WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: cond_merge_h_t_1(Pair(x5,x4),x3,x2,x1) -> T(merge#2(x5,x3),x2,merge#2(x4,x1)) 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) -> merge#2(x2,x1) merge#2(E(),x2) -> x2 merge#2(T(x8,x6,x4),x2) -> cond_merge_h_t_1(partition#2(x6,x2),x8,x6,x4) 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_merge_h_t_1/4,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,merge#2/2 ,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_merge_h_t_1,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 ,merge#2,partition#2} and constructors {0,E,Elem,False,Pair,S,T,True} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) -> c_2() cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3) ,leqElem#2#(x4,x6)) cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) -> c_4() cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1) ,leqElem#2#(x2,x6)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(cond_partition_pivot_heap_5#(partition#2(x6 ,x3) ,x5 ,x4 ,x2 ,x1) ,partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(cond_partition_pivot_heap_4#(partition#2(x6 ,x1) ,x5 ,x4 ,x3 ,x2) ,partition#2#(x6,x1)) cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) -> c_8() cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_9() cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(cond_partition_pivot_heap_9#(partition#2(x6 ,x3) ,x5 ,x4 ,x2 ,x1) ,partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(cond_partition_pivot_heap_8#(partition#2(x6 ,x1) ,x5 ,x4 ,x3 ,x2) ,partition#2#(x6,x1)) cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) -> c_12() cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) -> c_13() leq#2#(0(),x12) -> c_14() leq#2#(S(x16),0()) -> c_15() leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)) leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)) main#(x2,x1) -> c_18(merge#2#(x2,x1)) merge#2#(E(),x2) -> c_19() merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x2,E()) -> c_21() partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) -> c_2() cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3) ,leqElem#2#(x4,x6)) cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) -> c_4() cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1) ,leqElem#2#(x2,x6)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(cond_partition_pivot_heap_5#(partition#2(x6 ,x3) ,x5 ,x4 ,x2 ,x1) ,partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(cond_partition_pivot_heap_4#(partition#2(x6 ,x1) ,x5 ,x4 ,x3 ,x2) ,partition#2#(x6,x1)) cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) -> c_8() cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_9() cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(cond_partition_pivot_heap_9#(partition#2(x6 ,x3) ,x5 ,x4 ,x2 ,x1) ,partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(cond_partition_pivot_heap_8#(partition#2(x6 ,x1) ,x5 ,x4 ,x3 ,x2) ,partition#2#(x6,x1)) cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) -> c_12() cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) -> c_13() leq#2#(0(),x12) -> c_14() leq#2#(S(x16),0()) -> c_15() leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)) leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)) main#(x2,x1) -> c_18(merge#2#(x2,x1)) merge#2#(E(),x2) -> c_19() merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x2,E()) -> c_21() partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) - Weak TRS: cond_merge_h_t_1(Pair(x5,x4),x3,x2,x1) -> T(merge#2(x5,x3),x2,merge#2(x4,x1)) 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) -> merge#2(x2,x1) merge#2(E(),x2) -> x2 merge#2(T(x8,x6,x4),x2) -> cond_merge_h_t_1(partition#2(x6,x2),x8,x6,x4) 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/2,c_6/2,c_7/2 ,c_8/0,c_9/0,c_10/2,c_11/2,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) -> c_2() cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3) ,leqElem#2#(x4,x6)) cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) -> c_4() cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1) ,leqElem#2#(x2,x6)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(cond_partition_pivot_heap_5#(partition#2(x6 ,x3) ,x5 ,x4 ,x2 ,x1) ,partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(cond_partition_pivot_heap_4#(partition#2(x6 ,x1) ,x5 ,x4 ,x3 ,x2) ,partition#2#(x6,x1)) cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) -> c_8() cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_9() cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(cond_partition_pivot_heap_9#(partition#2(x6 ,x3) ,x5 ,x4 ,x2 ,x1) ,partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(cond_partition_pivot_heap_8#(partition#2(x6 ,x1) ,x5 ,x4 ,x3 ,x2) ,partition#2#(x6,x1)) cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) -> c_12() cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) -> c_13() leq#2#(0(),x12) -> c_14() leq#2#(S(x16),0()) -> c_15() leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)) leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)) main#(x2,x1) -> c_18(merge#2#(x2,x1)) merge#2#(E(),x2) -> c_19() merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x2,E()) -> c_21() partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) * Step 3: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) -> c_2() cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3) ,leqElem#2#(x4,x6)) cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) -> c_4() cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1) ,leqElem#2#(x2,x6)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(cond_partition_pivot_heap_5#(partition#2(x6 ,x3) ,x5 ,x4 ,x2 ,x1) ,partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(cond_partition_pivot_heap_4#(partition#2(x6 ,x1) ,x5 ,x4 ,x3 ,x2) ,partition#2#(x6,x1)) cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) -> c_8() cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_9() cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(cond_partition_pivot_heap_9#(partition#2(x6 ,x3) ,x5 ,x4 ,x2 ,x1) ,partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(cond_partition_pivot_heap_8#(partition#2(x6 ,x1) ,x5 ,x4 ,x3 ,x2) ,partition#2#(x6,x1)) cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) -> c_12() cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) -> c_13() leq#2#(0(),x12) -> c_14() leq#2#(S(x16),0()) -> c_15() leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)) leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)) main#(x2,x1) -> c_18(merge#2#(x2,x1)) merge#2#(E(),x2) -> c_19() merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x2,E()) -> c_21() partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/2,c_6/2,c_7/2 ,c_8/0,c_9/0,c_10/2,c_11/2,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 {2,4,8,9,12,13,14,15,19,21} by application of Pre({2,4,8,9,12,13,14,15,19,21}) = {1,6,7,10,11,16,17,18,20,22}. Here rules are labelled as follows: 1: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) 2: cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) -> c_2() 3: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3),leqElem#2#(x4,x6)) 4: cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) -> c_4() 5: cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1),leqElem#2#(x2,x6)) 6: cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(cond_partition_pivot_heap_5#(partition#2(x6,x3),x5,x4,x2,x1),partition#2#(x6,x3)) 7: cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(cond_partition_pivot_heap_4#(partition#2(x6 ,x1) ,x5 ,x4 ,x3 ,x2) ,partition#2#(x6,x1)) 8: cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) -> c_8() 9: cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_9() 10: cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(cond_partition_pivot_heap_9#(partition#2(x6,x3),x5,x4,x2,x1),partition#2#(x6,x3)) 11: cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(cond_partition_pivot_heap_8#(partition#2(x6,x1),x5,x4,x3,x2),partition#2#(x6,x1)) 12: cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) -> c_12() 13: cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) -> c_13() 14: leq#2#(0(),x12) -> c_14() 15: leq#2#(S(x16),0()) -> c_15() 16: leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)) 17: leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)) 18: main#(x2,x1) -> c_18(merge#2#(x2,x1)) 19: merge#2#(E(),x2) -> c_19() 20: merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) 21: partition#2#(x2,E()) -> c_21() 22: partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3) ,leqElem#2#(x4,x6)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1) ,leqElem#2#(x2,x6)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(cond_partition_pivot_heap_5#(partition#2(x6 ,x3) ,x5 ,x4 ,x2 ,x1) ,partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(cond_partition_pivot_heap_4#(partition#2(x6 ,x1) ,x5 ,x4 ,x3 ,x2) ,partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(cond_partition_pivot_heap_9#(partition#2(x6 ,x3) ,x5 ,x4 ,x2 ,x1) ,partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(cond_partition_pivot_heap_8#(partition#2(x6 ,x1) ,x5 ,x4 ,x3 ,x2) ,partition#2#(x6,x1)) leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)) leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)) main#(x2,x1) -> c_18(merge#2#(x2,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) - Weak DPs: cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) -> c_2() cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) -> c_4() cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) -> c_8() cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_9() cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) -> c_12() cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) -> c_13() leq#2#(0(),x12) -> c_14() leq#2#(S(x16),0()) -> c_15() merge#2#(E(),x2) -> c_19() partition#2#(x2,E()) -> c_21() - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/2,c_6/2,c_7/2 ,c_8/0,c_9/0,c_10/2,c_11/2,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) -->_2 merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) ,partition#2#(x6,x2)):11 -->_1 merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) ,partition#2#(x6,x2)):11 -->_2 merge#2#(E(),x2) -> c_19():21 -->_1 merge#2#(E(),x2) -> c_19():21 2:S:cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3),leqElem#2#(x4,x6)) -->_2 leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)):9 -->_1 cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(cond_partition_pivot_heap_8#(partition#2(x6,x1),x5,x4,x3,x2),partition#2#(x6,x1)):7 -->_1 cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(cond_partition_pivot_heap_9#(partition#2(x6,x3),x5,x4,x2,x1),partition#2#(x6,x3)):6 3:S:cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1),leqElem#2#(x2,x6)) -->_2 leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)):9 -->_1 cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(cond_partition_pivot_heap_4#(partition#2(x6,x1),x5,x4,x3,x2),partition#2#(x6,x1)):5 -->_1 cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(cond_partition_pivot_heap_5#(partition#2(x6,x3),x5,x4,x2,x1),partition#2#(x6,x3)):4 4:S:cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(cond_partition_pivot_heap_5#(partition#2(x6,x3),x5,x4,x2,x1),partition#2#(x6,x3)) -->_2 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):12 -->_2 partition#2#(x2,E()) -> c_21():22 -->_1 cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_9():16 5:S:cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(cond_partition_pivot_heap_4#(partition#2(x6,x1),x5,x4,x3,x2),partition#2#(x6,x1)) -->_2 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):12 -->_2 partition#2#(x2,E()) -> c_21():22 -->_1 cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) -> c_8():15 6:S:cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(cond_partition_pivot_heap_9#(partition#2(x6,x3),x5,x4,x2,x1),partition#2#(x6,x3)) -->_2 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):12 -->_2 partition#2#(x2,E()) -> c_21():22 -->_1 cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) -> c_13():18 7:S:cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(cond_partition_pivot_heap_8#(partition#2(x6,x1),x5,x4,x3,x2),partition#2#(x6,x1)) -->_2 