WORST_CASE(?,O(n^3)) * Step 1: DependencyPairs WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) main(x3) -> fold#3(insert_ord(leq()),x3) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0 ,insert_ord/1,leq/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1,fold#3,insert_ord#2,leq#2 ,main} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)) cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2() fold#3#(insert_ord(x2),Nil()) -> c_3() fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) ,fold#3#(insert_ord(x6),x2)) insert_ord#2#(leq(),x2,Nil()) -> c_5() insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(0(),x8) -> c_7() leq#2#(S(x12),0()) -> c_8() leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) main#(x3) -> c_10(fold#3#(insert_ord(leq()),x3)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)) cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2() fold#3#(insert_ord(x2),Nil()) -> c_3() fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) ,fold#3#(insert_ord(x6),x2)) insert_ord#2#(leq(),x2,Nil()) -> c_5() insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(0(),x8) -> c_7() leq#2#(S(x12),0()) -> c_8() leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) main#(x3) -> c_10(fold#3#(insert_ord(leq()),x3)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) main(x3) -> fold#3(insert_ord(leq()),x3) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2 ,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0 ,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,3,5,7,8} by application of Pre({2,3,5,7,8}) = {1,4,6,9,10}. Here rules are labelled as follows: 1: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)) 2: cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2() 3: fold#3#(insert_ord(x2),Nil()) -> c_3() 4: fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) ,fold#3#(insert_ord(x6),x2)) 5: insert_ord#2#(leq(),x2,Nil()) -> c_5() 6: insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) 7: leq#2#(0(),x8) -> c_7() 8: leq#2#(S(x12),0()) -> c_8() 9: leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) 10: main#(x3) -> c_10(fold#3#(insert_ord(leq()),x3)) * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)) fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) ,fold#3#(insert_ord(x6),x2)) insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) main#(x3) -> c_10(fold#3#(insert_ord(leq()),x3)) - Weak DPs: cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2() fold#3#(insert_ord(x2),Nil()) -> c_3() insert_ord#2#(leq(),x2,Nil()) -> c_5() leq#2#(0(),x8) -> c_7() leq#2#(S(x12),0()) -> c_8() - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) main(x3) -> fold#3(insert_ord(leq()),x3) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2 ,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0 ,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)) -->_1 insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2) ,leq#2#(x6,x4)):3 -->_1 insert_ord#2#(leq(),x2,Nil()) -> c_5():8 2:S:fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) ,fold#3#(insert_ord(x6),x2)) -->_1 insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2) ,leq#2#(x6,x4)):3 -->_1 insert_ord#2#(leq(),x2,Nil()) -> c_5():8 -->_2 fold#3#(insert_ord(x2),Nil()) -> c_3():7 -->_2 fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) ,fold#3#(insert_ord(x6),x2)):2 3:S:insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) -->_2 leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)):4 -->_2 leq#2#(S(x12),0()) -> c_8():10 -->_2 leq#2#(0(),x8) -> c_7():9 -->_1 cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2():6 -->_1 cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)):1 4:S:leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) -->_1 leq#2#(S(x12),0()) -> c_8():10 -->_1 leq#2#(0(),x8) -> c_7():9 -->_1 leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)):4 5:S:main#(x3) -> c_10(fold#3#(insert_ord(leq()),x3)) -->_1 fold#3#(insert_ord(x2),Nil()) -> c_3():7 -->_1 fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) ,fold#3#(insert_ord(x6),x2)):2 