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))