WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
main(x2,x1) -> partition#2(x1,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2} / {0/0,E/0,Elem/1,False/0,Pair/2
,S/1,T/3,True/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1,cond_partition_pivot_heap_3
,cond_partition_pivot_heap_4,cond_partition_pivot_heap_5,cond_partition_pivot_heap_7
,cond_partition_pivot_heap_8,cond_partition_pivot_heap_9,leq#2,leqElem#2,main
,partition#2} and constructors {0,E,Elem,False,Pair,S,T,True}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:
Strict DPs
cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) -> c_1()
cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_2(cond_partition_pivot_heap_7#(leqElem#2(x4
,x6)
,x6
,x2
,x1
,x5
,x4
,x3))
cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) -> c_3()
cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_4(cond_partition_pivot_heap_3#(leqElem#2(x2
,x6)
,x6
,x5
,x4
,x3
,x2
,x1))
cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_5(cond_partition_pivot_heap_5#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_6(cond_partition_pivot_heap_4#(partition#2(x6
,x1)
,x5
,x4
,x3
,x2))
cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) -> c_7()
cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9(cond_partition_pivot_heap_9#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_10(cond_partition_pivot_heap_8#(partition#2(x6
,x1)
,x5
,x4
,x3
,x2))
cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) -> c_11()
cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) -> c_12()
leq#2#(0(),x12) -> c_13()
leq#2#(S(x16),0()) -> c_14()
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
leqElem#2#(Elem(x4),Elem(x2)) -> c_16(leq#2#(x4,x2))
main#(x2,x1) -> c_17(partition#2#(x1,x2))
partition#2#(x2,E()) -> c_18()
partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
Weak DPs
and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) -> c_1()
cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_2(cond_partition_pivot_heap_7#(leqElem#2(x4
,x6)
,x6
,x2
,x1
,x5
,x4
,x3))
cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) -> c_3()
cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_4(cond_partition_pivot_heap_3#(leqElem#2(x2
,x6)
,x6
,x5
,x4
,x3
,x2
,x1))
cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_5(cond_partition_pivot_heap_5#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_6(cond_partition_pivot_heap_4#(partition#2(x6
,x1)
,x5
,x4
,x3
,x2))
cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) -> c_7()
cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9(cond_partition_pivot_heap_9#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_10(cond_partition_pivot_heap_8#(partition#2(x6
,x1)
,x5
,x4
,x3
,x2))
cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) -> c_11()
cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) -> c_12()
leq#2#(0(),x12) -> c_13()
leq#2#(S(x16),0()) -> c_14()
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
leqElem#2#(Elem(x4),Elem(x2)) -> c_16(leq#2#(x4,x2))
main#(x2,x1) -> c_17(partition#2#(x1,x2))
partition#2#(x2,E()) -> c_18()
partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
- Strict TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
main(x2,x1) -> partition#2(x1,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) -> c_1()
cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_2(cond_partition_pivot_heap_7#(leqElem#2(x4
,x6)
,x6
,x2
,x1
,x5
,x4
,x3))
cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) -> c_3()
cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_4(cond_partition_pivot_heap_3#(leqElem#2(x2
,x6)
,x6
,x5
,x4
,x3
,x2
,x1))
cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_5(cond_partition_pivot_heap_5#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_6(cond_partition_pivot_heap_4#(partition#2(x6
,x1)
,x5
,x4
,x3
,x2))
cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) -> c_7()
cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9(cond_partition_pivot_heap_9#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_10(cond_partition_pivot_heap_8#(partition#2(x6
,x1)
,x5
,x4
,x3
,x2))
cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) -> c_11()
cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) -> c_12()
leq#2#(0(),x12) -> c_13()
leq#2#(S(x16),0()) -> c_14()
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
leqElem#2#(Elem(x4),Elem(x2)) -> c_16(leq#2#(x4,x2))
main#(x2,x1) -> c_17(partition#2#(x1,x2))
partition#2#(x2,E()) -> c_18()
partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
* Step 3: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) -> c_1()
cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_2(cond_partition_pivot_heap_7#(leqElem#2(x4
,x6)
,x6
,x2
,x1
,x5
,x4
,x3))
cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) -> c_3()
cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_4(cond_partition_pivot_heap_3#(leqElem#2(x2
,x6)
,x6
,x5
,x4
,x3
,x2
,x1))
cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_5(cond_partition_pivot_heap_5#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_6(cond_partition_pivot_heap_4#(partition#2(x6
,x1)
,x5
,x4
,x3
,x2))
cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) -> c_7()
cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9(cond_partition_pivot_heap_9#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_10(cond_partition_pivot_heap_8#(partition#2(x6
,x1)
,x5
,x4
,x3
,x2))
cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) -> c_11()
cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) -> c_12()
leq#2#(0(),x12) -> c_13()
leq#2#(S(x16),0()) -> c_14()
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
leqElem#2#(Elem(x4),Elem(x2)) -> c_16(leq#2#(x4,x2))
main#(x2,x1) -> c_17(partition#2#(x1,x2))
partition#2#(x2,E()) -> c_18()
partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
- Strict TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
+ Details:
The weightgap principle applies using the following constant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(cond_partition_pivot_heap_1) = {1},
uargs(cond_partition_pivot_heap_3) = {1},
uargs(cond_partition_pivot_heap_4) = {1},
uargs(cond_partition_pivot_heap_5) = {1},
uargs(cond_partition_pivot_heap_7) = {1},
uargs(cond_partition_pivot_heap_8) = {1},
uargs(cond_partition_pivot_heap_9) = {1},
uargs(cond_partition_pivot_heap_1#) = {1},
uargs(cond_partition_pivot_heap_3#) = {1},
uargs(cond_partition_pivot_heap_4#) = {1},
uargs(cond_partition_pivot_heap_5#) = {1},
uargs(cond_partition_pivot_heap_7#) = {1},
uargs(cond_partition_pivot_heap_8#) = {1},
uargs(cond_partition_pivot_heap_9#) = {1},
uargs(c_2) = {1},
uargs(c_4) = {1},
uargs(c_5) = {1},
uargs(c_6) = {1},
uargs(c_9) = {1},
uargs(c_10) = {1},
uargs(c_15) = {1},
uargs(c_16) = {1},
uargs(c_17) = {1},
uargs(c_19) = {1}
Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [7]
p(E) = [0]
p(Elem) = [1] x1 + [2]
p(False) = [3]
p(Pair) = [0]
p(S) = [1] x1 + [4]
p(T) = [1] x1 + [1] x2 + [1] x3 + [0]
p(True) = [5]
p(cond_partition_pivot_heap_1) = [1] x1 + [1] x3 + [1] x5 + [0]
p(cond_partition_pivot_heap_3) = [1] x1 + [1] x5 + [1] x7 + [4]
p(cond_partition_pivot_heap_4) = [1] x1 + [6]
p(cond_partition_pivot_heap_5) = [1] x1 + [1] x5 + [4]
p(cond_partition_pivot_heap_7) = [1] x1 + [1] x4 + [1] x5 + [1] x7 + [0]
p(cond_partition_pivot_heap_8) = [1] x1 + [1] x3 + [1] x4 + [1]
p(cond_partition_pivot_heap_9) = [1] x1 + [1] x3 + [1] x5 + [1]
p(leq#2) = [1] x1 + [1]
p(leqElem#2) = [1] x1 + [0]
p(main) = [0]
p(partition#2) = [1] x2 + [1]
p(cond_partition_pivot_heap_1#) = [1] x1 + [4] x3 + [2] x4 + [4] x5 + [4]
p(cond_partition_pivot_heap_3#) = [1] x1 + [4] x3 + [4] x5 + [2] x6 + [4] x7 + [4]
p(cond_partition_pivot_heap_4#) = [1] x1 + [1] x2 + [1] x4 + [1]
p(cond_partition_pivot_heap_5#) = [1] x1 + [2] x4 + [4] x5 + [0]
p(cond_partition_pivot_heap_7#) = [1] x1 + [1] x3 + [4] x4 + [1] x5 + [1] x6 + [1] x7 + [3]
p(cond_partition_pivot_heap_8#) = [1] x1 + [1] x4 + [1] x5 + [1]
p(cond_partition_pivot_heap_9#) = [1] x1 + [1] x2 + [7]
p(leq#2#) = [1] x2 + [2]
p(leqElem#2#) = [1] x2 + [1]
p(main#) = [7] x1 + [5] x2 + [0]
p(partition#2#) = [3] x1 + [7] x2 + [0]
p(c_1) = [0]
p(c_2) = [1] x1 + [1]
p(c_3) = [0]
p(c_4) = [1] x1 + [2]
p(c_5) = [1] x1 + [4]
p(c_6) = [1] x1 + [1]
p(c_7) = [0]
p(c_8) = [0]
p(c_9) = [1] x1 + [4]
p(c_10) = [1] x1 + [0]
p(c_11) = [0]
p(c_12) = [1]
p(c_13) = [1]
p(c_14) = [0]
p(c_15) = [1] x1 + [6]
p(c_16) = [1] x1 + [1]
p(c_17) = [1] x1 + [0]
p(c_18) = [0]
p(c_19) = [1] x1 + [0]
Following rules are strictly oriented:
cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) = [4] x1 + [2] x2 + [7]
> [0]
= c_1()
cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) = [4] x1 + [2] x2 + [4] x3 + [4] x4 + [4] x5 + [7]
> [4] x1 + [1] x2 + [1] x3 + [2] x4 + [1] x5 + [4]
= c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3))
cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) = [2] x1 + [4] x2 + [9]
> [0]
= c_3()
cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) = [4] x1 + [4] x2 + [4] x3 + [2] x4 + [4] x5 + [9]
> [4] x1 + [3] x2 + [4] x3 + [4] x5 + [6]
= c_4(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1))
cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) = [4] x1 + [2] x2 + [4] x3 + [4] x5 + [7]
> [4] x1 + [2] x2 + [1] x3 + [5]
= c_5(cond_partition_pivot_heap_5#(partition#2(x6,x3),x5,x4,x2,x1))
cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) = [4] x1 + [2] x2 + [4] x3 + [4] x5 + [9]
> [1] x1 + [1] x3 + [1] x5 + [3]