6:W:cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2() 7:W:fold#3#(insert_ord(x2),Nil()) -> c_3() 8:W:insert_ord#2#(leq(),x2,Nil()) -> c_5() 9:W:leq#2#(0(),x8) -> c_7() 10:W:leq#2#(S(x12),0()) -> c_8() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: fold#3#(insert_ord(x2),Nil()) -> c_3() 8: insert_ord#2#(leq(),x2,Nil()) -> c_5() 6: cond_insert_ord_x_ys_1#(True(),x3,x2,x1) -> c_2() 9: leq#2#(0(),x8) -> c_7() 10: leq#2#(S(x12),0()) -> c_8() * Step 4: RemoveHeads WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)) fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) ,fold#3#(insert_ord(x6),x2)) insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) main#(x3) -> c_10(fold#3#(insert_ord(leq()),x3)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) main(x3) -> fold#3(insert_ord(leq()),x3) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2 ,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0 ,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)) -->_1 insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2) ,leq#2#(x6,x4)):3 2:S:fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) ,fold#3#(insert_ord(x6),x2)) -->_1 insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2) ,leq#2#(x6,x4)):3 -->_2 fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) ,fold#3#(insert_ord(x6),x2)):2 3:S:insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) -->_2 leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)):4 -->_1 cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)):1 4:S:leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) -->_1 leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)):4 5:S:main#(x3) -> c_10(fold#3#(insert_ord(leq()),x3)) -->_1 fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) ,fold#3#(insert_ord(x6),x2)):2 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(5,main#(x3) -> c_10(fold#3#(insert_ord(leq()),x3)))] * Step 5: UsableRules WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)) fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) ,fold#3#(insert_ord(x6),x2)) insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) main(x3) -> fold#3(insert_ord(leq()),x3) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2 ,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0 ,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)) fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) ,fold#3#(insert_ord(x6),x2)) insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) * Step 6: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)) fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) ,fold#3#(insert_ord(x6),x2)) insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2 ,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0 ,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) ,fold#3#(insert_ord(x6),x2)) and a lower component cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)) insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) Further, following extension rules are added to the lower component. fold#3#(insert_ord(x6),Cons(x4,x2)) -> fold#3#(insert_ord(x6),x2) fold#3#(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) ** Step 6.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) ,fold#3#(insert_ord(x6),x2)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2 ,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0 ,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) ,fold#3#(insert_ord(x6),x2)) -->_2 fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) ,fold#3#(insert_ord(x6),x2)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(fold#3#(insert_ord(x6),x2)) ** Step 6.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(fold#3#(insert_ord(x6),x2)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2 ,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0 ,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(fold#3#(insert_ord(x6),x2)) ** Step 6.