WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) ifselsort(false(),cons(N,L)) -> cons(min(cons(N,L)),selsort(replace(min(cons(N,L)),N,L))) ifselsort(true(),cons(N,L)) -> cons(N,selsort(L)) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() selsort(cons(N,L)) -> ifselsort(eq(N,min(cons(N,L))),cons(N,L)) selsort(nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq,ifmin,ifrepl,ifselsort,le,min,replace ,selsort} and constructors {0,cons,false,nil,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs eq#(0(),0()) -> c_1() eq#(0(),s(Y)) -> c_2() eq#(s(X),0()) -> c_3() eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifrepl#(true(),N,M,cons(K,L)) -> c_8() ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) le#(0(),Y) -> c_11() le#(s(X),0()) -> c_12() le#(s(X),s(Y)) -> c_13(le#(X,Y)) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) min#(cons(0(),nil())) -> c_15() min#(cons(s(N),nil())) -> c_16() replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) replace#(N,M,nil()) -> c_18() selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))) selsort#(nil()) -> c_20() Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: eq#(0(),0()) -> c_1() eq#(0(),s(Y)) -> c_2() eq#(s(X),0()) -> c_3() eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifrepl#(true(),N,M,cons(K,L)) -> c_8() ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) le#(0(),Y) -> c_11() le#(s(X),0()) -> c_12() le#(s(X),s(Y)) -> c_13(le#(X,Y)) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) min#(cons(0(),nil())) -> c_15() min#(cons(s(N),nil())) -> c_16() replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) replace#(N,M,nil()) -> c_18() selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))) selsort#(nil()) -> c_20() - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) ifselsort(false(),cons(N,L)) -> cons(min(cons(N,L)),selsort(replace(min(cons(N,L)),N,L))) ifselsort(true(),cons(N,L)) -> cons(N,selsort(L)) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() selsort(cons(N,L)) -> ifselsort(eq(N,min(cons(N,L))),cons(N,L)) selsort(nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/4,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/2,c_18/0,c_19/3,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() eq#(0(),0()) -> c_1() eq#(0(),s(Y)) -> c_2() eq#(s(X),0()) -> c_3() eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifrepl#(true(),N,M,cons(K,L)) -> c_8() ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) le#(0(),Y) -> c_11() le#(s(X),0()) -> c_12() le#(s(X),s(Y)) -> c_13(le#(X,Y)) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) min#(cons(0(),nil())) -> c_15() min#(cons(s(N),nil())) -> c_16() replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) replace#(N,M,nil()) -> c_18() selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))) selsort#(nil()) -> c_20() * Step 3: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: eq#(0(),0()) -> c_1() eq#(0(),s(Y)) -> c_2() eq#(s(X),0()) -> c_3() eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifrepl#(true(),N,M,cons(K,L)) -> c_8() ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) le#(0(),Y) -> c_11() le#(s(X),0()) -> c_12() le#(s(X),s(Y)) -> c_13(le#(X,Y)) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) min#(cons(0(),nil())) -> c_15() min#(cons(s(N),nil())) -> c_16() replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) replace#(N,M,nil()) -> c_18() selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))) selsort#(nil()) -> c_20() - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/4,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/2,c_18/0,c_19/3,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,8,11,12,15,16,18,20} by application of Pre({1,2,3,8,11,12,15,16,18,20}) = {4,5,6,7,9,10,13,14,17,19}. Here rules are labelled as follows: 1: eq#(0(),0()) -> c_1() 2: eq#(0(),s(Y)) -> c_2() 3: eq#(s(X),0()) -> c_3() 4: eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) 5: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) 6: ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) 7: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) 8: ifrepl#(true(),N,M,cons(K,L)) -> c_8() 9: ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) 10: ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) 11: le#(0(),Y) -> c_11() 12: le#(s(X),0()) -> c_12() 13: le#(s(X),s(Y)) -> c_13(le#(X,Y)) 14: min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) 15: min#(cons(0(),nil())) -> c_15() 16: min#(cons(s(N),nil())) -> c_16() 17: replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) 18: replace#(N,M,nil()) -> c_18() 19: selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))) 20: selsort#(nil()) -> c_20() * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) le#(s(X),s(Y)) -> c_13(le#(X,Y)) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))) - Weak DPs: eq#(0(),0()) -> c_1() eq#(0(),s(Y)) -> c_2() eq#(s(X),0()) -> c_3() ifrepl#(true(),N,M,cons(K,L)) -> c_8() le#(0(),Y) -> c_11() le#(s(X),0()) -> c_12() min#(cons(0(),nil())) -> c_15() min#(cons(s(N),nil())) -> c_16() replace#(N,M,nil()) -> c_18() selsort#(nil()) -> c_20() - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/4,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/2,c_18/0,c_19/3,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) -->_1 eq#(s(X),0()) -> c_3():13 -->_1 eq#(0(),s(Y)) -> c_2():12 -->_1 eq#(0(),0()) -> c_1():11 -->_1 eq#(s(X),s(Y)) -> c_4(eq#(X,Y)):1 2:S:ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):8 -->_1 min#(cons(s(N),nil())) -> c_16():18 -->_1 min#(cons(0(),nil())) -> c_15():17 3:S:ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):8 -->_1 min#(cons(s(N),nil())) -> c_16():18 -->_1 min#(cons(0(),nil())) -> c_15():17 4:S:ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) -->_1 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)):9 -->_1 replace#(N,M,nil()) -> c_18():19 5:S:ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) -->_2 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))):10 -->_3 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)):9 -->_4 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):8 -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):8 -->_2 selsort#(nil()) -> c_20():20 -->_3 replace#(N,M,nil()) -> c_18():19 -->_4 min#(cons(s(N),nil())) -> c_16():18 -->_1 min#(cons(s(N),nil())) -> c_16():18 -->_4 min#(cons(0(),nil())) -> c_15():17 -->_1 min#(cons(0(),nil())) -> c_15():17 6:S:ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) -->_1 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))):10 -->_1 selsort#(nil()) -> c_20():20 7:S:le#(s(X),s(Y)) -> c_13(le#(X,Y)) -->_1 le#(s(X),0()) -> c_12():16 -->_1 le#(0(),Y) -> c_11():15 -->_1 le#(s(X),s(Y)) -> c_13(le#(X,Y)):7 8:S:min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) -->_2 le#(s(X),0()) -> c_12():16 -->_2 le#(0(),Y) -> c_11():15 -->_2 le#(s(X),s(Y)) -> c_13(le#(X,Y)):7 -->_1 ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))):3 -->_1 ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))):2 9:S:replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) -->_1 ifrepl#(true(),N,M,cons(K,L)) -> c_8():14 -->_2 eq#(s(X),0()) -> c_3():13 -->_2 eq#(0(),s(Y)) -> c_2():12 -->_2 eq#(0(),0()) -> c_1():11 -->_1 ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)):4 -->_2 eq#(s(X),s(Y)) -> c_4(eq#(X,Y)):1 10:S:selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))) -->_3 min#(cons(s(N),nil())) -> c_16():18 -->_3 min#(cons(0(),nil())) -> c_15():17 -->_2 eq#(s(X),0()) -> c_3():13 -->_2 eq#(0(),s(Y)) -> c_2():12 -->_2 eq#(0(),0()) -> c_1():11 -->_3 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):8 -->_1 ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)):6 -->_1 ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))):5 -->_2 eq#(s(X),s(Y)) -> c_4(eq#(X,Y)):1 11:W:eq#(0(),0()) -> c_1() 12:W:eq#(0(),s(Y)) -> c_2() 13:W:eq#(s(X),0()) -> c_3() 14:W:ifrepl#(true(),N,M,cons(K,L)) -> c_8() 15:W:le#(0(),Y) -> c_11() 16:W:le#(s(X),0()) -> c_12() 17:W:min#(cons(0(),nil())) -> c_15() 18:W:min#(cons(s(N),nil())) -> c_16() 19:W:replace#(N,M,nil()) -> c_18() 20:W:selsort#(nil()) -> c_20() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 20: selsort#(nil()) -> c_20() 19: replace#(N,M,nil()) -> c_18() 14: ifrepl#(true(),N,M,cons(K,L)) -> c_8() 17: min#(cons(0(),nil())) -> c_15() 18: min#(cons(s(N),nil())) -> c_16() 15: le#(0(),Y) -> c_11() 16: le#(s(X),0()) -> c_12() 11: eq#(0(),0()) -> c_1() 12: eq#(0(),s(Y)) -> c_2() 13: eq#(s(X),0()) -> c_3() * Step 5: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) le#(s(X),s(Y)) -> c_13(le#(X,Y)) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/4,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/2,c_18/0,c_19/3,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) - Weak DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) le#(s(X),s(Y)) -> c_13(le#(X,Y)) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/4,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/2,c_18/0,c_19/3,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} Problem (S) - Strict DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) le#(s(X),s(Y)) -> c_13(le#(X,Y)) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))) - Weak DPs: eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/4,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/2,c_18/0,c_19/3,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} ** Step 5.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) - Weak DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) le#(s(X),s(Y)) -> c_13(le#(X,Y)) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/4,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/2,c_18/0,c_19/3,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) -->_1 eq#(s(X),s(Y)) -> c_4(eq#(X,Y)):1 2:W:ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):8 3:W:ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):8 4:W:ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) -->_1 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)):9 5:W:ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) -->_2 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))):10 -->_3 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)):9 -->_4 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):8 -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):8 6:W:ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) -->_1 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))):10 7:W:le#(s(X),s(Y)) -> c_13(le#(X,Y)) -->_1 le#(s(X),s(Y)) -> c_13(le#(X,Y)):7 8:W:min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) -->_1 ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))):2 -->_1 ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))):3 -->_2 le#(s(X),s(Y)) -> c_13(le#(X,Y)):7 9:W:replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) -->_2 eq#(s(X),s(Y)) -> c_4(eq#(X,Y)):1 -->_1 ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)):4 10:W:selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))) -->_2 eq#(s(X),s(Y)) -> c_4(eq#(X,Y)):1 -->_1 ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))):5 -->_1 ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)):6 -->_3 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):8 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) 8: min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) 3: ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) 7: le#(s(X),s(Y)) -> c_13(le#(X,Y)) ** Step 5.a:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) - Weak DPs: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/4,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/2,c_18/0,c_19/3,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) -->_1 eq#(s(X),s(Y)) -> c_4(eq#(X,Y)):1 4:W:ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) -->_1 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)):9 5:W:ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) -->_2 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))):10 -->_3 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)):9 6:W:ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) -->_1 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))):10 9:W:replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) -->_2 eq#(s(X),s(Y)) -> c_4(eq#(X,Y)):1 -->_1 ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)):4 10:W:selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))) -->_2 eq#(s(X),s(Y)) -> c_4(eq#(X,Y)):1 -->_1 ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))):5 -->_1 ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)):6 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),eq#(N,min(cons(N,L)))) ** Step 5.a:3: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) - Weak DPs: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),eq#(N,min(cons(N,L)))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/2,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,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 ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),eq#(N,min(cons(N,L)))) and a lower component eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) Further, following extension rules are added to the lower component. ifselsort#(false(),cons(N,L)) -> replace#(min(cons(N,L)),N,L) ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> selsort#(L) selsort#(cons(N,L)) -> eq#(N,min(cons(N,L))) selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) *** Step 5.a:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),eq#(N,min(cons(N,L)))) - Weak DPs: ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/2,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,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: 2: selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),eq#(N,min(cons(N,L)))) Consider the set of all dependency pairs 1: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L)) 2: selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),eq#(N,min(cons(N,L)))) 3: ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) 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 {2} These cover all (indirect) predecessors of dependency pairs {1,2,3} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 5.a:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),eq#(N,min(cons(N,L)))) - Weak DPs: ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/2,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,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_9) = {1,2}, uargs(c_10) = {1}, uargs(c_19) = {1,2} Following symbols are considered usable: {ifrepl,replace,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} TcT has computed the following interpretation: p(0) = [0] p(cons) = [1] x2 + [4] p(eq) = [2] x1 + [0] p(false) = [0] p(ifmin) = [1] x1 + [1] x2 + [0] p(ifrepl) = [1] x4 + [0] p(ifselsort) = [1] x2 + [1] p(le) = [0] p(min) = [0] p(nil) = [0] p(replace) = [1] x3 + [0] p(s) = [0] p(selsort) = [1] p(true) = [4] p(eq#) = [0] p(ifmin#) = [1] x1 + [2] p(ifrepl#) = [1] x1 + [1] x2 + [1] x3 + [0] p(ifselsort#) = [1] x2 + [2] p(le#) = [1] x1 + [2] p(min#) = [1] p(replace#) = [0] p(selsort#) = [1] x1 + [4] p(c_1) = [1] p(c_2) = [1] p(c_3) = [0] p(c_4) = [1] x1 + [4] p(c_5) = [4] x1 + [0] p(c_6) = [1] x1 + [1] p(c_7) = [1] p(c_8) = [4] p(c_9) = [1] x1 + [1] x2 + [2] p(c_10) = [1] x1 + [0] p(c_11) = [1] p(c_12) = [0] p(c_13) = [4] p(c_14) = [2] x1 + [1] p(c_15) = [2] p(c_16) = [0] p(c_17) = [4] x2 + [4] p(c_18) = [0] p(c_19) = [1] x1 + [4] x2 + [0] p(c_20) = [0] Following rules are strictly oriented: selsort#(cons(N,L)) = [1] L + [8] > [1] L + [6] = c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),eq#(N,min(cons(N,L)))) Following rules are (at-least) weakly oriented: ifselsort#(false(),cons(N,L)) = [1] L + [6] >= [1] L + [6] = c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) = [1] L + [6] >= [1] L + [4] = c_10(selsort#(L)) ifrepl(false(),N,M,cons(K,L)) = [1] L + [4] >= [1] L + [4] = cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) = [1] L + [4] >= [1] L + [4] = cons(M,L) replace(N,M,cons(K,L)) = [1] L + [4] >= [1] L + [4] = ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) = [0] >= [0] = nil() **** Step 5.a:3.