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):12 -->_2 partition#2#(x2,E()) -> c_21():22 -->_1 cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) -> c_12():17 8:S:leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)) -->_1 leq#2#(S(x16),0()) -> c_15():20 -->_1 leq#2#(0(),x12) -> c_14():19 -->_1 leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)):8 9:S:leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)) -->_1 leq#2#(S(x16),0()) -> c_15():20 -->_1 leq#2#(0(),x12) -> c_14():19 -->_1 leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)):8 10:S:main#(x2,x1) -> c_18(merge#2#(x2,x1)) -->_1 merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) ,partition#2#(x6,x2)):11 -->_1 merge#2#(E(),x2) -> c_19():21 11:S:merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) -->_2 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):12 -->_2 partition#2#(x2,E()) -> c_21():22 -->_1 cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)):1 12:S:partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) -->_1 cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) -> c_4():14 -->_1 cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) -> c_2():13 -->_2 leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)):9 -->_1 cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1),leqElem#2#(x2,x6)):3 -->_1 cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3),leqElem#2#(x4,x6)):2 13:W:cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) -> c_2() 14:W:cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) -> c_4() 15:W:cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) -> c_8() 16:W:cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_9() 17:W:cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) -> c_12() 18:W:cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) -> c_13() 19:W:leq#2#(0(),x12) -> c_14() 20:W:leq#2#(S(x16),0()) -> c_15() 21:W:merge#2#(E(),x2) -> c_19() 22:W:partition#2#(x2,E()) -> c_21() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 21: merge#2#(E(),x2) -> c_19() 18: cond_partition_pivot_heap_9#(Pair(x12,x13),x7,x8,x10,x11) -> c_13() 17: cond_partition_pivot_heap_8#(Pair(x12,x13),x7,x8,x9,x10) -> c_12() 16: cond_partition_pivot_heap_5#(Pair(x12,x13),x6,x7,x10,x11) -> c_9() 15: cond_partition_pivot_heap_4#(Pair(x12,x13),x6,x7,x9,x10) -> c_8() 22: partition#2#(x2,E()) -> c_21() 19: leq#2#(0(),x12) -> c_14() 20: leq#2#(S(x16),0()) -> c_15() 13: cond_partition_pivot_heap_1#(False(),x3,E(),x2,x1) -> c_2() 14: cond_partition_pivot_heap_1#(True(),x3,x2,x1,E()) -> c_4() * Step 5: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3) ,leqElem#2#(x4,x6)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1) ,leqElem#2#(x2,x6)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(cond_partition_pivot_heap_5#(partition#2(x6 ,x3) ,x5 ,x4 ,x2 ,x1) ,partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(cond_partition_pivot_heap_4#(partition#2(x6 ,x1) ,x5 ,x4 ,x3 ,x2) ,partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(cond_partition_pivot_heap_9#(partition#2(x6 ,x3) ,x5 ,x4 ,x2 ,x1) ,partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(cond_partition_pivot_heap_8#(partition#2(x6 ,x1) ,x5 ,x4 ,x3 ,x2) ,partition#2#(x6,x1)) leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)) leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)) main#(x2,x1) -> c_18(merge#2#(x2,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/2,c_6/2,c_7/2 ,c_8/0,c_9/0,c_10/2,c_11/2,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) -->_2 merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) ,partition#2#(x6,x2)):11 -->_1 merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) ,partition#2#(x6,x2)):11 2:S:cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3),leqElem#2#(x4,x6)) -->_2 leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)):9 -->_1 cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(cond_partition_pivot_heap_8#(partition#2(x6,x1),x5,x4,x3,x2),partition#2#(x6,x1)):7 -->_1 cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(cond_partition_pivot_heap_9#(partition#2(x6,x3),x5,x4,x2,x1),partition#2#(x6,x3)):6 3:S:cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1),leqElem#2#(x2,x6)) -->_2 leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)):9 -->_1 cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(cond_partition_pivot_heap_4#(partition#2(x6,x1),x5,x4,x3,x2),partition#2#(x6,x1)):5 -->_1 cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(cond_partition_pivot_heap_5#(partition#2(x6,x3),x5,x4,x2,x1),partition#2#(x6,x3)):4 4:S:cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(cond_partition_pivot_heap_5#(partition#2(x6,x3),x5,x4,x2,x1),partition#2#(x6,x3)) -->_2 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):12 5:S:cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(cond_partition_pivot_heap_4#(partition#2(x6,x1),x5,x4,x3,x2),partition#2#(x6,x1)) -->_2 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):12 6:S:cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(cond_partition_pivot_heap_9#(partition#2(x6,x3),x5,x4,x2,x1),partition#2#(x6,x3)) -->_2 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):12 7:S:cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(cond_partition_pivot_heap_8#(partition#2(x6,x1),x5,x4,x3,x2),partition#2#(x6,x1)) -->_2 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):12 8:S:leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)) -->_1 leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)):8 9:S:leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)) -->_1 leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)):8 10:S:main#(x2,x1) -> c_18(merge#2#(x2,x1)) -->_1 merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) ,partition#2#(x6,x2)):11 11:S:merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) -->_2 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):12 -->_1 cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)):1 12:S:partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) -->_2 leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)):9 -->_1 cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1),leqElem#2#(x2,x6)):3 -->_1 cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3),leqElem#2#(x4,x6)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) * Step 6: RemoveHeads WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3) ,leqElem#2#(x4,x6)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1) ,leqElem#2#(x2,x6)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)) leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)) main#(x2,x1) -> c_18(merge#2#(x2,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) -->_2 merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) ,partition#2#(x6,x2)):11 -->_1 merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) ,partition#2#(x6,x2)):11 2:S:cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3),leqElem#2#(x4,x6)) -->_2 leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)):9 -->_1 cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)):7 -->_1 cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)):6 3:S:cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1),leqElem#2#(x2,x6)) -->_2 leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)):9 -->_1 cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)):5 -->_1 cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)):4 4:S:cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):12 5:S:cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):12 6:S:cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):12 7:S:cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):12 8:S:leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)) -->_1 leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)):8 9:S:leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)) -->_1 leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)):8 10:S:main#(x2,x1) -> c_18(merge#2#(x2,x1)) -->_1 merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) ,partition#2#(x6,x2)):11 11:S:merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) -->_2 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):12 -->_1 cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)):1 12:S:partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) -->_2 leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)):9 -->_1 cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1),leqElem#2#(x2,x6)):3 -->_1 cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3),leqElem#2#(x4,x6)):2 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(10,main#(x2,x1) -> c_18(merge#2#(x2,x1)))] * Step 7: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3) ,leqElem#2#(x4,x6)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1) ,leqElem#2#(x2,x6)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)) leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 8: leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)) The strictly oriented rules are moved into the weak component. ** Step 7.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3) ,leqElem#2#(x4,x6)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1) ,leqElem#2#(x2,x6)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)) leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_1) = {1,2}, uargs(c_3) = {1,2}, uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_16) = {1}, uargs(c_17) = {1}, uargs(c_20) = {1,2}, uargs(c_22) = {1,2} 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,partition#2,cond_merge_h_t_1#,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#,merge#2#,partition#2#} TcT has computed the following interpretation: p(0) = 0 p(E) = 0 p(Elem) = 1 + x1 p(False) = 0 p(Pair) = x1 + x2 p(S) = 1 + x1 p(T) = x1 + x2 + x3 p(True) = 1 p(cond_merge_h_t_1) = 0 p(cond_partition_pivot_heap_1) = x3 + x4 + x5 p(cond_partition_pivot_heap_3) = x3 + x4 + x5 + x6 + x7 p(cond_partition_pivot_heap_4) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_5) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_7) = x3 + x4 + x5 + x6 + x7 p(cond_partition_pivot_heap_8) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_9) = x1 + x2 + x3 + x4 + x5 p(leq#2) = x2 p(leqElem#2) = 1 + x1^2 + x2 p(main) = 0 p(merge#2) = 0 p(partition#2) = x2 p(cond_merge_h_t_1#) = x1*x2 + x1*x4 p(cond_partition_pivot_heap_1#) = x2*x3 + x2*x5 p(cond_partition_pivot_heap_3#) = x2*x5 + x2*x7 p(cond_partition_pivot_heap_4#) = 0 p(cond_partition_pivot_heap_5#) = 0 p(cond_partition_pivot_heap_7#) = x2*x5 + x2*x7 p(cond_partition_pivot_heap_8#) = 0 p(cond_partition_pivot_heap_9#) = 0 p(leq#2#) = x2 p(leqElem#2#) = x1*x2 p(main#) = 0 p(merge#2#) = x1*x2 p(partition#2#) = x1*x2 p(c_1) = x1 + x2 p(c_2) = 0 p(c_3) = x1 + x2 p(c_4) = 0 p(c_5) = x1 + x2 p(c_6) = x1 p(c_7) = x1 p(c_8) = 0 p(c_9) = 0 p(c_10) = x1 p(c_11) = x1 p(c_12) = 0 p(c_13) = 0 p(c_14) = 0 p(c_15) = 0 p(c_16) = x1 p(c_17) = 1 + x1 p(c_18) = 0 p(c_19) = 0 p(c_20) = x1 + x2 p(c_21) = 0 p(c_22) = x1 + x2 Following rules are strictly oriented: leq#2#(S(x4),S(x2)) = 1 + x2 > x2 = c_16(leq#2#(x4,x2)) Following rules are (at-least) weakly oriented: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) = x1*x4 + x1*x5 + x3*x4 + x3*x5 >= x1*x4 + x3*x5 = c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) = x1*x6 + x3*x6 + x4*x6 + x5*x6 >= x3*x6 + x4*x6 + x5*x6 = c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3),leqElem#2#(x4,x6)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) = x1*x6 + x2*x6 + x3*x6 + x5*x6 >= x1*x6 + x2*x6 + x3*x6 = c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1),leqElem#2#(x2,x6)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) = x1*x6 + x3*x6 >= x3*x6 = c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) = x1*x6 + x3*x6 >= x1*x6 = c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) = x1*x6 + x3*x6 >= x3*x6 = c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) = x1*x6 + x3*x6 >= x1*x6 = c_11(partition#2#(x6,x1)) leqElem#2#(Elem(x4),Elem(x2)) = 1 + x2 + x2*x4 + x4 >= 1 + x2 = c_17(leq#2#(x4,x2)) merge#2#(T(x8,x6,x4),x2) = x2*x4 + x2*x6 + x2*x8 >= x2*x4 + x2*x6 + x2*x8 = c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) = x2*x8 + x4*x8 + x6*x8 >= x2*x8 + x4*x8 + x6*x8 = c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2),leqElem#2#(x4,x8)) cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) = x1 + x2 >= x1 + x2 = Pair(E(),T(E(),x2,x1)) cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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()) = x1 + x2 >= x1 + x2 = Pair(T(x2,x1,E()),E()) cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x10 + x12 + x13 + x6 + x7 + x9 >= x10 + x12 + x13 + x6 + x7 + x9 = Pair(T(T(x6,x7,x9),x10,x12),x13) cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) = x10 + x11 + x12 + x13 + x6 + x7 >= x10 + x11 + x12 + x13 + x6 + x7 = Pair(T(x6,x7,x12),T(x13,x10,x11)) cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x10 + x12 + x13 + x7 + x8 + x9 >= x10 + x12 + x13 + x7 + x8 + x9 = Pair(T(x9,x10,x12),T(x13,x7,x8)) cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) = x10 + x11 + x12 + x13 + x7 + x8 >= x10 + x11 + x12 + x13 + x7 + x8 = Pair(x12,T(x13,x10,T(x11,x7,x8))) partition#2(x2,E()) = 0 >= 0 = Pair(E(),E()) partition#2(x8,T(x6,x4,x2)) = x2 + x4 + x6 >= x2 + x4 + x6 = cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2) ** Step 7.