= c_6(cond_partition_pivot_heap_4#(partition#2(x6,x1),x5,x4,x3,x2))
cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) = [1] x6 + [1] x9 + [1]
> [0]
= c_7()
cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) = [1] x1 + [1] x2 + [1] x3 + [4] x4 + [1] x5 + [8]
> [1] x1 + [1] x2 + [1] x3 + [2]
= c_10(cond_partition_pivot_heap_8#(partition#2(x6,x1),x5,x4,x3,x2))
cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) = [1] x10 + [1] x9 + [1]
> [0]
= c_11()
cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) = [1] x7 + [7]
> [1]
= c_12()
leq#2#(0(),x12) = [1] x12 + [2]
> [1]
= c_13()
leq#2#(S(x16),0()) = [9]
> [0]
= c_14()
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) = [1] x1 + [3]
> [0]
= Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) = [1] x1 + [1] x3 + [1] x4 + [1] x5 + [3]
> [1] x1 + [1] x3 + [1] x4 + [1] x5 + [0]
= cond_partition_pivot_heap_7(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) = [1] x2 + [5]
> [0]
= Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) = [1] x1 + [1] x2 + [1] x3 + [1] x5 + [5]
> [1] x1 + [1] x2 + [1] x3 + [4]
= cond_partition_pivot_heap_3(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) = [1] x1 + [1] x3 + [7]
> [1] x1 + [1] x3 + [5]
= cond_partition_pivot_heap_5(partition#2(x6,x3),x5,x4,x2,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) = [1] x1 + [1] x3 + [9]
> [1] x1 + [7]
= cond_partition_pivot_heap_4(partition#2(x6,x1),x5,x4,x3,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) = [6]
> [0]
= Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) = [1] x11 + [4]
> [0]
= Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) = [1] x1 + [1] x3 + [1] x4 + [3]
> [1] x1 + [1] x3 + [1] x4 + [2]
= cond_partition_pivot_heap_9(partition#2(x6,x3),x5,x4,x2,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) = [1] x1 + [1] x3 + [1] x4 + [5]
> [1] x1 + [1] x3 + [1] x4 + [2]
= cond_partition_pivot_heap_8(partition#2(x6,x1),x5,x4,x3,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) = [1] x8 + [1] x9 + [1]
> [0]
= Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) = [1] x11 + [1] x8 + [1]
> [0]
= Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) = [8]
> [5]
= True()
leq#2(S(x16),0()) = [1] x16 + [5]
> [3]
= False()
leq#2(S(x4),S(x2)) = [1] x4 + [5]
> [1] x4 + [1]
= leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) = [1] x4 + [2]
> [1] x4 + [1]
= leq#2(x4,x2)
partition#2(x2,E()) = [1]
> [0]
= Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) = [1] x2 + [1] x4 + [1] x6 + [1]
> [1] x2 + [1] x4 + [1] x6 + [0]
= cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
Following rules are (at-least) weakly oriented:
cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) = [2] x10 + [4] x11 + [0]
>= [0]
= c_8()
cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) = [1] x1 + [1] x2 + [1] x3 + [4] x4 + [1] x5 + [6]
>= [1] x3 + [1] x5 + [12]
= c_9(cond_partition_pivot_heap_9#(partition#2(x6,x3),x5,x4,x2,x1))
leq#2#(S(x4),S(x2)) = [1] x2 + [6]
>= [1] x2 + [8]
= c_15(leq#2#(x4,x2))
leqElem#2#(Elem(x4),Elem(x2)) = [1] x2 + [3]
>= [1] x2 + [3]
= c_16(leq#2#(x4,x2))
main#(x2,x1) = [5] x1 + [7] x2 + [0]
>= [3] x1 + [7] x2 + [0]
= c_17(partition#2#(x1,x2))
partition#2#(x2,E()) = [3] x2 + [0]
>= [0]
= c_18()
partition#2#(x8,T(x6,x4,x2)) = [7] x2 + [7] x4 + [7] x6 + [3] x8 + [0]
>= [4] x2 + [3] x4 + [4] x6 + [4]
= c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9(cond_partition_pivot_heap_9#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
leqElem#2#(Elem(x4),Elem(x2)) -> c_16(leq#2#(x4,x2))
main#(x2,x1) -> c_17(partition#2#(x1,x2))
partition#2#(x2,E()) -> c_18()
partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
- Weak DPs:
cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) -> c_1()
cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_2(cond_partition_pivot_heap_7#(leqElem#2(x4
,x6)
,x6
,x2
,x1
,x5
,x4
,x3))
cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) -> c_3()
cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_4(cond_partition_pivot_heap_3#(leqElem#2(x2
,x6)
,x6
,x5
,x4
,x3
,x2
,x1))
cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_5(cond_partition_pivot_heap_5#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_6(cond_partition_pivot_heap_4#(partition#2(x6
,x1)
,x5
,x4
,x3
,x2))
cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) -> c_7()
cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_10(cond_partition_pivot_heap_8#(partition#2(x6
,x1)
,x5
,x4
,x3
,x2))
cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) -> c_11()
cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) -> c_12()
leq#2#(0(),x12) -> c_13()
leq#2#(S(x16),0()) -> c_14()
- Weak TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{6,7}
by application of
Pre({6,7}) = {5}.