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) - Weak DPs: ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),eq#(N,min(cons(N,L)))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/2,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 5.a:3.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),eq#(N,min(cons(N,L)))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/2,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L)) -->_1 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),eq#(N,min(cons(N,L)))):3 2:W:ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) -->_1 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),eq#(N,min(cons(N,L)))):3 3:W:selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),eq#(N,min(cons(N,L)))) -->_1 ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)):2 -->_1 ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L)) 3: selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),eq#(N,min(cons(N,L)))) 2: ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) **** Step 5.a:3.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/2,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 5.a:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) - Weak DPs: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> replace#(min(cons(N,L)),N,L) ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> selsort#(L) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) selsort#(cons(N,L)) -> eq#(N,min(cons(N,L))) selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/2,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,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: 1: eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) The strictly oriented rules are moved into the weak component. **** Step 5.a:3.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) - Weak DPs: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> replace#(min(cons(N,L)),N,L) ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> selsort#(L) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) selsort#(cons(N,L)) -> eq#(N,min(cons(N,L))) selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/2,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,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_4) = {1}, uargs(c_7) = {1}, uargs(c_17) = {1,2} Following symbols are considered usable: {ifmin,ifrepl,min,replace,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} TcT has computed the following interpretation: p(0) = [0] [2] p(cons) = [0 0] x1 + [0 2] x2 + [0] [0 1] [0 1] [0] p(eq) = [0 0] x1 + [0 1] x2 + [0] [0 1] [0 0] [2] p(false) = [0] [0] p(ifmin) = [0 0] x2 + [0] [0 1] [0] p(ifrepl) = [0 2] x3 + [1 1] x4 + [0] [0 1] [0 1] [0] p(ifselsort) = [0 2] x2 + [2] [1 0] [0] p(le) = [1 0] x1 + [0 3] x2 + [0] [2 1] [1 0] [3] p(min) = [0 0] x1 + [0] [0 1] [0] p(nil) = [0] [0] p(replace) = [0 2] x2 + [1 2] x3 + [0] [0 1] [0 1] [0] p(s) = [0 0] x1 + [0] [0 1] [2] p(selsort) = [2 1] x1 + [1] [0 0] [0] p(true) = [0] [0] p(eq#) = [0 2] x2 + [0] [0 0] [1] p(ifmin#) = [0 0] x1 + [1 2] x2 + [0] [0 2] [0 1] [0] p(ifrepl#) = [0 0] x2 + [0 1] x3 + [1 0] x4 + [0] [3 1] [0 2] [0 2] [0] p(ifselsort#) = [0 2] x2 + [0] [0 0] [1] p(le#) = [0 1] x1 + [0 0] x2 + [2] [0 0] [0 2] [0] p(min#) = [1] [0] p(replace#) = [0 1] x2 + [0 2] x3 + [0] [0 0] [0 0] [0] p(selsort#) = [0 2] x1 + [0] [0 0] [1] p(c_1) = [0] [0] p(c_2) = [0] [0] p(c_3) = [0] [0] p(c_4) = [1 0] x1 + [0] [0 0] [0] p(c_5) = [0 2] x1 + [0] [0 1] [0] p(c_6) = [2] [2] p(c_7) = [1 1] x1 + [0] [0 0] [0] p(c_8) = [0] [1] p(c_9) = [1 0] x1 + [0 0] x2 + [0] [0 0] [0 1] [0] p(c_10) = [1] [1] p(c_11) = [0] [1] p(c_12) = [1] [0] p(c_13) = [1] [0] p(c_14) = [1 2] x2 + [0] [1 2] [0] p(c_15) = [0] [2] p(c_16) = [2] [0] p(c_17) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] p(c_18) = [1] [0] p(c_19) = [2 2] x1 + [0] [0 2] [1] p(c_20) = [1] [1] Following rules are strictly oriented: eq#(s(X),s(Y)) = [0 2] Y + [4] [0 0] [1] > [0 2] Y + [0] [0 0] [0] = c_4(eq#(X,Y)) Following rules are (at-least) weakly oriented: ifrepl#(false(),N,M,cons(K,L)) = [0 0] K + [0 2] L + [0 1] M + [0 0] N + [0] [0 2] [0 2] [0 2] [3 1] [0] >= [0 2] L + [0 1] M + [0] [0 0] [0 0] [0] = c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) = [0 2] L + [0 2] N + [0] [0 0] [0 0] [1] >= [0 2] L + [0 1] N + [0] [0 0] [0 0] [0] = replace#(min(cons(N,L)),N,L) ifselsort#(false(),cons(N,L)) = [0 2] L + [0 2] N + [0] [0 0] [0 0] [1] >= [0 2] L + [0 2] N + [0] [0 0] [0 0] [1] = selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) = [0 2] L + [0 2] N + [0] [0 0] [0 0] [1] >= [0 2] L + [0] [0 0] [1] = selsort#(L) replace#(N,M,cons(K,L)) = [0 2] K + [0 2] L + [0 1] M + [0] [0 0] [0 0] [0 0] [0] >= [0 2] K + [0 2] L + [0 1] M + [0] [0 0] [0 0] [0 0] [0] = c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) selsort#(cons(N,L)) = [0 2] L + [0 2] N + [0] [0 0] [0 0] [1] >= [0 2] L + [0 2] N + [0] [0 0] [0 0] [1] = eq#(N,min(cons(N,L))) selsort#(cons(N,L)) = [0 2] L + [0 2] N + [0] [0 0] [0 0] [1] >= [0 2] L + [0 2] N + [0] [0 0] [0 0] [1] = ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ifmin(false(),cons(N,cons(M,L))) = [0 0] L + [0 0] M + [0 0] N + [0] [0 1] [0 1] [0 1] [0] >= [0 0] L + [0 0] M + [0] [0 1] [0 1] [0] = min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) = [0 0] L + [0 0] M + [0 0] N + [0] [0 1] [0 1] [0 1] [0] >= [0 0] L + [0 0] N + [0] [0 1] [0 1] [0] = min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) = [0 1] K + [0 3] L + [0 2] M + [0] [0 1] [0 1] [0 1] [0] >= [0 0] K + [0 2] L + [0 2] M + [0] [0 1] [0 1] [0 1] [0] = cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) = [0 1] K + [0 3] L + [0 2] M + [0] [0 1] [0 1] [0 1] [0] >= [0 2] L + [0 0] M + [0] [0 1] [0 1] [0] = cons(M,L) min(cons(N,cons(M,L))) = [0 0] L + [0 0] M + [0 0] N + [0] [0 1] [0 1] [0 1] [0] >= [0 0] L + [0 0] M + [0 0] N + [0] [0 1] [0 1] [0 1] [0] = ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) = [0] [2] >= [0] [2] = 0() min(cons(s(N),nil())) = [0 0] N + [0] [0 1] [2] >= [0 0] N + [0] [0 1] [2] = s(N) replace(N,M,cons(K,L)) = [0 2] K + [0 4] L + [0 2] M + [0] [0 1] [0 1] [0 1] [0] >= [0 1] K + [0 3] L + [0 2] M + [0] [0 1] [0 1] [0 1] [0] = ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) = [0 2] M + [0] [0 1] [0] >= [0] [0] = nil() **** Step 5.a:3.