2#/8
,leq#2#/2,lt#2#/2,main#/2} / {0/0,Cons/2,Elem/1,False/0,Nil/0,Node/3,S/1,True/0,c_1/2,c_2/0,c_3/0,c_4/0
,c_5/0,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_insTree_t_xs_1#,cond_link_t1_t2_2#,leq#2#,lt#2#
,main#} and constructors {0,Cons,Elem,False,Nil,Node,S,True}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) ->
c_1(cond_link_t1_t2_2#(leq#2(x8,x4)
,Node(x54,Elem(x8),x38)
,Node(x30,Elem(x4),x14)
,x54
,Elem(x8)
,x38
,Elem(x4)
,x14)
,leq#2#(x8,x4))
-->_2 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):2
2:S:leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
-->_1 leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2)):2
3:S:lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2))
-->_1 lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)):3
4:S:main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48)
,Node(x100,x132,x164)
,Node(x48,x64,x80)
,x28)
,lt#2#(x100,x48))
-->_2 lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2)):3
-->_1 cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) ->
c_1(cond_link_t1_t2_2#(leq#2(x8,x4)
,Node(x54,Elem(x8),x38)
,Node(x30,Elem(x4),x14)
,x54
,Elem(x8)
,x38
,Elem(x4)
,x14)
,leq#2#(x8,x4)):1
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(leq#2#(x8,x4))
* Step 5: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(leq#2#(x8,x4))
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2))
main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48)
,Node(x100,x132,x164)
,Node(x48,x64,x80)
,x28)
,lt#2#(x100,x48))
- Weak TRS:
cond_insTree_t_xs_1(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) ->
Cons(cond_link_t1_t2_2(leq#2(x8,x4)
,Node(x54,Elem(x8),x38)
,Node(x30,Elem(x4),x14)
,x54
,Elem(x8)
,x38
,Elem(x4)
,x14)
,Nil())
cond_insTree_t_xs_1(True(),Node(x5,x6,x7),Node(x2,x3,x4),x1) -> Cons(Node(x5,x6,x7),Cons(Node(x2,x3,x4),x1))
cond_link_t1_t2_2(False(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) ->
Node(S(x5),Elem(x2),Cons(Node(x11,Elem(x10),x9),x1))
cond_link_t1_t2_2(True(),Node(x11,Elem(x10),x9),Node(x8,Elem(x7),x6),x5,Elem(x4),x3,Elem(x2),x1) ->
Node(S(x5),Elem(x4),Cons(Node(x8,Elem(x7),x6),x3))
leq#2(0(),x20) -> True()
leq#2(S(x24),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
lt#2(0(),0()) -> False()
lt#2(0(),S(x20)) -> True()
lt#2(S(x20),0()) -> False()
lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
main(x4,Nil()) -> Cons(x4,Nil())
main(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> cond_insTree_t_xs_1(lt#2(x100,x48)
,Node(x100,x132,x164)
,Node(x48,x64,x80)
,x28)
- Signature:
{cond_insTree_t_xs_1/4,cond_link_t1_t2_2/8,leq#2/2,lt#2/2,main/2,cond_insTree_t_xs_1#/4,cond_link_t1_t2_2#/8
,leq#2#/2,lt#2#/2,main#/2} / {0/0,Cons/2,Elem/1,False/0,Nil/0,Node/3,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0
,c_5/0,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_insTree_t_xs_1#,cond_link_t1_t2_2#,leq#2#,lt#2#
,main#} and constructors {0,Cons,Elem,False,Nil,Node,S,True}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
lt#2(0(),0()) -> False()
lt#2(0(),S(x20)) -> True()
lt#2(S(x20),0()) -> False()
lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(leq#2#(x8,x4))
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2))
main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48)
,Node(x100,x132,x164)
,Node(x48,x64,x80)
,x28)
,lt#2#(x100,x48))
* Step 6: Ara WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(leq#2#(x8,x4))
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2))
main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48)
,Node(x100,x132,x164)
,Node(x48,x64,x80)
,x28)
,lt#2#(x100,x48))
- Weak TRS:
lt#2(0(),0()) -> False()
lt#2(0(),S(x20)) -> True()
lt#2(S(x20),0()) -> False()
lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
- Signature:
{cond_insTree_t_xs_1/4,cond_link_t1_t2_2/8,leq#2/2,lt#2/2,main/2,cond_insTree_t_xs_1#/4,cond_link_t1_t2_2#/8
,leq#2#/2,lt#2#/2,main#/2} / {0/0,Cons/2,Elem/1,False/0,Nil/0,Node/3,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0
,c_5/0,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_insTree_t_xs_1#,cond_link_t1_t2_2#,leq#2#,lt#2#
,main#} and constructors {0,Cons,Elem,False,Nil,Node,S,True}