Here rules are labelled as follows:
1: cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
2: cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) ->
c_9(cond_partition_pivot_heap_9#(partition#2(x6,x3),x5,x4,x2,x1))
3: leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
4: leqElem#2#(Elem(x4),Elem(x2)) -> c_16(leq#2#(x4,x2))
5: main#(x2,x1) -> c_17(partition#2#(x1,x2))
6: partition#2#(x2,E()) -> c_18()
7: partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
8: cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) -> c_1()
9: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3))
10: cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) -> c_3()
11: cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) ->
c_4(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1))
12: cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) ->
c_5(cond_partition_pivot_heap_5#(partition#2(x6,x3),x5,x4,x2,x1))
13: cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) ->
c_6(cond_partition_pivot_heap_4#(partition#2(x6,x1),x5,x4,x3,x2))
14: cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) -> c_7()
15: cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) ->
c_10(cond_partition_pivot_heap_8#(partition#2(x6,x1),x5,x4,x3,x2))
16: cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) -> c_11()
17: cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) -> c_12()
18: leq#2#(0(),x12) -> c_13()
19: leq#2#(S(x16),0()) -> c_14()
* Step 5: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9(cond_partition_pivot_heap_9#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
leqElem#2#(Elem(x4),Elem(x2)) -> c_16(leq#2#(x4,x2))
main#(x2,x1) -> c_17(partition#2#(x1,x2))
- Weak DPs:
cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) -> c_1()
cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_2(cond_partition_pivot_heap_7#(leqElem#2(x4
,x6)
,x6
,x2
,x1
,x5
,x4
,x3))
cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) -> c_3()
cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_4(cond_partition_pivot_heap_3#(leqElem#2(x2
,x6)
,x6
,x5
,x4
,x3
,x2
,x1))
cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_5(cond_partition_pivot_heap_5#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_6(cond_partition_pivot_heap_4#(partition#2(x6
,x1)
,x5
,x4
,x3
,x2))
cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) -> c_7()
cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_10(cond_partition_pivot_heap_8#(partition#2(x6
,x1)
,x5
,x4
,x3
,x2))
cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) -> c_11()
cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) -> c_12()
leq#2#(0(),x12) -> c_13()
leq#2#(S(x16),0()) -> c_14()
partition#2#(x2,E()) -> c_18()
partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
- Weak TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{5}
by application of
Pre({5}) = {}.
Here rules are labelled as follows:
1: cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
2: cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) ->
c_9(cond_partition_pivot_heap_9#(partition#2(x6,x3),x5,x4,x2,x1))
3: leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
4: leqElem#2#(Elem(x4),Elem(x2)) -> c_16(leq#2#(x4,x2))
5: main#(x2,x1) -> c_17(partition#2#(x1,x2))
6: cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) -> c_1()
7: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3))
8: cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) -> c_3()
9: cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) ->
c_4(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1))
10: cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) ->
c_5(cond_partition_pivot_heap_5#(partition#2(x6,x3),x5,x4,x2,x1))
11: cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) ->
c_6(cond_partition_pivot_heap_4#(partition#2(x6,x1),x5,x4,x3,x2))
12: cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) -> c_7()
13: cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) ->
c_10(cond_partition_pivot_heap_8#(partition#2(x6,x1),x5,x4,x3,x2))
14: cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) -> c_11()
15: cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) -> c_12()
16: leq#2#(0(),x12) -> c_13()
17: leq#2#(S(x16),0()) -> c_14()
18: partition#2#(x2,E()) -> c_18()
19: partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
* Step 6: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9(cond_partition_pivot_heap_9#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
leqElem#2#(Elem(x4),Elem(x2)) -> c_16(leq#2#(x4,x2))
- Weak DPs:
cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) -> c_1()
cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_2(cond_partition_pivot_heap_7#(leqElem#2(x4
,x6)
,x6
,x2
,x1
,x5
,x4
,x3))
cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) -> c_3()
cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_4(cond_partition_pivot_heap_3#(leqElem#2(x2
,x6)
,x6
,x5
,x4
,x3
,x2
,x1))
cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_5(cond_partition_pivot_heap_5#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_6(cond_partition_pivot_heap_4#(partition#2(x6
,x1)
,x5
,x4
,x3
,x2))
cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) -> c_7()
cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_10(cond_partition_pivot_heap_8#(partition#2(x6
,x1)
,x5
,x4
,x3
,x2))
cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) -> c_11()
cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) -> c_12()
leq#2#(0(),x12) -> c_13()
leq#2#(S(x16),0()) -> c_14()
main#(x2,x1) -> c_17(partition#2#(x1,x2))
partition#2#(x2,E()) -> c_18()
partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
- Weak TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
2:S:cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) ->
c_9(cond_partition_pivot_heap_9#(partition#2(x6,x3),x5,x4,x2,x1))
-->_1 cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) -> c_12():14
3:S:leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
-->_1 leq#2#(S(x16),0()) -> c_14():16
-->_1 leq#2#(0(),x12) -> c_13():15
-->_1 leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2)):3
4:S:leqElem#2#(Elem(x4),Elem(x2)) -> c_16(leq#2#(x4,x2))
-->_1 leq#2#(S(x16),0()) -> c_14():16
-->_1 leq#2#(0(),x12) -> c_13():15
-->_1 leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2)):3
5:W:cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) -> c_1()
6:W:cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3))