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3) ,leqElem#2#(x4,x6)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1) ,leqElem#2#(x2,x6)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) - Weak DPs: leq#2#(S(x4),S(x2)) -> c_16(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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 7.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3) ,leqElem#2#(x4,x6)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1) ,leqElem#2#(x2,x6)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) - Weak DPs: leq#2#(S(x4),S(x2)) -> c_16(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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) -->_2 merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) ,partition#2#(x6,x2)):9 -->_1 merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) ,partition#2#(x6,x2)):9 2:S:cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3),leqElem#2#(x4,x6)) -->_2 leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)):8 -->_1 cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)):7 -->_1 cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)):6 3:S:cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1),leqElem#2#(x2,x6)) -->_2 leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)):8 -->_1 cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)):5 -->_1 cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)):4 4:S:cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):10 5:S:cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):10 6:S:cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):10 7:S:cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):10 8:S:leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)) -->_1 leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)):11 9:S:merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) -->_2 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):10 -->_1 cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)):1 10:S:partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) -->_2 leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)):8 -->_1 cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1),leqElem#2#(x2,x6)):3 -->_1 cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3),leqElem#2#(x4,x6)):2 11:W:leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)) -->_1 leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)):11 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 11: leq#2#(S(x4),S(x2)) -> c_16(leq#2#(x4,x2)) ** Step 7.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3) ,leqElem#2#(x4,x6)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1) ,leqElem#2#(x2,x6)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/1,c_18/1,c_19/0,c_20/2,c_21/0,c_22/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) -->_2 merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) ,partition#2#(x6,x2)):9 -->_1 merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) ,partition#2#(x6,x2)):9 2:S:cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3),leqElem#2#(x4,x6)) -->_2 leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)):8 -->_1 cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)):7 -->_1 cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)):6 3:S:cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1),leqElem#2#(x2,x6)) -->_2 leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)):8 -->_1 cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)):5 -->_1 cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)):4 4:S:cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):10 5:S:cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):10 6:S:cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):10 7:S:cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):10 8:S:leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)) 9:S:merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) -->_2 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):10 -->_1 cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)):1 10:S:partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) -->_2 leqElem#2#(Elem(x4),Elem(x2)) -> c_17(leq#2#(x4,x2)):8 -->_1 cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1),leqElem#2#(x2,x6)):3 -->_1 cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3),leqElem#2#(x4,x6)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: leqElem#2#(Elem(x4),Elem(x2)) -> c_17() ** Step 7.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3) ,leqElem#2#(x4,x6)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1) ,leqElem#2#(x2,x6)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) leqElem#2#(Elem(x4),Elem(x2)) -> c_17() merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 8: leqElem#2#(Elem(x4),Elem(x2)) -> c_17() The strictly oriented rules are moved into the weak component. *** Step 7.b:3.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3) ,leqElem#2#(x4,x6)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1) ,leqElem#2#(x2,x6)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) leqElem#2#(Elem(x4),Elem(x2)) -> c_17() merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_1) = {1,2}, uargs(c_3) = {1,2}, uargs(c_5) = {1,2}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_20) = {1,2}, uargs(c_22) = {1,2} 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,partition#2,cond_merge_h_t_1#,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#,merge#2#,partition#2#} TcT has computed the following interpretation: p(0) = 1 p(E) = 0 p(Elem) = 1 p(False) = 0 p(Pair) = x1 + x2 p(S) = x1 p(T) = x1 + x2 + x3 p(True) = 0 p(cond_merge_h_t_1) = 0 p(cond_partition_pivot_heap_1) = x3 + x4 + x5 p(cond_partition_pivot_heap_3) = x3 + x4 + x5 + x6 + x7 p(cond_partition_pivot_heap_4) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_5) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_7) = x3 + x4 + x5 + x6 + x7 p(cond_partition_pivot_heap_8) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_9) = x1 + x2 + x3 + x4 + x5 p(leq#2) = 1 + x1 + x1*x2 + x1^2 + x2 + x2^2 p(leqElem#2) = 0 p(main) = 0 p(merge#2) = 0 p(partition#2) = x2 p(cond_merge_h_t_1#) = x1*x2 + x1*x4 p(cond_partition_pivot_heap_1#) = x2*x3 + x2*x5 p(cond_partition_pivot_heap_3#) = x2*x5 + x2*x7 p(cond_partition_pivot_heap_4#) = 0 p(cond_partition_pivot_heap_5#) = 0 p(cond_partition_pivot_heap_7#) = x2*x5 + x2*x7 p(cond_partition_pivot_heap_8#) = 0 p(cond_partition_pivot_heap_9#) = 0 p(leq#2#) = 0 p(leqElem#2#) = x1*x2 p(main#) = 0 p(merge#2#) = x1*x2 p(partition#2#) = x1*x2 p(c_1) = x1 + x2 p(c_2) = 0 p(c_3) = x1 + x2 p(c_4) = 0 p(c_5) = x1 + x2 p(c_6) = x1 p(c_7) = x1 p(c_8) = 0 p(c_9) = 0 p(c_10) = x1 p(c_11) = x1 p(c_12) = 0 p(c_13) = 0 p(c_14) = 0 p(c_15) = 0 p(c_16) = 0 p(c_17) = 0 p(c_18) = 0 p(c_19) = 0 p(c_20) = x1 + x2 p(c_21) = 0 p(c_22) = x1 + x2 Following rules are strictly oriented: leqElem#2#(Elem(x4),Elem(x2)) = 1 > 0 = c_17() Following rules are (at-least) weakly oriented: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) = x1*x4 + x1*x5 + x3*x4 + x3*x5 >= x1*x4 + x3*x5 = c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) = x1*x6 + x3*x6 + x4*x6 + x5*x6 >= x3*x6 + x4*x6 + x5*x6 = c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3),leqElem#2#(x4,x6)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) = x1*x6 + x2*x6 + x3*x6 + x5*x6 >= x1*x6 + x2*x6 + x3*x6 = c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1),leqElem#2#(x2,x6)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) = x1*x6 + x3*x6 >= x3*x6 = c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) = x1*x6 + x3*x6 >= x1*x6 = c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) = x1*x6 + x3*x6 >= x3*x6 = c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) = x1*x6 + x3*x6 >= x1*x6 = c_11(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) = x2*x4 + x2*x6 + x2*x8 >= x2*x4 + x2*x6 + x2*x8 = c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) = x2*x8 + x4*x8 + x6*x8 >= x2*x8 + x4*x8 + x6*x8 = c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2),leqElem#2#(x4,x8)) cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) = x1 + x2 >= x1 + x2 = Pair(E(),T(E(),x2,x1)) cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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()) = x1 + x2 >= x1 + x2 = Pair(T(x2,x1,E()),E()) cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x10 + x12 + x13 + x6 + x7 + x9 >= x10 + x12 + x13 + x6 + x7 + x9 = Pair(T(T(x6,x7,x9),x10,x12),x13) cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) = x10 + x11 + x12 + x13 + x6 + x7 >= x10 + x11 + x12 + x13 + x6 + x7 = Pair(T(x6,x7,x12),T(x13,x10,x11)) cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x10 + x12 + x13 + x7 + x8 + x9 >= x10 + x12 + x13 + x7 + x8 + x9 = Pair(T(x9,x10,x12),T(x13,x7,x8)) cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) = x10 + x11 + x12 + x13 + x7 + x8 >= x10 + x11 + x12 + x13 + x7 + x8 = Pair(x12,T(x13,x10,T(x11,x7,x8))) partition#2(x2,E()) = 0 >= 0 = Pair(E(),E()) partition#2(x8,T(x6,x4,x2)) = x2 + x4 + x6 >= x2 + x4 + x6 = cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2) *** Step 7.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3) ,leqElem#2#(x4,x6)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1) ,leqElem#2#(x2,x6)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) - Weak DPs: leqElem#2#(Elem(x4),Elem(x2)) -> c_17() - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 7.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3) ,leqElem#2#(x4,x6)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1) ,leqElem#2#(x2,x6)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) - Weak DPs: leqElem#2#(Elem(x4),Elem(x2)) -> c_17() - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) -->_2 merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) ,partition#2#(x6,x2)):8 -->_1 merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) ,partition#2#(x6,x2)):8 2:S:cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3),leqElem#2#(x4,x6)) -->_1 cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)):7 -->_1 cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)):6 -->_2 leqElem#2#(Elem(x4),Elem(x2)) -> c_17():10 3:S:cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1),leqElem#2#(x2,x6)) -->_1 cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)):5 -->_1 cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)):4 -->_2 leqElem#2#(Elem(x4),Elem(x2)) -> c_17():10 4:S:cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):9 5:S:cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):9 6:S:cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):9 7:S:cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):9 8:S:merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) -->_2 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):9 -->_1 cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)):1 9:S:partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) -->_2 leqElem#2#(Elem(x4),Elem(x2)) -> c_17():10 -->_1 cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1),leqElem#2#(x2,x6)):3 -->_1 cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3),leqElem#2#(x4,x6)):2 10:W:leqElem#2#(Elem(x4),Elem(x2)) -> c_17() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: leqElem#2#(Elem(x4),Elem(x2)) -> c_17() *** Step 7.