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> replace#(min(cons(N,L)),N,L) ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> selsort#(L) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) selsort#(cons(N,L)) -> eq#(N,min(cons(N,L))) selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/2,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 5.a:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> replace#(min(cons(N,L)),N,L) ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> selsort#(L) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) selsort#(cons(N,L)) -> eq#(N,min(cons(N,L))) selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/2,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) -->_1 eq#(s(X),s(Y)) -> c_4(eq#(X,Y)):1 2:W:ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) -->_1 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)):6 3:W:ifselsort#(false(),cons(N,L)) -> replace#(min(cons(N,L)),N,L) -->_1 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)):6 4:W:ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) -->_1 selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)):8 -->_1 selsort#(cons(N,L)) -> eq#(N,min(cons(N,L))):7 5:W:ifselsort#(true(),cons(N,L)) -> selsort#(L) -->_1 selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)):8 -->_1 selsort#(cons(N,L)) -> eq#(N,min(cons(N,L))):7 6:W:replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) -->_1 ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)):2 -->_2 eq#(s(X),s(Y)) -> c_4(eq#(X,Y)):1 7:W:selsort#(cons(N,L)) -> eq#(N,min(cons(N,L))) -->_1 eq#(s(X),s(Y)) -> c_4(eq#(X,Y)):1 8:W:selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) -->_1 ifselsort#(true(),cons(N,L)) -> selsort#(L):5 -->_1 ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)):4 -->_1 ifselsort#(false(),cons(N,L)) -> replace#(min(cons(N,L)),N,L):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) 8: selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) 5: ifselsort#(true(),cons(N,L)) -> selsort#(L) 7: selsort#(cons(N,L)) -> eq#(N,min(cons(N,L))) 3: ifselsort#(false(),cons(N,L)) -> replace#(min(cons(N,L)),N,L) 2: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) 6: replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) 1: eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) **** Step 5.a:3.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/2,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) le#(s(X),s(Y)) -> c_13(le#(X,Y)) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))) - Weak DPs: eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/4,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/2,c_18/0,c_19/3,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):7 2:S:ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):7 3:S:ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) -->_1 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)):8 4:S:ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) -->_2 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))):9 -->_3 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)):8 -->_4 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):7 -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):7 5:S:ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) -->_1 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))):9 6:S:le#(s(X),s(Y)) -> c_13(le#(X,Y)) -->_1 le#(s(X),s(Y)) -> c_13(le#(X,Y)):6 7:S:min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) -->_2 le#(s(X),s(Y)) -> c_13(le#(X,Y)):6 -->_1 ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))):2 -->_1 ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))):1 8:S:replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) -->_2 eq#(s(X),s(Y)) -> c_4(eq#(X,Y)):10 -->_1 ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)):3 9:S:selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))) -->_2 eq#(s(X),s(Y)) -> c_4(eq#(X,Y)):10 -->_3 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):7 -->_1 ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)):5 -->_1 ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))):4 10:W:eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) -->_1 eq#(s(X),s(Y)) -> c_4(eq#(X,Y)):10 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: eq#(s(X),s(Y)) -> c_4(eq#(X,Y)) ** Step 5.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) le#(s(X),s(Y)) -> c_13(le#(X,Y)) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/4,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/2,c_18/0,c_19/3,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):7 2:S:ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):7 3:S:ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) -->_1 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)):8 4:S:ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) -->_2 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))):9 -->_3 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)):8 -->_4 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):7 -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):7 5:S:ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) -->_1 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))):9 6:S:le#(s(X),s(Y)) -> c_13(le#(X,Y)) -->_1 le#(s(X),s(Y)) -> c_13(le#(X,Y)):6 7:S:min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) -->_2 le#(s(X),s(Y)) -> c_13(le#(X,Y)):6 -->_1 ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))):2 -->_1 ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))):1 8:S:replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L)),eq#(N,K)) -->_1 ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)):3 9:S:selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) ,eq#(N,min(cons(N,L))) ,min#(cons(N,L))) -->_3 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):7 -->_1 ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)):5 -->_1 ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))):4 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) ** Step 5.b:3: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) le#(s(X),s(Y)) -> c_13(le#(X,Y)) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/4,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) le#(s(X),s(Y)) -> c_13(le#(X,Y)) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) - Weak DPs: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/4,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} Problem (S) - Strict DPs: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) - Weak DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) le#(s(X),s(Y)) -> c_13(le#(X,Y)) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/4,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} *** Step 5.b:3.