-->_1 cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) ->
c_10(cond_partition_pivot_heap_8#(partition#2(x6,x1),x5,x4,x3,x2)):12
-->_1 cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) ->
c_9(cond_partition_pivot_heap_9#(partition#2(x6,x3),x5,x4,x2,x1)):2
7:W:cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) -> c_3()
8:W:cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) ->
c_4(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1))
-->_1 cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) ->
c_6(cond_partition_pivot_heap_4#(partition#2(x6,x1),x5,x4,x3,x2)):10
-->_1 cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) ->
c_5(cond_partition_pivot_heap_5#(partition#2(x6,x3),x5,x4,x2,x1)):9
9:W:cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) ->
c_5(cond_partition_pivot_heap_5#(partition#2(x6,x3),x5,x4,x2,x1))
-->_1 cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8():1
10:W:cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) ->
c_6(cond_partition_pivot_heap_4#(partition#2(x6,x1),x5,x4,x3,x2))
-->_1 cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) -> c_7():11
11:W:cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) -> c_7()
12:W:cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) ->
c_10(cond_partition_pivot_heap_8#(partition#2(x6,x1),x5,x4,x3,x2))
-->_1 cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) -> c_11():13
13:W:cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) -> c_11()
14:W:cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) -> c_12()
15:W:leq#2#(0(),x12) -> c_13()
16:W:leq#2#(S(x16),0()) -> c_14()
17:W:main#(x2,x1) -> c_17(partition#2#(x1,x2))
-->_1 partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)):19
-->_1 partition#2#(x2,E()) -> c_18():18
18:W:partition#2#(x2,E()) -> c_18()
19:W:partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
-->_1 cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) ->
c_4(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1)):8
-->_1 cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) -> c_3():7
-->_1 cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3)):6
-->_1 cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) -> c_1():5
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
18: partition#2#(x2,E()) -> c_18()
10: cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) ->
c_6(cond_partition_pivot_heap_4#(partition#2(x6,x1),x5,x4,x3,x2))
11: cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) -> c_7()
7: cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) -> c_3()
12: cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) ->
c_10(cond_partition_pivot_heap_8#(partition#2(x6,x1),x5,x4,x3,x2))
13: cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) -> c_11()
5: cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) -> c_1()
15: leq#2#(0(),x12) -> c_13()
16: leq#2#(S(x16),0()) -> c_14()
14: cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) -> c_12()
* Step 7: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9(cond_partition_pivot_heap_9#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
leqElem#2#(Elem(x4),Elem(x2)) -> c_16(leq#2#(x4,x2))
- Weak DPs:
cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_2(cond_partition_pivot_heap_7#(leqElem#2(x4
,x6)
,x6
,x2
,x1
,x5
,x4
,x3))
cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_4(cond_partition_pivot_heap_3#(leqElem#2(x2
,x6)
,x6
,x5
,x4
,x3
,x2
,x1))
cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_5(cond_partition_pivot_heap_5#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
main#(x2,x1) -> c_17(partition#2#(x1,x2))
partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
- Weak TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
2:S:cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) ->
c_9(cond_partition_pivot_heap_9#(partition#2(x6,x3),x5,x4,x2,x1))
3:S:leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
-->_1 leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2)):3
4:S:leqElem#2#(Elem(x4),Elem(x2)) -> c_16(leq#2#(x4,x2))
-->_1 leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2)):3
6:W:cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3))
-->_1 cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) ->
c_9(cond_partition_pivot_heap_9#(partition#2(x6,x3),x5,x4,x2,x1)):2
8:W:cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) ->
c_4(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1))
-->_1 cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) ->
c_5(cond_partition_pivot_heap_5#(partition#2(x6,x3),x5,x4,x2,x1)):9
9:W:cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) ->
c_5(cond_partition_pivot_heap_5#(partition#2(x6,x3),x5,x4,x2,x1))
-->_1 cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8():1
17:W:main#(x2,x1) -> c_17(partition#2#(x1,x2))
-->_1 partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)):19
19:W:partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
-->_1 cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) ->
c_4(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1)):8
-->_1 cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3)):6
Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9()
* Step 8: RemoveHeads WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9()
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
leqElem#2#(Elem(x4),Elem(x2)) -> c_16(leq#2#(x4,x2))
- Weak DPs:
cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_2(cond_partition_pivot_heap_7#(leqElem#2(x4
,x6)
,x6
,x2
,x1
,x5
,x4
,x3))
cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_4(cond_partition_pivot_heap_3#(leqElem#2(x2
,x6)
,x6
,x5
,x4
,x3
,x2
,x1))
cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_5(cond_partition_pivot_heap_5#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
main#(x2,x1) -> c_17(partition#2#(x1,x2))
partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
- Weak TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ Applied Processor:
RemoveHeads
+ Details:
Consider the dependency graph
1:S:cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
2:S:cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9()
3:S:leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
-->_1 leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2)):3
4:S:leqElem#2#(Elem(x4),Elem(x2)) -> c_16(leq#2#(x4,x2))
-->_1 leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2)):3
5:W:cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3))