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3) ,leqElem#2#(x4,x6)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1) ,leqElem#2#(x2,x6)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/2,c_4/0,c_5/2,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/2} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) -->_2 merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) ,partition#2#(x6,x2)):8 -->_1 merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) ,partition#2#(x6,x2)):8 2:S:cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3),leqElem#2#(x4,x6)) -->_1 cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)):7 -->_1 cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)):6 3:S:cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1),leqElem#2#(x2,x6)) -->_1 cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)):5 -->_1 cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)):4 4:S:cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):9 5:S:cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):9 6:S:cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):9 7:S:cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):9 8:S:merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) -->_2 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)):9 -->_1 cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)):1 9:S:partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2) ,leqElem#2#(x4,x8)) -->_1 cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1),leqElem#2#(x2,x6)):3 -->_1 cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3),leqElem#2#(x4,x6)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(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_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) *** Step 7.b:3.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(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_5(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_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 7: cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) The strictly oriented rules are moved into the weak component. **** Step 7.b:3.b:3.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(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_5(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_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_1) = {1,2}, uargs(c_3) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_20) = {1,2}, uargs(c_22) = {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,partition#2,cond_merge_h_t_1#,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#,merge#2#,partition#2#} TcT has computed the following interpretation: p(0) = 0 p(E) = 0 p(Elem) = 1 p(False) = 0 p(Pair) = x1 + x2 p(S) = 0 p(T) = x1 + x2 + x3 p(True) = 1 p(cond_merge_h_t_1) = 0 p(cond_partition_pivot_heap_1) = x3 + x4 + x5 p(cond_partition_pivot_heap_3) = x3 + x4 + x5 + x6 + x7 p(cond_partition_pivot_heap_4) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_5) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_7) = x3 + x4 + x5 + x6 + x7 p(cond_partition_pivot_heap_8) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_9) = x1 + x2 + x3 + x4 + x5 p(leq#2) = 1 p(leqElem#2) = x1*x2 p(main) = 0 p(merge#2) = 0 p(partition#2) = x2 p(cond_merge_h_t_1#) = x1*x2 + x1*x4 + x1^2 + x2^2 + x3*x4 + x4^2 p(cond_partition_pivot_heap_1#) = x2*x3 + x2*x5 + x2^2 p(cond_partition_pivot_heap_3#) = x2*x5 + x2*x6 + x2*x7 + x2^2 p(cond_partition_pivot_heap_4#) = 0 p(cond_partition_pivot_heap_5#) = 0 p(cond_partition_pivot_heap_7#) = x1 + x2*x5 + x2*x7 + x2^2 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(merge#2#) = x1*x2 + x1^2 + x2^2 p(partition#2#) = x1*x2 + x1^2 p(c_1) = x1 + x2 p(c_2) = 0 p(c_3) = x1 p(c_4) = 0 p(c_5) = x1 p(c_6) = x1 p(c_7) = x1 p(c_8) = 0 p(c_9) = 0 p(c_10) = x1 p(c_11) = x1 p(c_12) = 0 p(c_13) = 0 p(c_14) = 0 p(c_15) = 0 p(c_16) = 0 p(c_17) = 0 p(c_18) = 0 p(c_19) = 0 p(c_20) = x1 + x2 p(c_21) = 0 p(c_22) = x1 Following rules are strictly oriented: cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) = 1 + x1*x6 + x3*x6 + x6^2 > x1*x6 + x6^2 = c_11(partition#2#(x6,x1)) Following rules are (at-least) weakly oriented: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) = x1*x2 + x1*x4 + x1*x5 + x1^2 + x3*x4 + x3*x5 + x3^2 + 2*x4*x5 + x4^2 + x5^2 >= x1*x4 + x1^2 + x3*x5 + x3^2 + x4^2 + x5^2 = c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) = x1*x6 + x3*x6 + x4*x6 + x5*x6 + x6^2 >= x3*x6 + x4*x6 + x5*x6 + x6^2 = c_3(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)) = x1*x6 + x2*x6 + x3*x6 + x5*x6 + x6^2 >= x1*x6 + x2*x6 + x3*x6 + x6^2 = c_5(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) = x1*x6 + x2*x6 + x3*x6 + x6^2 >= x3*x6 + x6^2 = c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) = x1*x6 + x2*x6 + x3*x6 + x6^2 >= x1*x6 + x6^2 = c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) = x1*x6 + x3*x6 + x6^2 >= x3*x6 + x6^2 = c_10(partition#2#(x6,x3)) merge#2#(T(x8,x6,x4),x2) = x2*x4 + x2*x6 + x2*x8 + x2^2 + 2*x4*x6 + 2*x4*x8 + x4^2 + 2*x6*x8 + x6^2 + x8^2 >= x2*x4 + x2*x6 + x2*x8 + x2^2 + x4*x6 + x4^2 + x6^2 + x8^2 = c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) = x2*x8 + x4*x8 + x6*x8 + x8^2 >= x2*x8 + x6*x8 + x8^2 = c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) = x1 + x2 >= x1 + x2 = Pair(E(),T(E(),x2,x1)) cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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()) = x1 + x2 >= x1 + x2 = Pair(T(x2,x1,E()),E()) cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x10 + x12 + x13 + x6 + x7 + x9 >= x10 + x12 + x13 + x6 + x7 + x9 = Pair(T(T(x6,x7,x9),x10,x12),x13) cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) = x10 + x11 + x12 + x13 + x6 + x7 >= x10 + x11 + x12 + x13 + x6 + x7 = Pair(T(x6,x7,x12),T(x13,x10,x11)) cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x10 + x12 + x13 + x7 + x8 + x9 >= x10 + x12 + x13 + x7 + x8 + x9 = Pair(T(x9,x10,x12),T(x13,x7,x8)) cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) = x10 + x11 + x12 + x13 + x7 + x8 >= x10 + x11 + x12 + x13 + x7 + x8 = Pair(x12,T(x13,x10,T(x11,x7,x8))) leq#2(0(),x12) = 1 >= 1 = True() leq#2(S(x16),0()) = 1 >= 0 = False() leq#2(S(x4),S(x2)) = 1 >= 1 = leq#2(x4,x2) leqElem#2(Elem(x4),Elem(x2)) = 1 >= 1 = leq#2(x4,x2) partition#2(x2,E()) = 0 >= 0 = Pair(E(),E()) partition#2(x8,T(x6,x4,x2)) = x2 + x4 + x6 >= x2 + x4 + x6 = cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2) **** Step 7.b:3.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(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_5(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_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 7.b:3.b:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(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_5(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_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 6: cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) The strictly oriented rules are moved into the weak component. ***** Step 7.b:3.b:3.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(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_5(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_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_1) = {1,2}, uargs(c_3) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_20) = {1,2}, uargs(c_22) = {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,partition#2,cond_merge_h_t_1#,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#,merge#2#,partition#2#} TcT has computed the following interpretation: p(0) = 0 p(E) = 0 p(Elem) = 1 p(False) = 1 p(Pair) = x1 + x2 p(S) = 0 p(T) = x1 + x2 + x3 p(True) = 0 p(cond_merge_h_t_1) = 0 p(cond_partition_pivot_heap_1) = x3 + x4 + x5 p(cond_partition_pivot_heap_3) = x3 + x4 + x5 + x6 + x7 p(cond_partition_pivot_heap_4) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_5) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_7) = x3 + x4 + x5 + x6 + x7 p(cond_partition_pivot_heap_8) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_9) = x1 + x2 + x3 + x4 + x5 p(leq#2) = 1 p(leqElem#2) = x1*x2 p(main) = 0 p(merge#2) = 0 p(partition#2) = x2 p(cond_merge_h_t_1#) = x1*x2 + x1*x4 + x1^2 + x2^2 + x4^2 p(cond_partition_pivot_heap_1#) = x2*x3 + x2*x5 + x2^2 p(cond_partition_pivot_heap_3#) = x2*x5 + x2*x6 + x2*x7 + x2^2 p(cond_partition_pivot_heap_4#) = 0 p(cond_partition_pivot_heap_5#) = 0 p(cond_partition_pivot_heap_7#) = x1 + x2*x5 + x2*x7 + x2^2 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(merge#2#) = x1*x2 + x1^2 + x2^2 p(partition#2#) = x1*x2 + x1^2 p(c_1) = x1 + x2 p(c_2) = 0 p(c_3) = x1 p(c_4) = 0 p(c_5) = x1 p(c_6) = x1 p(c_7) = x1 p(c_8) = 0 p(c_9) = 0 p(c_10) = x1 p(c_11) = x1 p(c_12) = 0 p(c_13) = 0 p(c_14) = 0 p(c_15) = 0 p(c_16) = 0 p(c_17) = 0 p(c_18) = 0 p(c_19) = 0 p(c_20) = x1 + x2 p(c_21) = 0 p(c_22) = x1 Following rules are strictly oriented: cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) = 1 + x1*x6 + x3*x6 + x6^2 > x3*x6 + x6^2 = c_10(partition#2#(x6,x3)) Following rules are (at-least) weakly oriented: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) = x1*x4 + x1*x5 + x1^2 + x3*x4 + x3*x5 + x3^2 + 2*x4*x5 + x4^2 + x5^2 >= x1*x4 + x1^2 + x3*x5 + x3^2 + x4^2 + x5^2 = c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) = x1*x6 + x3*x6 + x4*x6 + x5*x6 + x6^2 >= x3*x6 + x4*x6 + x5*x6 + x6^2 = c_3(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)) = x1*x6 + x2*x6 + x3*x6 + x5*x6 + x6^2 >= x1*x6 + x2*x6 + x3*x6 + x6^2 = c_5(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) = x1*x6 + x2*x6 + x3*x6 + x6^2 >= x3*x6 + x6^2 = c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) = x1*x6 + x2*x6 + x3*x6 + x6^2 >= x1*x6 + x6^2 = c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) = x1*x6 + x3*x6 + x6^2 >= x1*x6 + x6^2 = c_11(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) = x2*x4 + x2*x6 + x2*x8 + x2^2 + 2*x4*x6 + 2*x4*x8 + x4^2 + 2*x6*x8 + x6^2 + x8^2 >= x2*x4 + x2*x6 + x2*x8 + x2^2 + x4^2 + x6^2 + x8^2 = c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) = x2*x8 + x4*x8 + x6*x8 + x8^2 >= x2*x8 + x6*x8 + x8^2 = c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) = x1 + x2 >= x1 + x2 = Pair(E(),T(E(),x2,x1)) cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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()) = x1 + x2 >= x1 + x2 = Pair(T(x2,x1,E()),E()) cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x10 + x12 + x13 + x6 + x7 + x9 >= x10 + x12 + x13 + x6 + x7 + x9 = Pair(T(T(x6,x7,x9),x10,x12),x13) cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) = x10 + x11 + x12 + x13 + x6 + x7 >= x10 + x11 + x12 + x13 + x6 + x7 = Pair(T(x6,x7,x12),T(x13,x10,x11)) cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x10 + x12 + x13 + x7 + x8 + x9 >= x10 + x12 + x13 + x7 + x8 + x9 = Pair(T(x9,x10,x12),T(x13,x7,x8)) cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) = x10 + x11 + x12 + x13 + x7 + x8 >= x10 + x11 + x12 + x13 + x7 + x8 = Pair(x12,T(x13,x10,T(x11,x7,x8))) leq#2(0(),x12) = 1 >= 0 = True() leq#2(S(x16),0()) = 1 >= 1 = False() leq#2(S(x4),S(x2)) = 1 >= 1 = leq#2(x4,x2) leqElem#2(Elem(x4),Elem(x2)) = 1 >= 1 = leq#2(x4,x2) partition#2(x2,E()) = 0 >= 0 = Pair(E(),E()) partition#2(x8,T(x6,x4,x2)) = x2 + x4 + x6 >= x2 + x4 + x6 = cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2) ***** Step 7.b:3.b:3.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(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_5(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_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 7.b:3.b:3.