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) le#(s(X),s(Y)) -> c_13(le#(X,Y)) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) - Weak DPs: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/4,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):7 2:S:ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):7 3:W:ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) -->_1 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))):8 4:W:ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) -->_2 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))):9 -->_3 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))):8 -->_4 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):7 -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):7 5:W:ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) -->_1 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))):9 6:S:le#(s(X),s(Y)) -> c_13(le#(X,Y)) -->_1 le#(s(X),s(Y)) -> c_13(le#(X,Y)):6 7:S:min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) -->_1 ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))):1 -->_1 ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))):2 -->_2 le#(s(X),s(Y)) -> c_13(le#(X,Y)):6 8:W:replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) -->_1 ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)):3 9:W:selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) -->_1 ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))):4 -->_1 ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)):5 -->_2 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):7 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) 8: replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) *** Step 5.b:3.a:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) le#(s(X),s(Y)) -> c_13(le#(X,Y)) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) - Weak DPs: ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/4,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):7 2:S:ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):7 4:W:ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) -->_2 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))):9 -->_4 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):7 -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):7 5:W:ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) -->_1 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))):9 6:S:le#(s(X),s(Y)) -> c_13(le#(X,Y)) -->_1 le#(s(X),s(Y)) -> c_13(le#(X,Y)):6 7:S:min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) -->_1 ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))):1 -->_1 ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))):2 -->_2 le#(s(X),s(Y)) -> c_13(le#(X,Y)):6 9:W:selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) -->_1 ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))):4 -->_1 ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)):5 -->_2 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):7 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,min#(cons(N,L))) *** Step 5.b:3.a:3: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) le#(s(X),s(Y)) -> c_13(le#(X,Y)) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) - Weak DPs: ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)),selsort#(replace(min(cons(N,L)),N,L)),min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,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 ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) and a lower component ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) le#(s(X),s(Y)) -> c_13(le#(X,Y)) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) Further, following extension rules are added to the lower component. ifselsort#(false(),cons(N,L)) -> min#(cons(N,L)) ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> selsort#(L) selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) selsort#(cons(N,L)) -> min#(cons(N,L)) **** Step 5.b:3.a:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)),selsort#(replace(min(cons(N,L)),N,L)),min#(cons(N,L))) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) - Weak DPs: ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,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: ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,min#(cons(N,L))) The strictly oriented rules are moved into the weak component. ***** Step 5.b:3.a:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)),selsort#(replace(min(cons(N,L)),N,L)),min#(cons(N,L))) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) - Weak DPs: ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,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_9) = {2}, uargs(c_10) = {1}, uargs(c_19) = {1} Following symbols are considered usable: {ifrepl,replace,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} TcT has computed the following interpretation: p(0) = [0] p(cons) = [1] x1 + [1] x2 + [1] p(eq) = [4] x1 + [0] p(false) = [0] p(ifmin) = [1] p(ifrepl) = [1] x3 + [1] x4 + [0] p(ifselsort) = [2] x2 + [1] p(le) = [0] p(min) = [0] p(nil) = [2] p(replace) = [1] x2 + [1] x3 + [0] p(s) = [0] p(selsort) = [1] x1 + [1] p(true) = [0] p(eq#) = [1] x2 + [0] p(ifmin#) = [2] p(ifrepl#) = [1] x1 + [0] p(ifselsort#) = [4] x2 + [2] p(le#) = [2] p(min#) = [5] x1 + [3] p(replace#) = [4] x2 + [0] p(selsort#) = [4] x1 + [2] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [1] x1 + [1] p(c_5) = [2] x1 + [0] p(c_6) = [1] x1 + [1] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] x2 + [0] p(c_10) = [1] x1 + [4] p(c_11) = [1] p(c_12) = [2] p(c_13) = [4] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [2] p(c_16) = [1] p(c_17) = [0] p(c_18) = [2] p(c_19) = [1] x1 + [0] p(c_20) = [2] Following rules are strictly oriented: ifselsort#(false(),cons(N,L)) = [4] L + [4] N + [6] > [4] L + [4] N + [2] = c_9(min#(cons(N,L)),selsort#(replace(min(cons(N,L)),N,L)),min#(cons(N,L))) Following rules are (at-least) weakly oriented: ifselsort#(true(),cons(N,L)) = [4] L + [4] N + [6] >= [4] L + [6] = c_10(selsort#(L)) selsort#(cons(N,L)) = [4] L + [4] N + [6] >= [4] L + [4] N + [6] = c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) ifrepl(false(),N,M,cons(K,L)) = [1] K + [1] L + [1] M + [1] >= [1] K + [1] L + [1] M + [1] = cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) = [1] K + [1] L + [1] M + [1] >= [1] L + [1] M + [1] = cons(M,L) replace(N,M,cons(K,L)) = [1] K + [1] L + [1] M + [1] >= [1] K + [1] L + [1] M + [1] = ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) = [1] M + [2] >= [2] = nil() ***** Step 5.b:3.a:3.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) - Weak DPs: ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)),selsort#(replace(min(cons(N,L)),N,L)),min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 5.b:3.a:3.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) - Weak DPs: ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)),selsort#(replace(min(cons(N,L)),N,L)),min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,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: selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) Consider the set of all dependency pairs 1: selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) 2: ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,min#(cons(N,L))) 3: ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) 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 {1,2,3} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ****** Step 5.b:3.a:3.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) - Weak DPs: ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)),selsort#(replace(min(cons(N,L)),N,L)),min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,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_9) = {2}, uargs(c_10) = {1}, uargs(c_19) = {1} Following symbols are considered usable: {ifrepl,replace,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} TcT has computed the following interpretation: p(0) = [0] p(cons) = [1] x2 + [4] p(eq) = [4] x1 + [0] p(false) = [0] p(ifmin) = [2] x1 + [1] p(ifrepl) = [1] x4 + [0] p(ifselsort) = [2] x1 + [2] x2 + [1] p(le) = [1] x2 + [5] p(min) = [0] p(nil) = [4] p(replace) = [1] x3 + [0] p(s) = [1] x1 + [0] p(selsort) = [0] p(true) = [0] p(eq#) = [4] x1 + [1] x2 + [4] p(ifmin#) = [1] x1 + [0] p(ifrepl#) = [4] x3 + [1] x4 + [4] p(ifselsort#) = [2] x2 + [0] p(le#) = [1] x1 + [0] p(min#) = [1] p(replace#) = [4] p(selsort#) = [2] x1 + [5] p(c_1) = [1] p(c_2) = [0] p(c_3) = [1] p(c_4) = [1] p(c_5) = [2] p(c_6) = [1] p(c_7) = [2] x1 + [4] p(c_8) = [0] p(c_9) = [1] x1 + [1] x2 + [1] p(c_10) = [1] x1 + [3] p(c_11) = [0] p(c_12) = [2] p(c_13) = [4] x1 + [1] p(c_14) = [1] x1 + [1] p(c_15) = [1] p(c_16) = [0] p(c_17) = [1] p(c_18) = [1] p(c_19) = [1] x1 + [4] p(c_20) = [2] Following rules are strictly oriented: selsort#(cons(N,L)) = [2] L + [13] > [2] L + [12] = c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) Following rules are (at-least) weakly oriented: ifselsort#(false(),cons(N,L)) = [2] L + [8] >= [2] L + [7] = c_9(min#(cons(N,L)),selsort#(replace(min(cons(N,L)),N,L)),min#(cons(N,L))) ifselsort#(true(),cons(N,L)) = [2] L + [8] >= [2] L + [8] = c_10(selsort#(L)) ifrepl(false(),N,M,cons(K,L)) = [1] L + [4] >= [1] L + [4] = cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) = [1] L + [4] >= [1] L + [4] = cons(M,L) replace(N,M,cons(K,L)) = [1] L + [4] >= [1] L + [4] = ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) = [4] >= [4] = nil() ****** Step 5.b:3.a:3.