-->_1 cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9():2
6:W:cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) ->
c_4(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1))
-->_1 cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) ->
c_5(cond_partition_pivot_heap_5#(partition#2(x6,x3),x5,x4,x2,x1)):7
7:W:cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) ->
c_5(cond_partition_pivot_heap_5#(partition#2(x6,x3),x5,x4,x2,x1))
-->_1 cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8():1
8:W:main#(x2,x1) -> c_17(partition#2#(x1,x2))
-->_1 partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)):9
9:W:partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
-->_1 cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) ->
c_4(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1)):6
-->_1 cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3)):5
Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
[(4,leqElem#2#(Elem(x4),Elem(x2)) -> c_16(leq#2#(x4,x2))),(8,main#(x2,x1) -> c_17(partition#2#(x1,x2)))]
* Step 9: Decompose WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9()
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
- Weak DPs:
cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_2(cond_partition_pivot_heap_7#(leqElem#2(x4
,x6)
,x6
,x2
,x1
,x5
,x4
,x3))
cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_4(cond_partition_pivot_heap_3#(leqElem#2(x2
,x6)
,x6
,x5
,x4
,x3
,x2
,x1))
cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_5(cond_partition_pivot_heap_5#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
- Weak TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ 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_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
- Weak DPs:
cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3))
cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_4(cond_partition_pivot_heap_3#(leqElem#2(x2
,x6)
,x6
,x5
,x4
,x3
,x2
,x1))
cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_5(cond_partition_pivot_heap_5#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9()
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
- Weak TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0
,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#
,cond_partition_pivot_heap_3#,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#
,cond_partition_pivot_heap_7#,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#
,main#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
Problem (S)
- Strict DPs:
cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9()
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
- Weak DPs:
cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3))
cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_4(cond_partition_pivot_heap_3#(leqElem#2(x2
,x6)
,x6
,x5
,x4
,x3
,x2
,x1))
cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_5(cond_partition_pivot_heap_5#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
- Weak TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0
,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#
,cond_partition_pivot_heap_3#,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#
,cond_partition_pivot_heap_7#,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#
,main#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
** Step 9.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
- Weak DPs:
cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_2(cond_partition_pivot_heap_7#(leqElem#2(x4
,x6)
,x6
,x2
,x1
,x5
,x4
,x3))
cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_4(cond_partition_pivot_heap_3#(leqElem#2(x2
,x6)
,x6
,x5
,x4
,x3
,x2
,x1))
cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_5(cond_partition_pivot_heap_5#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9()
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
- Weak TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
2:W:cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9()
3:W:leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
-->_1 leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2)):3
5:W:cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3))
-->_1 cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9():2
6:W:cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) ->
c_4(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1))
-->_1 cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) ->
c_5(cond_partition_pivot_heap_5#(partition#2(x6,x3),x5,x4,x2,x1)):7
7:W:cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) ->
c_5(cond_partition_pivot_heap_5#(partition#2(x6,x3),x5,x4,x2,x1))
-->_1 cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8():1
9:W:partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
-->_1 cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3)):5
-->_1 cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) ->
c_4(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1)):6
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
5: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3))
3: leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
2: cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9()
** Step 9.a:2: Trivial WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
- Weak DPs:
cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_4(cond_partition_pivot_heap_3#(leqElem#2(x2
,x6)
,x6
,x5
,x4
,x3
,x2
,x1))
cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_5(cond_partition_pivot_heap_5#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
- Weak TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ Applied Processor:
Trivial
+ Details:
Consider the dependency graph
1:S:cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
6:W:cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) ->
c_4(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1))
-->_1 cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) ->
c_5(cond_partition_pivot_heap_5#(partition#2(x6,x3),x5,x4,x2,x1)):7
7:W:cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) ->
c_5(cond_partition_pivot_heap_5#(partition#2(x6,x3),x5,x4,x2,x1))
-->_1 cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8():1
9:W:partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
-->_1 cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) ->
c_4(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1)):6
The dependency graph contains no loops, we remove all dependency pairs.