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(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_5(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_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3)) Consider the set of all dependency pairs 1: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) 2: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3)) 3: cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1)) 4: cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) 5: cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) 6: merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) 7: partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) 8: cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) 9: cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {2} These cover all (indirect) predecessors of dependency pairs {2,8,9} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ****** Step 7.b:3.b:3.b:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(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_5(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_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_1) = {1,2}, uargs(c_3) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_20) = {1,2}, uargs(c_22) = {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,partition#2,cond_merge_h_t_1#,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#,merge#2#,partition#2#} TcT has computed the following interpretation: p(0) = 1 p(E) = 0 p(Elem) = x1 p(False) = 1 p(Pair) = x1 + x2 p(S) = 1 + x1 p(T) = x1 + x2 + x3 p(True) = 0 p(cond_merge_h_t_1) = 0 p(cond_partition_pivot_heap_1) = x3 + x4 + x5 p(cond_partition_pivot_heap_3) = x3 + x4 + x5 + x6 + x7 p(cond_partition_pivot_heap_4) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_5) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_7) = x3 + x4 + x5 + x6 + x7 p(cond_partition_pivot_heap_8) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_9) = x1 + x2 + x3 + x4 + x5 p(leq#2) = x1*x2 p(leqElem#2) = x1*x2 p(main) = 0 p(merge#2) = 0 p(partition#2) = x2 p(cond_merge_h_t_1#) = x1 + x1*x2 + x1*x4 + x2 + x4 p(cond_partition_pivot_heap_1#) = x1 + x2 + x2*x3 + x2*x5 p(cond_partition_pivot_heap_3#) = x2 + x2*x5 + x2*x6 + x2*x7 p(cond_partition_pivot_heap_4#) = 0 p(cond_partition_pivot_heap_5#) = 0 p(cond_partition_pivot_heap_7#) = x2 + x2*x5 + x2*x7 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(merge#2#) = x1 + x1*x2 + x2 p(partition#2#) = x1 + x1*x2 p(c_1) = x1 + x2 p(c_2) = 0 p(c_3) = x1 p(c_4) = 0 p(c_5) = x1 p(c_6) = x1 p(c_7) = x1 p(c_8) = 0 p(c_9) = 0 p(c_10) = x1 p(c_11) = x1 p(c_12) = 0 p(c_13) = 0 p(c_14) = 0 p(c_15) = 0 p(c_16) = 0 p(c_17) = 0 p(c_18) = 0 p(c_19) = 0 p(c_20) = x1 + x2 p(c_21) = 0 p(c_22) = x1 Following rules are strictly oriented: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) = 1 + x1*x6 + x3*x6 + x4*x6 + x5*x6 + x6 > x3*x6 + x5*x6 + x6 = c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3)) Following rules are (at-least) weakly oriented: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) = x1 + x1*x4 + x1*x5 + x3 + x3*x4 + x3*x5 + x4 + x5 >= x1 + x1*x4 + x3 + x3*x5 + x4 + x5 = c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) = x1*x6 + x2*x6 + x3*x6 + x5*x6 + x6 >= x1*x6 + x2*x6 + x3*x6 + x6 = c_5(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) = x1*x6 + x2*x6 + x3*x6 + x6 >= x3*x6 + x6 = c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) = x1*x6 + x2*x6 + x3*x6 + x6 >= x1*x6 + x6 = c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) = x1*x6 + x3*x6 + x6 >= x3*x6 + x6 = c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) = x1*x6 + x3*x6 + x6 >= x1*x6 + x6 = c_11(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) = x2 + x2*x4 + x2*x6 + x2*x8 + x4 + x6 + x8 >= x2 + x2*x4 + x2*x6 + x2*x8 + x4 + x6 + x8 = c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) = x2*x8 + x4*x8 + x6*x8 + x8 >= x2*x8 + x4*x8 + x6*x8 + x8 = c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) = x1 + x2 >= x1 + x2 = Pair(E(),T(E(),x2,x1)) cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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()) = x1 + x2 >= x1 + x2 = Pair(T(x2,x1,E()),E()) cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x10 + x12 + x13 + x6 + x7 + x9 >= x10 + x12 + x13 + x6 + x7 + x9 = Pair(T(T(x6,x7,x9),x10,x12),x13) cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) = x10 + x11 + x12 + x13 + x6 + x7 >= x10 + x11 + x12 + x13 + x6 + x7 = Pair(T(x6,x7,x12),T(x13,x10,x11)) cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x10 + x12 + x13 + x7 + x8 + x9 >= x10 + x12 + x13 + x7 + x8 + x9 = Pair(T(x9,x10,x12),T(x13,x7,x8)) cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) = x10 + x11 + x12 + x13 + x7 + x8 >= x10 + x11 + x12 + x13 + x7 + x8 = Pair(x12,T(x13,x10,T(x11,x7,x8))) leq#2(0(),x12) = x12 >= 0 = True() leq#2(S(x16),0()) = 1 + x16 >= 1 = False() leq#2(S(x4),S(x2)) = 1 + x2 + x2*x4 + x4 >= x2*x4 = leq#2(x4,x2) leqElem#2(Elem(x4),Elem(x2)) = x2*x4 >= x2*x4 = leq#2(x4,x2) partition#2(x2,E()) = 0 >= 0 = Pair(E(),E()) partition#2(x8,T(x6,x4,x2)) = x2 + x4 + x6 >= x2 + x4 + x6 = cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2) ****** Step 7.b:3.b:3.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(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_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(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_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 7.b:3.b:3.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(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_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(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_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 3: cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) The strictly oriented rules are moved into the weak component. ******* Step 7.b:3.b:3.b:1.b:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(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_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(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_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_1) = {1,2}, uargs(c_3) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_20) = {1,2}, uargs(c_22) = {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,partition#2,cond_merge_h_t_1#,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#,merge#2#,partition#2#} TcT has computed the following interpretation: p(0) = 1 p(E) = 0 p(Elem) = x1 p(False) = 1 p(Pair) = x1 + x2 p(S) = 1 + x1 p(T) = x1 + x2 + x3 p(True) = 0 p(cond_merge_h_t_1) = 0 p(cond_partition_pivot_heap_1) = x3 + x4 + x5 p(cond_partition_pivot_heap_3) = x3 + x4 + x5 + x6 + x7 p(cond_partition_pivot_heap_4) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_5) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_7) = x3 + x4 + x5 + x6 + x7 p(cond_partition_pivot_heap_8) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_9) = x1 + x2 + x3 + x4 + x5 p(leq#2) = x1*x2 p(leqElem#2) = x1*x2 p(main) = 0 p(merge#2) = 0 p(partition#2) = x2 p(cond_merge_h_t_1#) = x1*x2 + x1*x4 p(cond_partition_pivot_heap_1#) = x2*x3 + x2*x5 p(cond_partition_pivot_heap_3#) = x1 + x2*x5 + x2*x7 p(cond_partition_pivot_heap_4#) = 0 p(cond_partition_pivot_heap_5#) = 0 p(cond_partition_pivot_heap_7#) = x2*x5 + x2*x7 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(merge#2#) = x1*x2 p(partition#2#) = x1*x2 p(c_1) = x1 + x2 p(c_2) = 0 p(c_3) = x1 p(c_4) = 0 p(c_5) = x1 p(c_6) = x1 p(c_7) = x1 p(c_8) = 0 p(c_9) = 0 p(c_10) = x1 p(c_11) = x1 p(c_12) = 0 p(c_13) = 0 p(c_14) = 0 p(c_15) = 0 p(c_16) = 0 p(c_17) = 0 p(c_18) = 0 p(c_19) = 0 p(c_20) = x1 + x2 p(c_21) = 0 p(c_22) = x1 Following rules are strictly oriented: cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) = 1 + x1*x6 + x3*x6 > x3*x6 = c_6(partition#2#(x6,x3)) Following rules are (at-least) weakly oriented: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) = x1*x4 + x1*x5 + x3*x4 + x3*x5 >= x1*x4 + x3*x5 = c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) = x1*x6 + x3*x6 + x4*x6 + x5*x6 >= x3*x6 + x5*x6 = c_3(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)) = x1*x6 + x2*x6 + x3*x6 + x5*x6 >= x1*x6 + x2*x6 + x3*x6 = c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) = x1*x6 + x3*x6 >= x1*x6 = c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) = x1*x6 + x3*x6 >= x3*x6 = c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) = x1*x6 + x3*x6 >= x1*x6 = c_11(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) = x2*x4 + x2*x6 + x2*x8 >= x2*x4 + x2*x6 + x2*x8 = c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) = x2*x8 + x4*x8 + x6*x8 >= x2*x8 + x6*x8 = c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) = x1 + x2 >= x1 + x2 = Pair(E(),T(E(),x2,x1)) cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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()) = x1 + x2 >= x1 + x2 = Pair(T(x2,x1,E()),E()) cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x10 + x12 + x13 + x6 + x7 + x9 >= x10 + x12 + x13 + x6 + x7 + x9 = Pair(T(T(x6,x7,x9),x10,x12),x13) cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) = x10 + x11 + x12 + x13 + x6 + x7 >= x10 + x11 + x12 + x13 + x6 + x7 = Pair(T(x6,x7,x12),T(x13,x10,x11)) cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x10 + x12 + x13 + x7 + x8 + x9 >= x10 + x12 + x13 + x7 + x8 + x9 = Pair(T(x9,x10,x12),T(x13,x7,x8)) cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) = x10 + x11 + x12 + x13 + x7 + x8 >= x10 + x11 + x12 + x13 + x7 + x8 = Pair(x12,T(x13,x10,T(x11,x7,x8))) leq#2(0(),x12) = x12 >= 0 = True() leq#2(S(x16),0()) = 1 + x16 >= 1 = False() leq#2(S(x4),S(x2)) = 1 + x2 + x2*x4 + x4 >= x2*x4 = leq#2(x4,x2) leqElem#2(Elem(x4),Elem(x2)) = x2*x4 >= x2*x4 = leq#2(x4,x2) partition#2(x2,E()) = 0 >= 0 = Pair(E(),E()) partition#2(x8,T(x6,x4,x2)) = x2 + x4 + x6 >= x2 + x4 + x6 = cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2) ******* Step 7.b:3.b:3.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 7.b:3.b:3.b:1.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 3: cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) The strictly oriented rules are moved into the weak component. ******** Step 7.b:3.b:3.b:1.b:1.b:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_1) = {1,2}, uargs(c_3) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_20) = {1,2}, uargs(c_22) = {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,partition#2,cond_merge_h_t_1#,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#,merge#2#,partition#2#} TcT has computed the following interpretation: p(0) = 0 p(E) = 0 p(Elem) = 1 p(False) = 1 p(Pair) = x1 + x2 p(S) = 0 p(T) = x1 + x2 + x3 p(True) = 1 p(cond_merge_h_t_1) = 0 p(cond_partition_pivot_heap_1) = x3 + x4 + x5 p(cond_partition_pivot_heap_3) = x3 + x4 + x5 + x6 + x7 p(cond_partition_pivot_heap_4) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_5) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_7) = x3 + x4 + x5 + x6 + x7 p(cond_partition_pivot_heap_8) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_9) = x1 + x2 + x3 + x4 + x5 p(leq#2) = 1 p(leqElem#2) = x1*x2 p(main) = 0 p(merge#2) = 0 p(partition#2) = x2 p(cond_merge_h_t_1#) = x1 + x1*x2 + x1*x4 + x1^2 + x2 + x2^2 + x4 + x4^2 p(cond_partition_pivot_heap_1#) = x2 + x2*x3 + x2*x4 + x2*x5 + x2^2 p(cond_partition_pivot_heap_3#) = x1 + x2 + x2*x4 + x2*x5 + x2*x7 + x2^2 p(cond_partition_pivot_heap_4#) = 0 p(cond_partition_pivot_heap_5#) = 0 p(cond_partition_pivot_heap_7#) = x1 + x2 + x2*x5 + x2*x7 + x2^2 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(merge#2#) = x1 + x1*x2 + x1^2 + x2 + x2^2 p(partition#2#) = x1 + x1*x2 + x1^2 p(c_1) = x1 + x2 p(c_2) = 0 p(c_3) = x1 p(c_4) = 0 p(c_5) = x1 p(c_6) = x1 p(c_7) = x1 p(c_8) = 0 p(c_9) = 0 p(c_10) = 1 + x1 p(c_11) = x1 p(c_12) = 0 p(c_13) = 0 p(c_14) = 0 p(c_15) = 0 p(c_16) = 0 p(c_17) = 0 p(c_18) = 0 p(c_19) = 0 p(c_20) = x1 + x2 p(c_21) = 0 p(c_22) = x1 Following rules are strictly oriented: cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) = 1 + x1*x6 + x3*x6 + x4*x6 + x6 + x6^2 > x1*x6 + x6 + x6^2 = c_7(partition#2#(x6,x1)) Following rules are (at-least) weakly oriented: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) = x1 + x1*x4 + x1*x5 + x1^2 + x3 + x3*x4 + x3*x5 + x3^2 + x4 + 2*x4*x5 + x4^2 + x5 + x5^2 >= x1 + x1*x4 + x1^2 + x3 + x3*x5 + x3^2 + x4 + x4^2 + x5 + x5^2 = c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) = x1*x6 + x2*x6 + x3*x6 + x4*x6 + x5*x6 + x6 + x6^2 >= x3*x6 + x4*x6 + x5*x6 + x6 + x6^2 = c_3(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)) = x1*x6 + x2*x6 + x3*x6 + x4*x6 + x5*x6 + x6 + x6^2 >= x1*x6 + x2*x6 + x3*x6 + x4*x6 + x6 + x6^2 = c_5(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*x6 + x3*x6 + x4*x6 + x6 + x6^2 >= x3*x6 + x6 + x6^2 = c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) = 1 + x1*x6 + x3*x6 + x6 + x6^2 >= 1 + x3*x6 + x6 + x6^2 = c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) = 1 + x1*x6 + x3*x6 + x6 + x6^2 >= x1*x6 + x6 + x6^2 = c_11(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) = x2 + x2*x4 + x2*x6 + x2*x8 + x2^2 + x4 + 2*x4*x6 + 2*x4*x8 + x4^2 + x6 + 2*x6*x8 + x6^2 + x8 + x8^2 >= x2 + x2*x4 + x2*x6 + x2*x8 + x2^2 + x4 + x4^2 + x6 + x6^2 + x8 + x8^2 = c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) = x2*x8 + x4*x8 + x6*x8 + x8 + x8^2 >= x2*x8 + x4*x8 + x6*x8 + x8 + x8^2 = c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) = x1 + x2 >= x1 + x2 = Pair(E(),T(E(),x2,x1)) cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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()) = x1 + x2 >= x1 + x2 = Pair(T(x2,x1,E()),E()) cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x10 + x12 + x13 + x6 + x7 + x9 >= x10 + x12 + x13 + x6 + x7 + x9 = Pair(T(T(x6,x7,x9),x10,x12),x13) cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) = x10 + x11 + x12 + x13 + x6 + x7 >= x10 + x11 + x12 + x13 + x6 + x7 = Pair(T(x6,x7,x12),T(x13,x10,x11)) cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x10 + x12 + x13 + x7 + x8 + x9 >= x10 + x12 + x13 + x7 + x8 + x9 = Pair(T(x9,x10,x12),T(x13,x7,x8)) cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) = x10 + x11 + x12 + x13 + x7 + x8 >= x10 + x11 + x12 + x13 + x7 + x8 = Pair(x12,T(x13,x10,T(x11,x7,x8))) leq#2(0(),x12) = 1 >= 1 = True() leq#2(S(x16),0()) = 1 >= 1 = False() leq#2(S(x4),S(x2)) = 1 >= 1 = leq#2(x4,x2) leqElem#2(Elem(x4),Elem(x2)) = 1 >= 1 = leq#2(x4,x2) partition#2(x2,E()) = 0 >= 0 = Pair(E(),E()) partition#2(x8,T(x6,x4,x2)) = x2 + x4 + x6 >= x2 + x4 + x6 = cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2) ******** Step 7.b:3.b:3.b:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 7.b:3.b:3.b:1.b:1.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 3: merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) Consider the set of all dependency pairs 1: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) 2: cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1)) 3: merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) 4: partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) 5: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3)) 6: cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) 7: cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) 8: cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) 9: cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) Processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {3} These cover all (indirect) predecessors of dependency pairs {1,3} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ********* Step 7.b:3.b:3.b:1.b:1.b:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True} + Applied Processor: NaturalMI {miDimension = 2, 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 (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_1) = {1,2}, uargs(c_3) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_20) = {1,2}, uargs(c_22) = {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,partition#2,cond_merge_h_t_1#,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#,merge#2#,partition#2#} TcT has computed the following interpretation: p(0) = [0] [0] p(E) = [0] [0] p(Elem) = [0] [0] p(False) = [0] [1] p(Pair) = [0 1] x1 + [0 1] x2 + [0] [0 0] [0 0] [1] p(S) = [0] [0] p(T) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0] [0 1] [0 1] [0 1] [1] p(True) = [0] [1] p(cond_merge_h_t_1) = [0] [0] p(cond_partition_pivot_heap_1) = [0 1] x3 + [0 1] x4 + [0 1] x5 + [1] [0 0] [0 0] [0 0] [1] p(cond_partition_pivot_heap_3) = [0 1] x1 + [0 1] x3 + [0 1] x4 + [0 1] x5 + [0 1] x6 + [0 1] x7 + [1] [0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [1] p(cond_partition_pivot_heap_4) = [1 1] x1 + [0 1] x2 + [0 1] x3 + [0 1] x4 + [0 1] x5 + [1] [0 0] [0 0] [0 0] [0 0] [0 0] [1] p(cond_partition_pivot_heap_5) = [1 1] x1 + [0 1] x2 + [0 1] x3 + [0 1] x4 + [0 1] x5 + [1] [0 0] [0 0] [0 0] [0 0] [0 0] [1] p(cond_partition_pivot_heap_7) = [0 1] x1 + [0 1] x3 + [0 1] x4 + [0 1] x5 + [0 1] x6 + [0 1] x7 + [1] [0 1] [0 0] [0 0] [0 0] [0 0] [0 0] [0] p(cond_partition_pivot_heap_8) = [1 1] x1 + [0 1] x2 + [0 1] x3 + [0 1] x4 + [0 1] x5 + [1] [0 0] [0 0] [0 0] [0 0] [0 0] [1] p(cond_partition_pivot_heap_9) = [1 1] x1 + [0 1] x2 + [0 1] x3 + [0 1] x4 + [0 1] x5 + [1] [0 0] [0 0] [0 0] [0 0] [0 0] [1] p(leq#2) = [0] [1] p(leqElem#2) = [0] [1] p(main) = [0] [0] p(merge#2) = [0] [0] p(partition#2) = [0 1] x2 + [0] [0 0] [1] p(cond_merge_h_t_1#) = [1 0] x1 + [0 1] x2 + [0 1] x3 + [0 1] x4 + [0] [0 0] [0 1] [0 0] [0 0] [1] p(cond_partition_pivot_heap_1#) = [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0] [1 1] [0 1] [0 1] [0 1] [1] p(cond_partition_pivot_heap_3#) = [0 0] x2 + [0] [0 1] [0] p(cond_partition_pivot_heap_4#) = [0] [0] p(cond_partition_pivot_heap_5#) = [0] [0] p(cond_partition_pivot_heap_7#) = [0 0] x1 + [0 0] x2 + [0 0] x4 + [0] [0 1] [0 1] [0 1] [1] p(cond_partition_pivot_heap_8#) = [0] [0] p(cond_partition_pivot_heap_9#) = [0] [0] p(leq#2#) = [0] [0] p(leqElem#2#) = [0] [0] p(main#) = [0] [0] p(merge#2#) = [0 1] x1 + [0 1] x2 + [0] [0 0] [0 0] [1] p(partition#2#) = [0 0] x1 + [0] [0 1] [0] p(c_1) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 0] [0] p(c_2) = [0] [0] p(c_3) = [1 0] x1 + [0] [0 1] [0] p(c_4) = [0] [0] p(c_5) = [1 0] x1 + [0] [0 0] [1] p(c_6) = [1 0] x1 + [0] [0 1] [0] p(c_7) = [1 0] x1 + [0] [0 1] [0] p(c_8) = [0] [0] p(c_9) = [0] [0] p(c_10) = [1 0] x1 + [0] [0 1] [1] p(c_11) = [1 0] x1 + [0] [0 0] [0] p(c_12) = [0] [0] p(c_13) = [0] [0] p(c_14) = [0] [0] p(c_15) = [0] [0] p(c_16) = [0] [0] p(c_17) = [0] [0] p(c_18) = [0] [0] p(c_19) = [0] [0] p(c_20) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [1] p(c_21) = [0] [0] p(c_22) = [1 0] x1 + [0] [0 0] [0] Following rules are strictly oriented: merge#2#(T(x8,x6,x4),x2) = [0 1] x2 + [0 1] x4 + [0 1] x6 + [0 1] x8 + [1] [0 0] [0 0] [0 0] [0 0] [1] > [0 1] x2 + [0 1] x4 + [0 1] x6 + [0 1] x8 + [0] [0 0] [0 0] [0 0] [0 0] [1] = c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) Following rules are (at-least) weakly oriented: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) = [0 1] x1 + [0 1] x2 + [0 1] x3 + [0 1] x4 + [0 1] x5 + [0] [0 0] [0 0] [0 1] [0 0] [0 0] [1] >= [0 1] x1 + [0 1] x3 + [0 1] x4 + [0 1] x5 + [0] [0 0] [0 0] [0 0] [0 0] [1] = c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0] [0 1] [0 1] [0 1] [0 1] [0 1] [1 1] [2] >= [0 0] x1 + [0 0] x6 + [0] [0 1] [0 1] [2] = c_3(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)) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0 0] x4 + [0 0] x5 + [0 0] x6 + [0] [0 1] [0 1] [0 1] [0 1] [0 1] [1 1] [2] >= [0] [1] = c_5(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) = [0 0] x6 + [0] [0 1] [0] >= [0 0] x6 + [0] [0 1] [0] = c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) = [0 0] x6 + [0] [0 1] [0] >= [0 0] x6 + [0] [0 1] [0] = c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) = [0 0] x4 + [0 0] x6 + [0] [0 1] [0 1] [2] >= [0 0] x6 + [0] [0 1] [1] = c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) = [0 0] x4 + [0 0] x6 + [0] [0 1] [0 1] [2] >= [0] [0] = c_11(partition#2#(x6,x1)) partition#2#(x8,T(x6,x4,x2)) = [0 0] x8 + [0] [0 1] [0] >= [0] [0] = c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) = [0 1] x1 + [0 1] x2 + [1] [0 0] [0 0] [1] >= [0 1] x1 + [0 1] x2 + [1] [0 0] [0 0] [1] = Pair(E(),T(E(),x2,x1)) cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) = [0 1] x1 + [0 1] x2 + [0 1] x3 + [0 1] x4 + [0 1] x5 + [2] [0 0] [0 0] [0 0] [0 0] [0 0] [1] >= [0 1] x1 + [0 1] x2 + [0 1] x3 + [0 1] x4 + [0 1] x5 + [2] [0 0] [0 0] [0 0] [0 0] [0 0] [1] = 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()) = [0 1] x1 + [0 1] x2 + [1] [0 0] [0 0] [1] >= [0 1] x1 + [0 1] x2 + [1] [0 0] [0 0] [1] = Pair(T(x2,x1,E()),E()) cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) = [0 1] x1 + [0 1] x2 + [0 1] x3 + [0 1] x4 + [0 1] x5 + [2] [0 0] [0 0] [0 0] [0 0] [0 0] [1] >= [0 1] x1 + [0 1] x2 + [0 1] x3 + [0 1] x4 + [0 1] x5 + [2] [0 0] [0 0] [0 0] [0 0] [0 0] [1] = 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) = [0 1] x1 + [0 1] x2 + [0 1] x3 + [0 1] x4 + [0 1] x5 + [2] [0 0] [0 0] [0 0] [0 0] [0 0] [1] >= [0 1] x1 + [0 1] x2 + [0 1] x3 + [0 1] x4 + [0 1] x5 + [2] [0 0] [0 0] [0 0] [0 0] [0 0] [1] = 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) = [0 1] x1 + [0 1] x2 + [0 1] x3 + [0 1] x4 + [0 1] x5 + [2] [0 0] [0 0] [0 0] [0 0] [0 0] [1] >= [0 1] x1 + [0 1] x2 + [0 1] x3 + [0 1] x4 + [0 1] x5 + [2] [0 0] [0 0] [0 0] [0 0] [0 0] [1] = 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) = [0 1] x10 + [0 1] x12 + [0 1] x13 + [0 1] x6 + [0 1] x7 + [0 1] x9 + [2] [0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [1] >= [0 1] x10 + [0 1] x12 + [0 1] x13 + [0 1] x6 + [0 1] x7 + [0 1] x9 + [2] [0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [1] = Pair(T(T(x6,x7,x9),x10,x12),x13) cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) = [0 1] x10 + [0 1] x11 + [0 1] x12 + [0 1] x13 + [0 1] x6 + [0 1] x7 + [2] [0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [1] >= [0 1] x10 + [0 1] x11 + [0 1] x12 + [0 1] x13 + [0 1] x6 + [0 1] x7 + [2] [0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [1] = Pair(T(x6,x7,x12),T(x13,x10,x11)) cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) = [0 1] x1 + [0 1] x2 + [0 1] x3 + [0 1] x4 + [0 1] x5 + [2] [0 0] [0 0] [0 0] [0 0] [0 0] [1] >= [0 1] x1 + [0 1] x2 + [0 1] x3 + [0 1] x4 + [0 1] x5 + [2] [0 0] [0 0] [0 0] [0 0] [0 0] [1] = 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) = [0 1] x1 + [0 1] x2 + [0 1] x3 + [0 1] x4 + [0 1] x5 + [2] [0 0] [0 0] [0 0] [0 0] [0 0] [1] >= [0 1] x1 + [0 1] x2 + [0 1] x3 + [0 1] x4 + [0 1] x5 + [2] [0 0] [0 0] [0 0] [0 0] [0 0] [1] = 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) = [0 1] x10 + [0 1] x12 + [0 1] x13 + [0 1] x7 + [0 1] x8 + [0 1] x9 + [2] [0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [1] >= [0 1] x10 + [0 1] x12 + [0 1] x13 + [0 1] x7 + [0 1] x8 + [0 1] x9 + [2] [0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [1] = Pair(T(x9,x10,x12),T(x13,x7,x8)) cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) = [0 1] x10 + [0 1] x11 + [0 1] x12 + [0 1] x13 + [0 1] x7 + [0 1] x8 + [2] [0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [1] >= [0 1] x10 + [0 1] x11 + [0 1] x12 + [0 1] x13 + [0 1] x7 + [0 1] x8 + [2] [0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [1] = Pair(x12,T(x13,x10,T(x11,x7,x8))) leq#2(0(),x12) = [0] [1] >= [0] [1] = True() leq#2(S(x16),0()) = [0] [1] >= [0] [1] = False() leq#2(S(x4),S(x2)) = [0] [1] >= [0] [1] = leq#2(x4,x2) leqElem#2(Elem(x4),Elem(x2)) = [0] [1] >= [0] [1] = leq#2(x4,x2) partition#2(x2,E()) = [0] [1] >= [0] [1] = Pair(E(),E()) partition#2(x8,T(x6,x4,x2)) = [0 1] x2 + [0 1] x4 + [0 1] x6 + [1] [0 0] [0 0] [0 0] [1] >= [0 1] x2 + [0 1] x4 + [0 1] x6 + [1] [0 0] [0 0] [0 0] [1] = cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2) ********* Step 7.b:3.b:3.b:1.b:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 7.b:3.b:3.b:1.b:1.b:1.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1)) Consider the set of all dependency pairs 1: cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1)) 2: partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) 3: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) 4: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3)) 5: cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) 6: cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) 7: cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) 8: cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) 9: merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,5,6} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ********** Step 7.b:3.b:3.b:1.b:1.b:1.b:1.b:1.