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)),selsort#(replace(min(cons(N,L)),N,L)),min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 5.b:3.a:3.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)),selsort#(replace(min(cons(N,L)),N,L)),min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,min#(cons(N,L))) -->_2 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))):3 2:W:ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) -->_1 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))):3 3:W:selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) -->_1 ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)):2 -->_1 ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,min#(cons(N,L))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,min#(cons(N,L))) 3: selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) 2: ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) ****** Step 5.b:3.a:3.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 5.b:3.a:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) le#(s(X),s(Y)) -> c_13(le#(X,Y)) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) - Weak DPs: ifselsort#(false(),cons(N,L)) -> min#(cons(N,L)) ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> selsort#(L) selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) selsort#(cons(N,L)) -> min#(cons(N,L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) 3: le#(s(X),s(Y)) -> c_13(le#(X,Y)) The strictly oriented rules are moved into the weak component. ***** Step 5.b:3.a:3.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) le#(s(X),s(Y)) -> c_13(le#(X,Y)) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) - Weak DPs: ifselsort#(false(),cons(N,L)) -> min#(cons(N,L)) ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> selsort#(L) selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) selsort#(cons(N,L)) -> min#(cons(N,L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: NaturalMI {miDimension = 3, 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_5) = {1}, uargs(c_6) = {1}, uargs(c_13) = {1}, uargs(c_14) = {1,2} Following symbols are considered usable: {ifrepl,replace,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} TcT has computed the following interpretation: p(0) = [0] [0] [1] p(cons) = [0 0 0] [0 1 0] [1] [0 0 0] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [1] p(eq) = [0 0 1] [1] [0 0 0] x1 + [0] [1 0 0] [0] p(false) = [0] [0] [0] p(ifmin) = [0 0 1] [0] [0 0 1] x2 + [0] [0 0 0] [0] p(ifrepl) = [0 1 1] [1 0 0] [0] [0 1 1] x3 + [0 1 0] x4 + [0] [0 0 1] [0 0 1] [0] p(ifselsort) = [0] [0] [0] p(le) = [0 0 0] [0] [0 0 0] x2 + [0] [0 0 1] [0] p(min) = [0 0 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] p(nil) = [0] [1] [0] p(replace) = [0 1 1] [1 0 0] [0] [0 1 1] x2 + [0 1 0] x3 + [0] [0 0 1] [0 0 1] [0] p(s) = [0 1 0] [1] [0 0 1] x1 + [0] [0 0 1] [1] p(selsort) = [0] [0] [0] p(true) = [0] [0] [0] p(eq#) = [0] [0] [0] p(ifmin#) = [1 0 0] [1] [0 0 1] x2 + [0] [1 0 0] [0] p(ifrepl#) = [0] [0] [0] p(ifselsort#) = [0 0 1] [1] [0 0 0] x2 + [0] [0 0 1] [0] p(le#) = [0 0 0] [0 0 1] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 1 0] [0] p(min#) = [0 1 0] [1] [0 0 0] x1 + [0] [0 1 0] [1] p(replace#) = [0] [0] [0] p(selsort#) = [0 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [1] p(c_1) = [0] [0] [0] p(c_2) = [0] [0] [0] p(c_3) = [0] [0] [0] p(c_4) = [0] [0] [0] p(c_5) = [1 0 0] [0] [0 0 1] x1 + [0] [1 0 0] [0] p(c_6) = [1 0 0] [1] [1 0 0] x1 + [0] [0 0 1] [0] p(c_7) = [0] [0] [0] p(c_8) = [0] [0] [0] p(c_9) = [0] [0] [0] p(c_10) = [0] [0] [0] p(c_11) = [0] [0] [0] p(c_12) = [0] [0] [0] p(c_13) = [1 0 0] [0] [0 0 0] x1 + [0] [1 0 0] [0] p(c_14) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [1 0 0] [0] p(c_15) = [0] [0] [0] p(c_16) = [0] [0] [0] p(c_17) = [0] [0] [0] p(c_18) = [0] [0] [0] p(c_19) = [0] [0] [0] p(c_20) = [0] [0] [0] Following rules are strictly oriented: ifmin#(false(),cons(N,cons(M,L))) = [0 0 1] [0 0 0] [0 0 0] [2] [0 0 1] L + [0 0 1] M + [0 0 1] N + [2] [0 0 1] [0 0 0] [0 0 0] [1] > [0 0 1] [1] [0 0 1] L + [1] [0 0 1] [1] = c_5(min#(cons(M,L))) le#(s(X),s(Y)) = [0 0 0] [0 0 1] [1] [0 0 1] X + [0 0 0] Y + [0] [0 0 0] [0 0 1] [0] > [0 0 1] [0] [0 0 0] Y + [0] [0 0 1] [0] = c_13(le#(X,Y)) Following rules are (at-least) weakly oriented: ifmin#(true(),cons(N,cons(M,L))) = [0 0 1] [0 0 0] [0 0 0] [2] [0 0 1] L + [0 0 1] M + [0 0 1] N + [2] [0 0 1] [0 0 0] [0 0 0] [1] >= [0 0 1] [2] [0 0 1] L + [1] [0 0 1] [1] = c_6(min#(cons(N,L))) ifselsort#(false(),cons(N,L)) = [0 0 1] [0 0 1] [2] [0 0 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [1] >= [0 0 1] [1] [0 0 0] L + [0] [0 0 1] [1] = min#(cons(N,L)) ifselsort#(false(),cons(N,L)) = [0 0 1] [0 0 1] [2] [0 0 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [1] >= [0 0 1] [0 0 1] [1] [0 0 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [1] = selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) = [0 0 1] [0 0 1] [2] [0 0 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [1] >= [0 0 1] [1] [0 0 0] L + [0] [0 0 1] [1] = selsort#(L) min#(cons(N,cons(M,L))) = [0 0 1] [0 0 1] [2] [0 0 0] L + [0 0 0] M + [0] [0 0 1] [0 0 1] [2] >= [0 0 1] [0 0 1] [2] [0 0 0] L + [0 0 0] M + [0] [0 0 1] [0 0 1] [1] = c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) selsort#(cons(N,L)) = [0 0 1] [0 0 1] [2] [0 0 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [2] >= [0 0 1] [0 0 1] [2] [0 0 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [1] = ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) selsort#(cons(N,L)) = [0 0 1] [0 0 1] [2] [0 0 0] L + [0 0 0] N + [0] [0 0 1] [0 0 1] [2] >= [0 0 1] [1] [0 0 0] L + [0] [0 0 1] [1] = min#(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) = [0 0 0] [0 1 0] [0 1 1] [1] [0 0 0] K + [0 0 1] L + [0 1 1] M + [0] [0 0 1] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 1 0] [0 1 1] [1] [0 0 0] K + [0 0 1] L + [0 0 1] M + [0] [0 0 1] [0 0 1] [0 0 1] [1] = cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) = [0 0 0] [0 1 0] [0 1 1] [1] [0 0 0] K + [0 0 1] L + [0 1 1] M + [0] [0 0 1] [0 0 1] [0 0 1] [1] >= [0 1 0] [0 0 0] [1] [0 0 1] L + [0 0 0] M + [0] [0 0 1] [0 0 1] [1] = cons(M,L) replace(N,M,cons(K,L)) = [0 0 0] [0 1 0] [0 1 1] [1] [0 0 0] K + [0 0 1] L + [0 1 1] M + [0] [0 0 1] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 1 0] [0 1 1] [1] [0 0 0] K + [0 0 1] L + [0 1 1] M + [0] [0 0 1] [0 0 1] [0 0 1] [1] = ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) = [0 1 1] [0] [0 1 1] M + [1] [0 0 1] [0] >= [0] [1] [0] = nil() ***** Step 5.b:3.a:3.