** Step 9.a:3: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
** Step 9.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9()
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
- Weak DPs:
cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_2(cond_partition_pivot_heap_7#(leqElem#2(x4
,x6)
,x6
,x2
,x1
,x5
,x4
,x3))
cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_4(cond_partition_pivot_heap_3#(leqElem#2(x2
,x6)
,x6
,x5
,x4
,x3
,x2
,x1))
cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_5(cond_partition_pivot_heap_5#(partition#2(x6
,x3)
,x5
,x4
,x2
,x1))
cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
- Weak TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9()
2:S:leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
-->_1 leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2)):2
3:W:cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3))
-->_1 cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9():1
4:W:cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) ->
c_4(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1))
-->_1 cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) ->
c_5(cond_partition_pivot_heap_5#(partition#2(x6,x3),x5,x4,x2,x1)):5
5:W:cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) ->
c_5(cond_partition_pivot_heap_5#(partition#2(x6,x3),x5,x4,x2,x1))
-->_1 cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8():6
6:W:cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
7:W:partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
-->_1 cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) ->
c_4(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1)):4
-->_1 cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) ->
c_4(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1))
5: cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) ->
c_5(cond_partition_pivot_heap_5#(partition#2(x6,x3),x5,x4,x2,x1))
6: cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_8()
** Step 9.b:2: Decompose WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9()
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
- Weak DPs:
cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_2(cond_partition_pivot_heap_7#(leqElem#2(x4
,x6)
,x6
,x2
,x1
,x5
,x4
,x3))
partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
- Weak TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ 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_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9()
- Weak DPs:
cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3))
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
- Weak TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0
,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#
,cond_partition_pivot_heap_3#,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#
,cond_partition_pivot_heap_7#,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#
,main#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
Problem (S)
- Strict DPs:
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
- Weak DPs:
cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3))
cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9()
partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
- Weak TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0
,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#
,cond_partition_pivot_heap_3#,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#
,cond_partition_pivot_heap_7#,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#
,main#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
*** Step 9.b:2.a:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9()
- Weak DPs:
cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_2(cond_partition_pivot_heap_7#(leqElem#2(x4
,x6)
,x6
,x2
,x1
,x5
,x4
,x3))
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
- Weak TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9()
2:W:leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
-->_1 leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2)):2
3:W:cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3))
-->_1 cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9():1
7:W:partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
-->_1 cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3)):3
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
2: leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
*** Step 9.b:2.a:2: Trivial WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9()
- Weak DPs:
cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_2(cond_partition_pivot_heap_7#(leqElem#2(x4
,x6)
,x6
,x2
,x1
,x5
,x4
,x3))
partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
- Weak TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ Applied Processor:
Trivial
+ Details:
Consider the dependency graph
1:S:cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9()
3:W:cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3))
-->_1 cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9():1
7:W:partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
-->_1 cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3)):3
The dependency graph contains no loops, we remove all dependency pairs.
*** Step 9.b:2.a:3: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
*** Step 9.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
- Weak DPs:
cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_2(cond_partition_pivot_heap_7#(leqElem#2(x4
,x6)
,x6
,x2
,x1
,x5
,x4
,x3))
cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9()
partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
- Weak TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
-->_1 leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2)):1
2:W:cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3))
-->_1 cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9():3
3:W:cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9()
4:W:partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
-->_1 cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3)):2
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: partition#2#(x8,T(x6,x4,x2)) -> c_19(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2))
2: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) ->
c_2(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3))
3: cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_9()
*** Step 9.b:2.b:2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
- Weak TRS:
cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) -> Pair(E(),T(E(),x2,x1))
cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) -> cond_partition_pivot_heap_7(leqElem#2(x4,x6)
,x6
,x2
,x1
,x5
,x4
,x3)
cond_partition_pivot_heap_1(True(),x3,x2,x1,E()) -> Pair(T(x2,x1,E()),E())
cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) -> cond_partition_pivot_heap_3(leqElem#2(x2,x6)
,x6
,x5
,x4
,x3
,x2
,x1)
cond_partition_pivot_heap_3(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_5(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_3(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_4(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_4(Pair(x12,x13),x6,x7,x9,x10) -> Pair(T(T(x6,x7,x9),x10,x12),x13)
cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) -> Pair(T(x6,x7,x12),T(x13,x10,x11))
cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_9(partition#2(x6,x3)
,x5
,x4
,x2
,x1)
cond_partition_pivot_heap_7(True(),x6,x5,x4,x3,x2,x1) -> cond_partition_pivot_heap_8(partition#2(x6,x1)
,x5
,x4
,x3
,x2)
cond_partition_pivot_heap_8(Pair(x12,x13),x7,x8,x9,x10) -> Pair(T(x9,x10,x12),T(x13,x7,x8))
cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) -> Pair(x12,T(x13,x10,T(x11,x7,x8)))
leq#2(0(),x12) -> True()
leq#2(S(x16),0()) -> False()
leq#2(S(x4),S(x2)) -> leq#2(x4,x2)
leqElem#2(Elem(x4),Elem(x2)) -> leq#2(x4,x2)
partition#2(x2,E()) -> Pair(E(),E())
partition#2(x8,T(x6,x4,x2)) -> cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2)
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
*** Step 9.b:2.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ 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: leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
The strictly oriented rules are moved into the weak component.