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2 ,x6) ,x6 ,x5 ,x4 ,x3 ,x2 ,x1)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4 ,x6) ,x6 ,x2 ,x1 ,x5 ,x4 ,x3)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_1) = {1,2}, uargs(c_3) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_20) = {1,2}, uargs(c_22) = {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,partition#2,cond_merge_h_t_1#,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#,merge#2#,partition#2#} TcT has computed the following interpretation: p(0) = 0 p(E) = 0 p(Elem) = 1 p(False) = 0 p(Pair) = x1 + x2 p(S) = 0 p(T) = x1 + x2 + x3 p(True) = 1 p(cond_merge_h_t_1) = 0 p(cond_partition_pivot_heap_1) = x3 + x4 + x5 p(cond_partition_pivot_heap_3) = x3 + x4 + x5 + x6 + x7 p(cond_partition_pivot_heap_4) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_5) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_7) = x3 + x4 + x5 + x6 + x7 p(cond_partition_pivot_heap_8) = x1 + x2 + x3 + x4 + x5 p(cond_partition_pivot_heap_9) = x1 + x2 + x3 + x4 + x5 p(leq#2) = 1 p(leqElem#2) = x1*x2 p(main) = 0 p(merge#2) = 0 p(partition#2) = x2 p(cond_merge_h_t_1#) = x1*x2 + x1*x4 p(cond_partition_pivot_heap_1#) = x1 + x2*x3 + x2*x5 p(cond_partition_pivot_heap_3#) = x2*x5 + x2*x7 p(cond_partition_pivot_heap_4#) = 0 p(cond_partition_pivot_heap_5#) = 0 p(cond_partition_pivot_heap_7#) = x2*x4 + x2*x5 + x2*x6 + x2*x7 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(merge#2#) = x1*x2 p(partition#2#) = x1*x2 p(c_1) = x1 + x2 p(c_2) = 0 p(c_3) = x1 p(c_4) = 0 p(c_5) = x1 p(c_6) = x1 p(c_7) = x1 p(c_8) = 0 p(c_9) = 0 p(c_10) = x1 p(c_11) = x1 p(c_12) = 0 p(c_13) = 0 p(c_14) = 0 p(c_15) = 0 p(c_16) = 0 p(c_17) = 0 p(c_18) = 0 p(c_19) = 0 p(c_20) = x1 + x2 p(c_21) = 0 p(c_22) = x1 Following rules are strictly oriented: cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) = 1 + x1*x6 + x2*x6 + x3*x6 + x5*x6 > x1*x6 + x3*x6 = c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1)) Following rules are (at-least) weakly oriented: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) = x1*x4 + x1*x5 + x3*x4 + x3*x5 >= x1*x4 + x3*x5 = c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) = x1*x6 + x3*x6 + x4*x6 + x5*x6 >= x1*x6 + x3*x6 + x4*x6 + x5*x6 = c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3)) cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) = x1*x6 + x3*x6 >= x3*x6 = c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) = x1*x6 + x3*x6 >= x1*x6 = c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) = x1*x6 + x2*x6 + x3*x6 + x4*x6 >= x3*x6 = c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) = x1*x6 + x2*x6 + x3*x6 + x4*x6 >= x1*x6 = c_11(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) = x2*x4 + x2*x6 + x2*x8 >= x2*x4 + x2*x6 + x2*x8 = c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) partition#2#(x8,T(x6,x4,x2)) = x2*x8 + x4*x8 + x6*x8 >= x2*x8 + x4*x8 + x6*x8 = c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) = x1 + x2 >= x1 + x2 = Pair(E(),T(E(),x2,x1)) cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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()) = x1 + x2 >= x1 + x2 = Pair(T(x2,x1,E()),E()) cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x10 + x12 + x13 + x6 + x7 + x9 >= x10 + x12 + x13 + x6 + x7 + x9 = Pair(T(T(x6,x7,x9),x10,x12),x13) cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) = x10 + x11 + x12 + x13 + x6 + x7 >= x10 + x11 + x12 + x13 + x6 + x7 = Pair(T(x6,x7,x12),T(x13,x10,x11)) cond_partition_pivot_heap_7(False(),x6,x5,x4,x3,x2,x1) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x1 + x2 + x3 + x4 + x5 >= x1 + x2 + x3 + x4 + x5 = 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) = x10 + x12 + x13 + x7 + x8 + x9 >= x10 + x12 + x13 + x7 + x8 + x9 = Pair(T(x9,x10,x12),T(x13,x7,x8)) cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) = x10 + x11 + x12 + x13 + x7 + x8 >= x10 + x11 + x12 + x13 + x7 + x8 = Pair(x12,T(x13,x10,T(x11,x7,x8))) leq#2(0(),x12) = 1 >= 1 = True() leq#2(S(x16),0()) = 1 >= 0 = False() leq#2(S(x4),S(x2)) = 1 >= 1 = leq#2(x4,x2) leqElem#2(Elem(x4),Elem(x2)) = 1 >= 1 = leq#2(x4,x2) partition#2(x2,E()) = 0 >= 0 = Pair(E(),E()) partition#2(x8,T(x6,x4,x2)) = x2 + x4 + x6 >= x2 + x4 + x6 = cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2) ********** Step 7.b:3.b:3.b:1.b:1.b:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(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_5(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_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 7.b:3.b:3.b:1.b:1.b:1.b:1.b:1.b:1.b:1: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(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_5(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_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) and a lower component cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(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_5(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_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) Further, following extension rules are added to the lower component. cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> merge#2#(x4,x1) cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> merge#2#(x5,x3) merge#2#(T(x8,x6,x4),x2) -> cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) merge#2#(T(x8,x6,x4),x2) -> partition#2#(x6,x2) *********** Step 7.b:3.b:3.b:1.b:1.b:1.b:1.b:1.b:1.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) - Weak DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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: merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) Consider the set of all dependency pairs 1: merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) 2: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ************ Step 7.b:3.b:3.b:1.b:1.b:1.b:1.b:1.b:1.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) - Weak DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) - 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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_1) = {1,2}, uargs(c_20) = {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,partition#2,cond_merge_h_t_1#,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#,merge#2#,partition#2#} TcT has computed the following interpretation: p(0) = [2] p(E) = [0] p(Elem) = [0] p(False) = [0] p(Pair) = [1] x1 + [1] x2 + [0] p(S) = [1] p(T) = [1] x1 + [1] x3 + [1] p(True) = [0] p(cond_merge_h_t_1) = [1] x2 + [2] x3 + [1] p(cond_partition_pivot_heap_1) = [1] x3 + [1] x5 + [1] p(cond_partition_pivot_heap_3) = [1] x3 + [1] x5 + [1] x7 + [2] p(cond_partition_pivot_heap_4) = [1] x1 + [1] x2 + [1] x4 + [2] p(cond_partition_pivot_heap_5) = [1] x1 + [1] x2 + [1] x5 + [2] p(cond_partition_pivot_heap_7) = [1] x4 + [1] x5 + [1] x7 + [2] p(cond_partition_pivot_heap_8) = [1] x1 + [1] x3 + [1] x4 + [2] p(cond_partition_pivot_heap_9) = [1] x1 + [1] x3 + [1] x5 + [2] p(leq#2) = [1] x2 + [0] p(leqElem#2) = [0] p(main) = [2] x1 + [1] x2 + [0] p(merge#2) = [0] p(partition#2) = [1] x2 + [0] p(cond_merge_h_t_1#) = [1] x1 + [1] x2 + [1] x4 + [0] p(cond_partition_pivot_heap_1#) = [1] x1 + [4] x2 + [1] x3 + [4] x4 + [1] x5 + [0] p(cond_partition_pivot_heap_3#) = [4] x2 + [1] x3 + [1] x4 + [1] x5 + [2] p(cond_partition_pivot_heap_4#) = [1] x2 + [1] x4 + [1] p(cond_partition_pivot_heap_5#) = [4] x4 + [1] x5 + [4] p(cond_partition_pivot_heap_7#) = [1] x3 + [1] x5 + [1] x7 + [0] p(cond_partition_pivot_heap_8#) = [1] x1 + [4] x3 + [1] x5 + [2] p(cond_partition_pivot_heap_9#) = [2] x2 + [1] x4 + [1] p(leq#2#) = [1] x2 + [4] p(leqElem#2#) = [1] p(main#) = [1] x2 + [0] p(merge#2#) = [1] x1 + [1] x2 + [0] p(partition#2#) = [2] x1 + [0] p(c_1) = [1] x1 + [1] x2 + [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] p(c_5) = [0] p(c_6) = [1] p(c_7) = [2] p(c_8) = [1] p(c_9) = [1] p(c_10) = [0] p(c_11) = [2] p(c_12) = [1] p(c_13) = [0] p(c_14) = [0] p(c_15) = [1] p(c_16) = [1] x1 + [2] p(c_17) = [0] p(c_18) = [0] p(c_19) = [1] p(c_20) = [1] x1 + [0] p(c_21) = [1] p(c_22) = [0] Following rules are strictly oriented: merge#2#(T(x8,x6,x4),x2) = [1] x2 + [1] x4 + [1] x8 + [1] > [1] x2 + [1] x4 + [1] x8 + [0] = c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) Following rules are (at-least) weakly oriented: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) = [1] x1 + [1] x3 + [1] x4 + [1] x5 + [0] >= [1] x1 + [1] x3 + [1] x4 + [1] x5 + [0] = c_1(merge#2#(x5,x3),merge#2#(x4,x1)) cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) = [1] x1 + [1] >= [1] x1 + [1] = Pair(E(),T(E(),x2,x1)) cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) = [1] x1 + [1] x3 + [1] x5 + [2] >= [1] x1 + [1] x3 + [1] x5 + [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()) = [1] x2 + [1] >= [1] x2 + [1] = Pair(T(x2,x1,E()),E()) cond_partition_pivot_heap_1(True(),x6,x5,x4,T(x3,x2,x1)) = [1] x1 + [1] x3 + [1] x5 + [2] >= [1] x1 + [1] x3 + [1] x5 + [2] = 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 + [1] x5 + [2] >= [1] x1 + [1] x3 + [1] x5 + [2] = 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 + [1] x5 + [2] >= [1] x1 + [1] x3 + [1] x5 + [2] = 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] x12 + [1] x13 + [1] x6 + [1] x9 + [2] >= [1] x12 + [1] x13 + [1] x6 + [1] x9 + [2] = Pair(T(T(x6,x7,x9),x10,x12),x13) cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) = [1] x11 + [1] x12 + [1] x13 + [1] x6 + [2] >= [1] x11 + [1] x12 + [1] x13 + [1] x6 + [2] = 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 + [2] >= [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 + [2] >= [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] x12 + [1] x13 + [1] x8 + [1] x9 + [2] >= [1] x12 + [1] x13 + [1] x8 + [1] x9 + [2] = Pair(T(x9,x10,x12),T(x13,x7,x8)) cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) = [1] x11 + [1] x12 + [1] x13 + [1] x8 + [2] >= [1] x11 + [1] x12 + [1] x13 + [1] x8 + [2] = Pair(x12,T(x13,x10,T(x11,x7,x8))) partition#2(x2,E()) = [0] >= [0] = Pair(E(),E()) partition#2(x8,T(x6,x4,x2)) = [1] x2 + [1] x6 + [1] >= [1] x2 + [1] x6 + [1] = cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2) ************ Step 7.b:3.b:3.b:1.b:1.b:1.b:1.b:1.b:1.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 7.b:3.b:3.b:1.b:1.b:1.b:1.b:1.b:1.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) -->_2 merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) ,partition#2#(x6,x2)):2 -->_1 merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) ,partition#2#(x6,x2)):2 2:W:merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) -->_1 cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> c_1(merge#2#(x5,x3),merge#2#(x4,x1)) 2: merge#2#(T(x8,x6,x4),x2) -> c_20(cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4),partition#2#(x6,x2)) ************ Step 7.b:3.b:3.b:1.b:1.b:1.b:1.b:1.b:1.b:1.a:1.b:2: 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 7.b:3.b:3.b:1.b:1.b:1.b:1.b:1.