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) - Weak DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifselsort#(false(),cons(N,L)) -> min#(cons(N,L)) ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> selsort#(L) le#(s(X),s(Y)) -> c_13(le#(X,Y)) selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) selsort#(cons(N,L)) -> min#(cons(N,L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 5.b:3.a:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) - Weak DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifselsort#(false(),cons(N,L)) -> min#(cons(N,L)) ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> selsort#(L) le#(s(X),s(Y)) -> c_13(le#(X,Y)) selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) selsort#(cons(N,L)) -> min#(cons(N,L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):2 2:S:min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) -->_2 le#(s(X),s(Y)) -> c_13(le#(X,Y)):7 -->_1 ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))):3 -->_1 ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))):1 3:W:ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):2 4:W:ifselsort#(false(),cons(N,L)) -> min#(cons(N,L)) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):2 5:W:ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) -->_1 selsort#(cons(N,L)) -> min#(cons(N,L)):9 -->_1 selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)):8 6:W:ifselsort#(true(),cons(N,L)) -> selsort#(L) -->_1 selsort#(cons(N,L)) -> min#(cons(N,L)):9 -->_1 selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)):8 7:W:le#(s(X),s(Y)) -> c_13(le#(X,Y)) -->_1 le#(s(X),s(Y)) -> c_13(le#(X,Y)):7 8:W:selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) -->_1 ifselsort#(true(),cons(N,L)) -> selsort#(L):6 -->_1 ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)):5 -->_1 ifselsort#(false(),cons(N,L)) -> min#(cons(N,L)):4 9:W:selsort#(cons(N,L)) -> min#(cons(N,L)) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: le#(s(X),s(Y)) -> c_13(le#(X,Y)) ***** Step 5.b:3.a:3.b:1.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) - Weak DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifselsort#(false(),cons(N,L)) -> min#(cons(N,L)) ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> selsort#(L) selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) selsort#(cons(N,L)) -> min#(cons(N,L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):2 2:S:min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) -->_1 ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))):3 -->_1 ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))):1 3:W:ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):2 4:W:ifselsort#(false(),cons(N,L)) -> min#(cons(N,L)) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):2 5:W:ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) -->_1 selsort#(cons(N,L)) -> min#(cons(N,L)):9 -->_1 selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)):8 6:W:ifselsort#(true(),cons(N,L)) -> selsort#(L) -->_1 selsort#(cons(N,L)) -> min#(cons(N,L)):9 -->_1 selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)):8 8:W:selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) -->_1 ifselsort#(true(),cons(N,L)) -> selsort#(L):6 -->_1 ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)):5 -->_1 ifselsort#(false(),cons(N,L)) -> min#(cons(N,L)):4 9:W:selsort#(cons(N,L)) -> min#(cons(N,L)) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L)))) ***** Step 5.b:3.a:3.b:1.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L)))) - Weak DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifselsort#(false(),cons(N,L)) -> min#(cons(N,L)) ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> selsort#(L) selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) selsort#(cons(N,L)) -> min#(cons(N,L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,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: ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) The strictly oriented rules are moved into the weak component. ****** Step 5.b:3.a:3.b:1.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L)))) - Weak DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifselsort#(false(),cons(N,L)) -> min#(cons(N,L)) ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> selsort#(L) selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) selsort#(cons(N,L)) -> min#(cons(N,L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,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_5) = {1}, uargs(c_6) = {1}, uargs(c_14) = {1} Following symbols are considered usable: {ifrepl,replace,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} TcT has computed the following interpretation: p(0) = [0] p(cons) = [1] x2 + [2] p(eq) = [1] x1 + [6] p(false) = [0] p(ifmin) = [4] p(ifrepl) = [1] x4 + [0] p(ifselsort) = [1] p(le) = [1] x2 + [0] p(min) = [0] p(nil) = [0] p(replace) = [1] x3 + [0] p(s) = [1] p(selsort) = [1] p(true) = [0] p(eq#) = [4] x1 + [4] x2 + [1] p(ifmin#) = [2] x2 + [0] p(ifrepl#) = [1] x1 + [1] x2 + [2] x4 + [1] p(ifselsort#) = [4] x2 + [0] p(le#) = [4] x1 + [4] x2 + [4] p(min#) = [2] x1 + [0] p(replace#) = [1] x1 + [1] x3 + [0] p(selsort#) = [4] x1 + [1] p(c_1) = [4] p(c_2) = [2] p(c_3) = [2] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [1] p(c_6) = [1] x1 + [2] p(c_7) = [0] p(c_8) = [4] p(c_9) = [2] x3 + [1] p(c_10) = [2] x1 + [0] p(c_11) = [1] p(c_12) = [0] p(c_13) = [1] x1 + [1] p(c_14) = [1] x1 + [0] p(c_15) = [0] p(c_16) = [1] p(c_17) = [1] p(c_18) = [2] p(c_19) = [1] p(c_20) = [0] Following rules are strictly oriented: ifmin#(true(),cons(N,cons(M,L))) = [2] L + [8] > [2] L + [6] = c_6(min#(cons(N,L))) Following rules are (at-least) weakly oriented: ifmin#(false(),cons(N,cons(M,L))) = [2] L + [8] >= [2] L + [5] = c_5(min#(cons(M,L))) ifselsort#(false(),cons(N,L)) = [4] L + [8] >= [2] L + [4] = min#(cons(N,L)) ifselsort#(false(),cons(N,L)) = [4] L + [8] >= [4] L + [1] = selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) = [4] L + [8] >= [4] L + [1] = selsort#(L) min#(cons(N,cons(M,L))) = [2] L + [8] >= [2] L + [8] = c_14(ifmin#(le(N,M),cons(N,cons(M,L)))) selsort#(cons(N,L)) = [4] L + [9] >= [4] L + [8] = ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) selsort#(cons(N,L)) = [4] L + [9] >= [2] L + [4] = min#(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) = [1] L + [2] >= [1] L + [2] = cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) = [1] L + [2] >= [1] L + [2] = cons(M,L) replace(N,M,cons(K,L)) = [1] L + [2] >= [1] L + [2] = ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) = [0] >= [0] = nil() ****** Step 5.b:3.a:3.b:1.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L)))) - Weak DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) ifselsort#(false(),cons(N,L)) -> min#(cons(N,L)) ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> selsort#(L) selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) selsort#(cons(N,L)) -> min#(cons(N,L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 5.b:3.a:3.b:1.b:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L)))) - Weak DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) ifselsort#(false(),cons(N,L)) -> min#(cons(N,L)) ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> selsort#(L) selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) selsort#(cons(N,L)) -> min#(cons(N,L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,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: min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L)))) Consider the set of all dependency pairs 1: min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L)))) 2: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) 3: ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) 4: ifselsort#(false(),cons(N,L)) -> min#(cons(N,L)) 5: ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) 6: ifselsort#(true(),cons(N,L)) -> selsort#(L) 7: selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) 8: selsort#(cons(N,L)) -> min#(cons(N,L)) 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 {1,2,3} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ******* Step 5.b:3.a:3.b:1.b:3.