+ Applied Processor:
Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1}
+ Details:
Signatures used:
----------------
0 :: [] -(0)-> "A"(0)
Cons :: ["A"(1) x "A"(0)] -(0)-> "A"(1)
Elem :: ["A"(0)] -(0)-> "A"(0)
False :: [] -(0)-> "A"(0)
Nil :: [] -(0)-> "A"(0)
Node :: ["A"(0) x "A"(0) x "A"(0)] -(0)-> "A"(0)
Node :: ["A"(1) x "A"(0) x "A"(0)] -(0)-> "A"(1)
S :: ["A"(0)] -(0)-> "A"(0)
S :: ["A"(1)] -(1)-> "A"(1)
True :: [] -(0)-> "A"(0)
lt#2 :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
cond_insTree_t_xs_1# :: ["A"(0) x "A"(0) x "A"(0) x "A"(0)] -(1)-> "A"(0)
leq#2# :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
lt#2# :: ["A"(0) x "A"(1)] -(0)-> "A"(0)
main# :: ["A"(0) x "A"(1)] -(2)-> "A"(0)
c_1 :: ["A"(0)] -(0)-> "A"(0)
c_7 :: ["A"(0)] -(0)-> "A"(0)
c_11 :: ["A"(0)] -(0)-> "A"(0)
c_13 :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
Cost-free Signatures used:
--------------------------
Base Constructor Signatures used:
---------------------------------
Following Still Strict Rules were Typed as:
-------------------------------------------
1. Strict:
cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(leq#2#(x8,x4))
lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2))
main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48)
,Node(x100,x132,x164)
,Node(x48,x64,x80)
,x28)
,lt#2#(x100,x48))
2. Weak:
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
* Step 7: Ara WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
- Weak DPs:
cond_insTree_t_xs_1#(False(),Node(x54,Elem(x8),x38),Node(x30,Elem(x4),x14),Nil()) -> c_1(leq#2#(x8,x4))
lt#2#(S(x4),S(x2)) -> c_11(lt#2#(x4,x2))
main#(Node(x100,x132,x164),Cons(Node(x48,x64,x80),x28)) -> c_13(cond_insTree_t_xs_1#(lt#2(x100,x48)
,Node(x100,x132,x164)
,Node(x48,x64,x80)
,x28)
,lt#2#(x100,x48))
- Weak TRS:
lt#2(0(),0()) -> False()
lt#2(0(),S(x20)) -> True()
lt#2(S(x20),0()) -> False()
lt#2(S(x4),S(x2)) -> lt#2(x4,x2)
- Signature:
{cond_insTree_t_xs_1/4,cond_link_t1_t2_2/8,leq#2/2,lt#2/2,main/2,cond_insTree_t_xs_1#/4,cond_link_t1_t2_2#/8
,leq#2#/2,lt#2#/2,main#/2} / {0/0,Cons/2,Elem/1,False/0,Nil/0,Node/3,S/1,True/0,c_1/1,c_2/0,c_3/0,c_4/0
,c_5/0,c_6/0,c_7/1,c_8/0,c_9/0,c_10/0,c_11/1,c_12/0,c_13/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_insTree_t_xs_1#,cond_link_t1_t2_2#,leq#2#,lt#2#
,main#} and constructors {0,Cons,Elem,False,Nil,Node,S,True}
+ Applied Processor:
Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1}
+ Details:
Signatures used:
----------------
0 :: [] -(0)-> "A"(1)
0 :: [] -(0)-> "A"(8)
Cons :: ["A"(15) x "A"(15)] -(15)-> "A"(15)
Elem :: ["A"(12)] -(0)-> "A"(12)
Elem :: ["A"(1)] -(0)-> "A"(1)
False :: [] -(0)-> "A"(0)
False :: [] -(0)-> "A"(14)
Nil :: [] -(0)-> "A"(0)
Node :: ["A"(12) x "A"(12) x "A"(0)] -(0)-> "A"(12)
Node :: ["A"(1) x "A"(1) x "A"(0)] -(0)-> "A"(1)
Node :: ["A"(15) x "A"(15) x "A"(0)] -(0)-> "A"(15)
Node :: ["A"(5) x "A"(5) x "A"(0)] -(0)-> "A"(5)
S :: ["A"(12)] -(12)-> "A"(12)
S :: ["A"(1)] -(1)-> "A"(1)
S :: ["A"(8)] -(8)-> "A"(8)
S :: ["A"(2)] -(2)-> "A"(2)
True :: [] -(0)-> "A"(14)
lt#2 :: ["A"(1) x "A"(8)] -(1)-> "A"(0)
cond_insTree_t_xs_1# :: ["A"(0) x "A"(12) x "A"(1) x "A"(0)] -(2)-> "A"(0)
leq#2# :: ["A"(12) x "A"(1)] -(1)-> "A"(0)
lt#2# :: ["A"(1) x "A"(2)] -(5)-> "A"(0)
main# :: ["A"(15) x "A"(15)] -(15)-> "A"(0)
c_1 :: ["A"(0)] -(0)-> "A"(12)
c_7 :: ["A"(0)] -(0)-> "A"(0)
c_11 :: ["A"(0)] -(0)-> "A"(0)
c_13 :: ["A"(0) x "A"(0)] -(0)-> "A"(14)
Cost-free Signatures used:
--------------------------
Base Constructor Signatures used:
---------------------------------
"0_A" :: [] -(0)-> "A"(1)
"Cons_A" :: ["A"(1) x "A"(1)] -(1)-> "A"(1)
"Elem_A" :: ["A"(1)] -(0)-> "A"(1)
"False_A" :: [] -(0)-> "A"(1)
"Nil_A" :: [] -(0)-> "A"(1)
"Node_A" :: ["A"(1) x "A"(1) x "A"(0)] -(0)-> "A"(1)
"S_A" :: ["A"(1)] -(1)-> "A"(1)
"True_A" :: [] -(0)-> "A"(1)
"c_11_A" :: ["A"(0)] -(0)-> "A"(1)
"c_13_A" :: ["A"(0) x "A"(0)] -(0)-> "A"(1)
"c_1_A" :: ["A"(0)] -(0)-> "A"(1)
"c_7_A" :: ["A"(0)] -(0)-> "A"(1)
Following Still Strict Rules were Typed as:
-------------------------------------------
1. Strict:
leq#2#(S(x4),S(x2)) -> c_7(leq#2#(x4,x2))
2. Weak:
WORST_CASE(?,O(n^1))