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> merge#2#(x4,x1) cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> merge#2#(x5,x3) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(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_5(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_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) merge#2#(T(x8,x6,x4),x2) -> partition#2#(x6,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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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: partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) Consider the set of all dependency pairs 1: partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) 2: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> merge#2#(x4,x1) 3: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> merge#2#(x5,x3) 4: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3)) 5: cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(cond_partition_pivot_heap_3#(leqElem#2(x2,x6),x6,x5,x4,x3,x2,x1)) 6: cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) 7: cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) 8: cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) 9: cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) 10: merge#2#(T(x8,x6,x4),x2) -> cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) 11: merge#2#(T(x8,x6,x4),x2) -> partition#2#(x6,x2) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,4,5,6,7,8,9} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ************ Step 7.b:3.b:3.b:1.b:1.b:1.b:1.b:1.b:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) - Weak DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> merge#2#(x4,x1) cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> merge#2#(x5,x3) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(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_5(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_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) merge#2#(T(x8,x6,x4),x2) -> partition#2#(x6,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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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_3) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1}, uargs(c_7) = {1}, uargs(c_10) = {1}, uargs(c_11) = {1}, uargs(c_22) = {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,partition#2,cond_merge_h_t_1#,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#,merge#2#,partition#2#} TcT has computed the following interpretation: p(0) = [0] p(E) = [0] p(Elem) = [2] p(False) = [1] p(Pair) = [1] x1 + [1] x2 + [0] p(S) = [1] x1 + [4] p(T) = [1] x1 + [1] x2 + [1] x3 + [0] p(True) = [1] p(cond_merge_h_t_1) = [1] x1 + [1] x3 + [1] p(cond_partition_pivot_heap_1) = [1] x3 + [1] x4 + [1] x5 + [0] p(cond_partition_pivot_heap_3) = [1] x3 + [1] x4 + [1] x5 + [1] x6 + [1] x7 + [0] p(cond_partition_pivot_heap_4) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [0] p(cond_partition_pivot_heap_5) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [0] p(cond_partition_pivot_heap_7) = [1] x3 + [1] x4 + [1] x5 + [1] x6 + [1] x7 + [0] p(cond_partition_pivot_heap_8) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [0] p(cond_partition_pivot_heap_9) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [0] p(leq#2) = [2] p(leqElem#2) = [1] x1 + [0] p(main) = [0] p(merge#2) = [1] x1 + [1] x2 + [0] p(partition#2) = [1] x2 + [0] p(cond_merge_h_t_1#) = [4] x1 + [4] x2 + [4] x3 + [4] x4 + [2] p(cond_partition_pivot_heap_1#) = [4] x2 + [4] x3 + [2] x4 + [4] x5 + [0] p(cond_partition_pivot_heap_3#) = [1] x1 + [4] x2 + [2] x4 + [4] x5 + [1] x6 + [4] x7 + [0] p(cond_partition_pivot_heap_4#) = [1] x1 + [1] x2 + [4] x4 + [4] x5 + [0] p(cond_partition_pivot_heap_5#) = [1] x1 + [4] x2 + [2] x5 + [2] p(cond_partition_pivot_heap_7#) = [4] x1 + [4] x2 + [1] x3 + [3] x4 + [4] x5 + [4] x7 + [0] p(cond_partition_pivot_heap_8#) = [4] x1 + [1] p(cond_partition_pivot_heap_9#) = [1] x2 + [1] x4 + [1] p(leq#2#) = [1] x2 + [2] p(leqElem#2#) = [2] x1 + [0] p(main#) = [1] x1 + [0] p(merge#2#) = [4] x1 + [4] x2 + [2] p(partition#2#) = [4] x1 + [4] x2 + [1] p(c_1) = [1] x2 + [4] p(c_2) = [4] p(c_3) = [1] x1 + [0] p(c_4) = [1] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [4] p(c_10) = [1] x1 + [3] p(c_11) = [1] x1 + [0] p(c_12) = [2] p(c_13) = [2] p(c_14) = [4] p(c_15) = [1] p(c_16) = [1] x1 + [1] p(c_17) = [1] p(c_18) = [0] p(c_19) = [0] p(c_20) = [2] x1 + [1] x2 + [0] p(c_21) = [0] p(c_22) = [1] x1 + [0] Following rules are strictly oriented: partition#2#(x8,T(x6,x4,x2)) = [4] x2 + [4] x4 + [4] x6 + [4] x8 + [1] > [4] x2 + [2] x4 + [4] x6 + [4] x8 + [0] = c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) Following rules are (at-least) weakly oriented: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) = [4] x1 + [4] x2 + [4] x3 + [4] x4 + [4] x5 + [2] >= [4] x1 + [4] x4 + [2] = merge#2#(x4,x1) cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) = [4] x1 + [4] x2 + [4] x3 + [4] x4 + [4] x5 + [2] >= [4] x3 + [4] x5 + [2] = merge#2#(x5,x3) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) = [4] x1 + [2] x2 + [4] x3 + [4] x4 + [4] x5 + [4] x6 + [0] >= [3] x1 + [1] x2 + [4] x3 + [4] x4 + [4] x5 + [4] x6 + [0] = c_3(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)) = [4] x1 + [4] x2 + [4] x3 + [2] x4 + [4] x5 + [4] x6 + [0] >= [4] x1 + [2] x2 + [4] x3 + [2] x4 + [4] x6 + [0] = c_5(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 + [1] x2 + [4] x3 + [2] x4 + [4] x6 + [1] >= [4] x3 + [4] x6 + [1] = c_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) = [4] x1 + [1] x2 + [4] x3 + [2] x4 + [4] x6 + [1] >= [4] x1 + [4] x6 + [1] = c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) = [4] x1 + [4] x3 + [3] x4 + [1] x5 + [4] x6 + [4] >= [4] x3 + [4] x6 + [4] = c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) = [4] x1 + [4] x3 + [3] x4 + [1] x5 + [4] x6 + [4] >= [4] x1 + [4] x6 + [1] = c_11(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) = [4] x2 + [4] x4 + [4] x6 + [4] x8 + [2] >= [4] x2 + [4] x4 + [4] x6 + [4] x8 + [2] = cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) merge#2#(T(x8,x6,x4),x2) = [4] x2 + [4] x4 + [4] x6 + [4] x8 + [2] >= [4] x2 + [4] x6 + [1] = partition#2#(x6,x2) cond_partition_pivot_heap_1(False(),x3,E(),x2,x1) = [1] x1 + [1] x2 + [0] >= [1] x1 + [1] x2 + [0] = Pair(E(),T(E(),x2,x1)) cond_partition_pivot_heap_1(False(),x6,T(x5,x4,x3),x2,x1) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [0] >= [1] x1 + [1] x2 + [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] x1 + [1] x2 + [0] >= [1] x1 + [1] x2 + [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] x4 + [1] x5 + [0] >= [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [0] = 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] x2 + [1] x3 + [1] x4 + [1] x5 + [0] >= [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [0] = 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] x2 + [1] x3 + [1] x4 + [1] x5 + [0] >= [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [0] = 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] x10 + [1] x12 + [1] x13 + [1] x6 + [1] x7 + [1] x9 + [0] >= [1] x10 + [1] x12 + [1] x13 + [1] x6 + [1] x7 + [1] x9 + [0] = Pair(T(T(x6,x7,x9),x10,x12),x13) cond_partition_pivot_heap_5(Pair(x12,x13),x6,x7,x10,x11) = [1] x10 + [1] x11 + [1] x12 + [1] x13 + [1] x6 + [1] x7 + [0] >= [1] x10 + [1] x11 + [1] x12 + [1] x13 + [1] x6 + [1] x7 + [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] x2 + [1] x3 + [1] x4 + [1] x5 + [0] >= [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [0] = 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] x2 + [1] x3 + [1] x4 + [1] x5 + [0] >= [1] x1 + [1] x2 + [1] x3 + [1] x4 + [1] x5 + [0] = 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] x12 + [1] x13 + [1] x7 + [1] x8 + [1] x9 + [0] >= [1] x10 + [1] x12 + [1] x13 + [1] x7 + [1] x8 + [1] x9 + [0] = Pair(T(x9,x10,x12),T(x13,x7,x8)) cond_partition_pivot_heap_9(Pair(x12,x13),x7,x8,x10,x11) = [1] x10 + [1] x11 + [1] x12 + [1] x13 + [1] x7 + [1] x8 + [0] >= [1] x10 + [1] x11 + [1] x12 + [1] x13 + [1] x7 + [1] x8 + [0] = Pair(x12,T(x13,x10,T(x11,x7,x8))) leq#2(0(),x12) = [2] >= [1] = True() leq#2(S(x16),0()) = [2] >= [1] = False() leq#2(S(x4),S(x2)) = [2] >= [2] = leq#2(x4,x2) leqElem#2(Elem(x4),Elem(x2)) = [2] >= [2] = leq#2(x4,x2) partition#2(x2,E()) = [0] >= [0] = Pair(E(),E()) partition#2(x8,T(x6,x4,x2)) = [1] x2 + [1] x4 + [1] x6 + [0] >= [1] x2 + [1] x4 + [1] x6 + [0] = cond_partition_pivot_heap_1(leqElem#2(x4,x8),x8,x6,x4,x2) ************ Step 7.b:3.b:3.b:1.b:1.b:1.b:1.b:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> merge#2#(x4,x1) cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> merge#2#(x5,x3) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(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_5(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_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) merge#2#(T(x8,x6,x4),x2) -> partition#2#(x6,x2) partition#2#(x8,T(x6,x4,x2)) -> c_22(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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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 7.b:3.b:3.b:1.b:1.b:1.b:1.b:1.b:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> merge#2#(x4,x1) cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> merge#2#(x5,x3) cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(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_5(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_6(partition#2#(x6,x3)) cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) merge#2#(T(x8,x6,x4),x2) -> cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) merge#2#(T(x8,x6,x4),x2) -> partition#2#(x6,x2) partition#2#(x8,T(x6,x4,x2)) -> c_22(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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,partition#2#} and constructors {0,E,Elem,False,Pair,S,T,True} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> merge#2#(x4,x1) -->_1 merge#2#(T(x8,x6,x4),x2) -> partition#2#(x6,x2):10 -->_1 merge#2#(T(x8,x6,x4),x2) -> cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4):9 2:W:cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> merge#2#(x5,x3) -->_1 merge#2#(T(x8,x6,x4),x2) -> partition#2#(x6,x2):10 -->_1 merge#2#(T(x8,x6,x4),x2) -> cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4):9 3:W:cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(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_11(partition#2#(x6,x1)):8 -->_1 cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)):7 4:W:cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(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_7(partition#2#(x6,x1)):6 -->_1 cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)):5 5:W:cond_partition_pivot_heap_3#(False(),x6,x5,x4,x3,x2,x1) -> c_6(partition#2#(x6,x3)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)):11 6:W:cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)):11 7:W:cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)):11 8:W:cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)):11 9:W:merge#2#(T(x8,x6,x4),x2) -> cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) -->_1 cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> merge#2#(x5,x3):2 -->_1 cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> merge#2#(x4,x1):1 10:W:merge#2#(T(x8,x6,x4),x2) -> partition#2#(x6,x2) -->_1 partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)):11 11:W:partition#2#(x8,T(x6,x4,x2)) -> c_22(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_5(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_3(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. 1: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> merge#2#(x4,x1) 9: merge#2#(T(x8,x6,x4),x2) -> cond_merge_h_t_1#(partition#2(x6,x2),x8,x6,x4) 2: cond_merge_h_t_1#(Pair(x5,x4),x3,x2,x1) -> merge#2#(x5,x3) 10: merge#2#(T(x8,x6,x4),x2) -> partition#2#(x6,x2) 11: partition#2#(x8,T(x6,x4,x2)) -> c_22(cond_partition_pivot_heap_1#(leqElem#2(x4,x8),x8,x6,x4,x2)) 8: cond_partition_pivot_heap_7#(True(),x6,x5,x4,x3,x2,x1) -> c_11(partition#2#(x6,x1)) 3: cond_partition_pivot_heap_1#(False(),x6,T(x5,x4,x3),x2,x1) -> c_3(cond_partition_pivot_heap_7#(leqElem#2(x4,x6),x6,x2,x1,x5,x4,x3)) 7: cond_partition_pivot_heap_7#(False(),x6,x5,x4,x3,x2,x1) -> c_10(partition#2#(x6,x3)) 6: cond_partition_pivot_heap_3#(True(),x6,x5,x4,x3,x2,x1) -> c_7(partition#2#(x6,x1)) 4: cond_partition_pivot_heap_1#(True(),x6,x5,x4,T(x3,x2,x1)) -> c_5(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_6(partition#2#(x6,x3)) ************ Step 7.b:3.b:3.b:1.b:1.b:1.b:1.b:1.b:1.b:1.b:1.b:2: 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_merge_h_t_1/4,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,merge#2/2 ,partition#2/2,cond_merge_h_t_1#/4,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,merge#2#/2 ,partition#2#/2} / {0/0,E/0,Elem/1,False/0,Pair/2,S/1,T/3,True/0,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1 ,c_8/0,c_9/0,c_10/1,c_11/1,c_12/0,c_13/0,c_14/0,c_15/0,c_16/1,c_17/0,c_18/1,c_19/0,c_20/2,c_21/0,c_22/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_merge_h_t_1#,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#,merge#2#,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^2))