**** Step 9.b:2.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ 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_15) = {1}
Following symbols are considered usable:
{cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#,cond_partition_pivot_heap_4#
,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#,cond_partition_pivot_heap_8#
,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#,partition#2#}
TcT has computed the following interpretation:
p(0) = [0]
p(E) = [0]
p(Elem) = [0]
p(False) = [0]
p(Pair) = [0]
p(S) = [1] x1 + [1]
p(T) = [1] x1 + [0]
p(True) = [0]
p(cond_partition_pivot_heap_1) = [1] x2 + [2] x3 + [1]
p(cond_partition_pivot_heap_3) = [2] x1 + [2] x2 + [8] x3 + [1] x4 + [1] x5 + [1] x6 + [1] x7 + [1]
p(cond_partition_pivot_heap_4) = [1] x3 + [1]
p(cond_partition_pivot_heap_5) = [4] x2 + [4] x3 + [1] x4 + [1] x5 + [8]
p(cond_partition_pivot_heap_7) = [0]
p(cond_partition_pivot_heap_8) = [0]
p(cond_partition_pivot_heap_9) = [0]
p(leq#2) = [0]
p(leqElem#2) = [0]
p(main) = [0]
p(partition#2) = [0]
p(cond_partition_pivot_heap_1#) = [0]
p(cond_partition_pivot_heap_3#) = [0]
p(cond_partition_pivot_heap_4#) = [4] x3 + [0]
p(cond_partition_pivot_heap_5#) = [8] x5 + [8]
p(cond_partition_pivot_heap_7#) = [4] x2 + [8] x5 + [1] x6 + [8] x7 + [0]
p(cond_partition_pivot_heap_8#) = [1] x1 + [4] x2 + [8] x4 + [4]
p(cond_partition_pivot_heap_9#) = [8] x1 + [2] x2 + [8] x3 + [2] x4 + [1] x5 + [8]
p(leq#2#) = [2] x1 + [8] x2 + [0]
p(leqElem#2#) = [2] x1 + [1] x2 + [8]
p(main#) = [2] x1 + [2] x2 + [0]
p(partition#2#) = [1] x1 + [4] x2 + [1]
p(c_1) = [0]
p(c_2) = [1] x1 + [2]
p(c_3) = [0]
p(c_4) = [2] x1 + [0]
p(c_5) = [1] x1 + [0]
p(c_6) = [1]
p(c_7) = [1]
p(c_8) = [1]
p(c_9) = [1]
p(c_10) = [8]
p(c_11) = [1]
p(c_12) = [0]
p(c_13) = [2]
p(c_14) = [2]
p(c_15) = [1] x1 + [8]
p(c_16) = [1] x1 + [1]
p(c_17) = [1]
p(c_18) = [0]
p(c_19) = [1] x1 + [0]
Following rules are strictly oriented:
leq#2#(S(x4),S(x2)) = [8] x2 + [2] x4 + [10]
> [8] x2 + [2] x4 + [8]
= c_15(leq#2#(x4,x2))
Following rules are (at-least) weakly oriented:
**** Step 9.b:2.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()
**** Step 9.b:2.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
-->_1 leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2)):1
The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: leq#2#(S(x4),S(x2)) -> c_15(leq#2#(x4,x2))
**** Step 9.b:2.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Signature:
{cond_partition_pivot_heap_1/5,cond_partition_pivot_heap_3/7,cond_partition_pivot_heap_4/5
,cond_partition_pivot_heap_5/5,cond_partition_pivot_heap_7/7,cond_partition_pivot_heap_8/5
,cond_partition_pivot_heap_9/5,leq#2/2,leqElem#2/2,main/2,partition#2/2,cond_partition_pivot_heap_1#/5
,cond_partition_pivot_heap_3#/7,cond_partition_pivot_heap_4#/5,cond_partition_pivot_heap_5#/5
,cond_partition_pivot_heap_7#/7,cond_partition_pivot_heap_8#/5,cond_partition_pivot_heap_9#/5,leq#2#/2
,leqElem#2#/2,main#/2,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/0,c_2/1,c_3/0
,c_4/1,c_5/1,c_6/1,c_7/0,c_8/0,c_9/0,c_10/1,c_11/0,c_12/0,c_13/0,c_14/0,c_15/1,c_16/1,c_17/1,c_18/0,c_19/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {cond_partition_pivot_heap_1#,cond_partition_pivot_heap_3#
,cond_partition_pivot_heap_4#,cond_partition_pivot_heap_5#,cond_partition_pivot_heap_7#
,cond_partition_pivot_heap_8#,cond_partition_pivot_heap_9#,leq#2#,leqElem#2#,main#
,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).
WORST_CASE(?,O(n^1))