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(fold#3#(insert_ord(x6),x2)) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2 ,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0 ,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_4) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x1 + [1] x2 + [3] p(False) = [0] p(Nil) = [0] p(S) = [1] x1 + [0] p(True) = [0] p(cond_insert_ord_x_ys_1) = [0] p(fold#3) = [0] p(insert_ord) = [1] x1 + [0] p(insert_ord#2) = [0] p(leq) = [0] p(leq#2) = [0] p(main) = [0] p(cond_insert_ord_x_ys_1#) = [0] p(fold#3#) = [3] x2 + [0] p(insert_ord#2#) = [0] p(leq#2#) = [0] p(main#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] Following rules are strictly oriented: fold#3#(insert_ord(x6),Cons(x4,x2)) = [3] x2 + [3] x4 + [9] > [3] x2 + [0] = c_4(fold#3#(insert_ord(x6),x2)) Following rules are (at-least) weakly oriented: Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.a:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: fold#3#(insert_ord(x6),Cons(x4,x2)) -> c_4(fold#3#(insert_ord(x6),x2)) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2 ,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0 ,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 6.b:1: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)) insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) - Weak DPs: fold#3#(insert_ord(x6),Cons(x4,x2)) -> fold#3#(insert_ord(x6),x2) fold#3#(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2 ,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0 ,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)) fold#3#(insert_ord(x6),Cons(x4,x2)) -> fold#3#(insert_ord(x6),x2) fold#3#(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) and a lower component leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) Further, following extension rules are added to the lower component. cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> insert_ord#2#(leq(),x0,x2) fold#3#(insert_ord(x6),Cons(x4,x2)) -> fold#3#(insert_ord(x6),x2) fold#3#(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2#(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2) insert_ord#2#(leq(),x6,Cons(x4,x2)) -> leq#2#(x6,x4) *** Step 6.b:1.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)) insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) - Weak DPs: fold#3#(insert_ord(x6),Cons(x4,x2)) -> fold#3#(insert_ord(x6),x2) fold#3#(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2 ,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0 ,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)) -->_1 insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2) ,leq#2#(x6,x4)):2 2:S:insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2),leq#2#(x6,x4)) -->_1 cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)):1 3:W:fold#3#(insert_ord(x6),Cons(x4,x2)) -> fold#3#(insert_ord(x6),x2) -->_1 fold#3#(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)):4 -->_1 fold#3#(insert_ord(x6),Cons(x4,x2)) -> fold#3#(insert_ord(x6),x2):3 4:W:fold#3#(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) -->_1 insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2) ,leq#2#(x6,x4)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2)) *** Step 6.b:1.a:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)) insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2)) - Weak DPs: fold#3#(insert_ord(x6),Cons(x4,x2)) -> fold#3#(insert_ord(x6),x2) fold#3#(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2 ,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0 ,c_3/0,c_4/2,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(Cons) = {2}, uargs(cond_insert_ord_x_ys_1) = {1}, uargs(insert_ord#2) = {3}, uargs(cond_insert_ord_x_ys_1#) = {1}, uargs(insert_ord#2#) = {3}, uargs(c_1) = {1}, uargs(c_6) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x2 + [0] p(False) = [0] p(Nil) = [5] p(S) = [1] x1 + [0] p(True) = [0] p(cond_insert_ord_x_ys_1) = [1] x1 + [1] x4 + [0] p(fold#3) = [6] x1 + [1] x2 + [0] p(insert_ord) = [1] x1 + [0] p(insert_ord#2) = [1] x3 + [0] p(leq) = [0] p(leq#2) = [0] p(main) = [0] p(cond_insert_ord_x_ys_1#) = [1] x1 + [1] x4 + [1] p(fold#3#) = [7] x1 + [1] x2 + [5] p(insert_ord#2#) = [1] x3 + [0] p(leq#2#) = [0] p(main#) = [0] p(c_1) = [1] x1 + [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [0] Following rules are strictly oriented: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) = [1] x2 + [1] > [1] x2 + [0] = c_1(insert_ord#2#(leq(),x0,x2)) Following rules are (at-least) weakly oriented: fold#3#(insert_ord(x6),Cons(x4,x2)) = [1] x2 + [7] x6 + [5] >= [1] x2 + [7] x6 + [5] = fold#3#(insert_ord(x6),x2) fold#3#(insert_ord(x6),Cons(x4,x2)) = [1] x2 + [7] x6 + [5] >= [1] x2 + [6] x6 + [0] = insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2#(leq(),x6,Cons(x4,x2)) = [1] x2 + [0] >= [1] x2 + [1] = c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2)) cond_insert_ord_x_ys_1(False(),x0,x5,x2) = [1] x2 + [0] >= [1] x2 + [0] = Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) = [1] x1 + [0] >= [1] x1 + [0] = Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) = [6] x2 + [5] >= [5] = Nil() fold#3(insert_ord(x6),Cons(x4,x2)) = [1] x2 + [6] x6 + [0] >= [1] x2 + [6] x6 + [0] = insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) = [5] >= [5] = Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) = [1] x2 + [0] >= [1] x2 + [0] = cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) = [0] >= [0] = True() leq#2(S(x12),0()) = [0] >= [0] = False() leq#2(S(x4),S(x2)) = [0] >= [0] = leq#2(x4,x2) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 6.b:1.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2)) - Weak DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)) fold#3#(insert_ord(x6),Cons(x4,x2)) -> fold#3#(insert_ord(x6),x2) fold#3#(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2 ,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0 ,c_3/0,c_4/2,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(Cons) = {2}, uargs(cond_insert_ord_x_ys_1) = {1}, uargs(insert_ord#2) = {3}, uargs(cond_insert_ord_x_ys_1#) = {1}, uargs(insert_ord#2#) = {3}, uargs(c_1) = {1}, uargs(c_6) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x2 + [1] p(False) = [0] p(Nil) = [0] p(S) = [0] p(True) = [0] p(cond_insert_ord_x_ys_1) = [1] x1 + [1] x4 + [5] p(fold#3) = [4] x1 + [4] x2 + [4] p(insert_ord) = [1] x1 + [0] p(insert_ord#2) = [1] x3 + [4] p(leq) = [4] p(leq#2) = [0] p(main) = [0] p(cond_insert_ord_x_ys_1#) = [1] x1 + [1] x4 + [0] p(fold#3#) = [4] x1 + [4] x2 + [0] p(insert_ord#2#) = [1] x3 + [0] p(leq#2#) = [1] p(main#) = [1] x1 + [1] p(c_1) = [1] x1 + [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [2] x1 + [1] p(c_5) = [1] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [2] p(c_9) = [2] p(c_10) = [0] Following rules are strictly oriented: insert_ord#2#(leq(),x6,Cons(x4,x2)) = [1] x2 + [1] > [1] x2 + [0] = c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2)) Following rules are (at-least) weakly oriented: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) = [1] x2 + [0] >= [1] x2 + [0] = c_1(insert_ord#2#(leq(),x0,x2)) fold#3#(insert_ord(x6),Cons(x4,x2)) = [4] x2 + [4] x6 + [4] >= [4] x2 + [4] x6 + [0] = fold#3#(insert_ord(x6),x2) fold#3#(insert_ord(x6),Cons(x4,x2)) = [4] x2 + [4] x6 + [4] >= [4] x2 + [4] x6 + [4] = insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) cond_insert_ord_x_ys_1(False(),x0,x5,x2) = [1] x2 + [5] >= [1] x2 + [5] = Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) = [1] x1 + [5] >= [1] x1 + [2] = Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) = [4] x2 + [4] >= [0] = Nil() fold#3(insert_ord(x6),Cons(x4,x2)) = [4] x2 + [4] x6 + [8] >= [4] x2 + [4] x6 + [8] = insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) = [4] >= [1] = Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) = [1] x2 + [5] >= [1] x2 + [5] = cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) = [0] >= [0] = True() leq#2(S(x12),0()) = [0] >= [0] = False() leq#2(S(x4),S(x2)) = [0] >= [0] = leq#2(x4,x2) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 6.b:1.a:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> c_1(insert_ord#2#(leq(),x0,x2)) fold#3#(insert_ord(x6),Cons(x4,x2)) -> fold#3#(insert_ord(x6),x2) fold#3#(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2#(leq(),x6,Cons(x4,x2)) -> c_6(cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2 ,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0 ,c_3/0,c_4/2,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 6.b:1.b:1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) - Weak DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> insert_ord#2#(leq(),x0,x2) fold#3#(insert_ord(x6),Cons(x4,x2)) -> fold#3#(insert_ord(x6),x2) fold#3#(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2#(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2) insert_ord#2#(leq(),x6,Cons(x4,x2)) -> leq#2#(x6,x4) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2 ,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0 ,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(Cons) = {2}, uargs(cond_insert_ord_x_ys_1) = {1}, uargs(insert_ord#2) = {3}, uargs(cond_insert_ord_x_ys_1#) = {1}, uargs(insert_ord#2#) = {3}, uargs(c_9) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(Cons) = [1] x1 + [1] x2 + [0] p(False) = [0] p(Nil) = [2] p(S) = [1] x1 + [3] p(True) = [0] p(cond_insert_ord_x_ys_1) = [1] x1 + [2] x2 + [1] x3 + [1] x4 + [0] p(fold#3) = [2] x2 + [0] p(insert_ord) = [1] p(insert_ord#2) = [2] x2 + [1] x3 + [0] p(leq) = [0] p(leq#2) = [0] p(main) = [1] p(cond_insert_ord_x_ys_1#) = [1] x1 + [1] x2 + [1] x4 + [7] p(fold#3#) = [4] x1 + [2] x2 + [4] p(insert_ord#2#) = [1] x2 + [1] x3 + [7] p(leq#2#) = [1] x1 + [5] p(main#) = [2] x1 + [2] p(c_1) = [4] p(c_2) = [0] p(c_3) = [1] p(c_4) = [1] x2 + [0] p(c_5) = [4] p(c_6) = [1] x1 + [4] x2 + [0] p(c_7) = [1] p(c_8) = [0] p(c_9) = [1] x1 + [2] p(c_10) = [1] x1 + [1] Following rules are strictly oriented: leq#2#(S(x4),S(x2)) = [1] x4 + [8] > [1] x4 + [7] = c_9(leq#2#(x4,x2)) Following rules are (at-least) weakly oriented: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) = [1] x0 + [1] x2 + [7] >= [1] x0 + [1] x2 + [7] = insert_ord#2#(leq(),x0,x2) fold#3#(insert_ord(x6),Cons(x4,x2)) = [2] x2 + [2] x4 + [8] >= [2] x2 + [8] = fold#3#(insert_ord(x6),x2) fold#3#(insert_ord(x6),Cons(x4,x2)) = [2] x2 + [2] x4 + [8] >= [2] x2 + [1] x4 + [7] = insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2#(leq(),x6,Cons(x4,x2)) = [1] x2 + [1] x4 + [1] x6 + [7] >= [1] x2 + [1] x6 + [7] = cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2) insert_ord#2#(leq(),x6,Cons(x4,x2)) = [1] x2 + [1] x4 + [1] x6 + [7] >= [1] x6 + [5] = leq#2#(x6,x4) cond_insert_ord_x_ys_1(False(),x0,x5,x2) = [2] x0 + [1] x2 + [1] x5 + [0] >= [2] x0 + [1] x2 + [1] x5 + [0] = Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) = [1] x1 + [1] x2 + [2] x3 + [0] >= [1] x1 + [1] x2 + [1] x3 + [0] = Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) = [4] >= [2] = Nil() fold#3(insert_ord(x6),Cons(x4,x2)) = [2] x2 + [2] x4 + [0] >= [2] x2 + [2] x4 + [0] = insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) = [2] x2 + [2] >= [1] x2 + [2] = Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) = [1] x2 + [1] x4 + [2] x6 + [0] >= [1] x2 + [1] x4 + [2] x6 + [0] = cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) = [0] >= [0] = True() leq#2(S(x12),0()) = [0] >= [0] = False() leq#2(S(x4),S(x2)) = [0] >= [0] = leq#2(x4,x2) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 6.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: cond_insert_ord_x_ys_1#(False(),x0,x5,x2) -> insert_ord#2#(leq(),x0,x2) fold#3#(insert_ord(x6),Cons(x4,x2)) -> fold#3#(insert_ord(x6),x2) fold#3#(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2#(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2#(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1#(leq#2(x6,x4),x6,x4,x2) insert_ord#2#(leq(),x6,Cons(x4,x2)) -> leq#2#(x6,x4) leq#2#(S(x4),S(x2)) -> c_9(leq#2#(x4,x2)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1,cond_insert_ord_x_ys_1#/4,fold#3#/2 ,insert_ord#2#/3,leq#2#/2,main#/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0,insert_ord/1,leq/0,c_1/1,c_2/0 ,c_3/0,c_4/2,c_5/0,c_6/2,c_7/0,c_8/0,c_9/1,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1#,fold#3#,insert_ord#2#,leq#2# ,main#} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))