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L)))) - Weak DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) ifselsort#(false(),cons(N,L)) -> min#(cons(N,L)) ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> selsort#(L) selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) selsort#(cons(N,L)) -> min#(cons(N,L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,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_5) = {1}, uargs(c_6) = {1}, uargs(c_14) = {1} Following symbols are considered usable: {ifrepl,replace,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} TcT has computed the following interpretation: p(0) = [1] p(cons) = [1] x2 + [4] p(eq) = [5] x1 + [4] p(false) = [0] p(ifmin) = [4] p(ifrepl) = [1] x4 + [4] p(ifselsort) = [1] x1 + [4] x2 + [4] p(le) = [5] x2 + [0] p(min) = [5] p(nil) = [1] p(replace) = [1] x3 + [4] p(s) = [0] p(selsort) = [4] x1 + [1] p(true) = [0] p(eq#) = [1] p(ifmin#) = [1] x2 + [0] p(ifrepl#) = [1] x1 + [2] x2 + [1] p(ifselsort#) = [1] x2 + [1] p(le#) = [4] p(min#) = [1] x1 + [1] p(replace#) = [2] p(selsort#) = [1] x1 + [1] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] p(c_4) = [0] p(c_5) = [1] x1 + [1] p(c_6) = [1] x1 + [3] p(c_7) = [1] x1 + [0] p(c_8) = [1] p(c_9) = [1] p(c_10) = [1] p(c_11) = [0] p(c_12) = [0] p(c_13) = [1] x1 + [1] p(c_14) = [1] x1 + [0] p(c_15) = [2] p(c_16) = [1] p(c_17) = [2] p(c_18) = [4] p(c_19) = [1] p(c_20) = [2] Following rules are strictly oriented: min#(cons(N,cons(M,L))) = [1] L + [9] > [1] L + [8] = c_14(ifmin#(le(N,M),cons(N,cons(M,L)))) Following rules are (at-least) weakly oriented: ifmin#(false(),cons(N,cons(M,L))) = [1] L + [8] >= [1] L + [6] = c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) = [1] L + [8] >= [1] L + [8] = c_6(min#(cons(N,L))) ifselsort#(false(),cons(N,L)) = [1] L + [5] >= [1] L + [5] = min#(cons(N,L)) ifselsort#(false(),cons(N,L)) = [1] L + [5] >= [1] L + [5] = selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) = [1] L + [5] >= [1] L + [1] = selsort#(L) selsort#(cons(N,L)) = [1] L + [5] >= [1] L + [5] = ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) selsort#(cons(N,L)) = [1] L + [5] >= [1] L + [5] = min#(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) = [1] L + [8] >= [1] L + [8] = cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) = [1] L + [8] >= [1] L + [4] = cons(M,L) replace(N,M,cons(K,L)) = [1] L + [8] >= [1] L + [8] = ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) = [5] >= [1] = nil() ******* Step 5.b:3.a:3.b:1.b:3.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) ifselsort#(false(),cons(N,L)) -> min#(cons(N,L)) ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> selsort#(L) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L)))) selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) selsort#(cons(N,L)) -> min#(cons(N,L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******* Step 5.b:3.a:3.b:1.b:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) ifselsort#(false(),cons(N,L)) -> min#(cons(N,L)) ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> selsort#(L) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L)))) selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) selsort#(cons(N,L)) -> min#(cons(N,L)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L)))):6 2:W:ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L)))):6 3:W:ifselsort#(false(),cons(N,L)) -> min#(cons(N,L)) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L)))):6 4:W:ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) -->_1 selsort#(cons(N,L)) -> min#(cons(N,L)):8 -->_1 selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)):7 5:W:ifselsort#(true(),cons(N,L)) -> selsort#(L) -->_1 selsort#(cons(N,L)) -> min#(cons(N,L)):8 -->_1 selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)):7 6:W:min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L)))) -->_1 ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))):2 -->_1 ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))):1 7:W:selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) -->_1 ifselsort#(true(),cons(N,L)) -> selsort#(L):5 -->_1 ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)):4 -->_1 ifselsort#(false(),cons(N,L)) -> min#(cons(N,L)):3 8:W:selsort#(cons(N,L)) -> min#(cons(N,L)) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L)))):6 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: ifselsort#(false(),cons(N,L)) -> selsort#(replace(min(cons(N,L)),N,L)) 7: selsort#(cons(N,L)) -> ifselsort#(eq(N,min(cons(N,L))),cons(N,L)) 5: ifselsort#(true(),cons(N,L)) -> selsort#(L) 8: selsort#(cons(N,L)) -> min#(cons(N,L)) 3: ifselsort#(false(),cons(N,L)) -> min#(cons(N,L)) 1: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) 6: min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L)))) 2: ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) ******* Step 5.b:3.a:3.b:1.b:3.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/3,c_10/1,c_11/0,c_12/0,c_13/1,c_14/1,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 5.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) - Weak DPs: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) le#(s(X),s(Y)) -> c_13(le#(X,Y)) min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/4,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) -->_1 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))):4 2:S:ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) -->_4 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):9 -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):9 -->_2 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))):5 -->_3 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))):4 3:S:ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) -->_1 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))):5 4:S:replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) -->_1 ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)):1 5:S:selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) -->_2 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):9 -->_1 ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)):3 -->_1 ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))):2 6:W:ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):9 7:W:ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) -->_1 min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)):9 8:W:le#(s(X),s(Y)) -> c_13(le#(X,Y)) -->_1 le#(s(X),s(Y)) -> c_13(le#(X,Y)):8 9:W:min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) -->_2 le#(s(X),s(Y)) -> c_13(le#(X,Y)):8 -->_1 ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))):7 -->_1 ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))):6 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: min#(cons(N,cons(M,L))) -> c_14(ifmin#(le(N,M),cons(N,cons(M,L))),le#(N,M)) 7: ifmin#(true(),cons(N,cons(M,L))) -> c_6(min#(cons(N,L))) 6: ifmin#(false(),cons(N,cons(M,L))) -> c_5(min#(cons(M,L))) 8: le#(s(X),s(Y)) -> c_13(le#(X,Y)) *** Step 5.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/4,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/2,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) -->_1 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))):4 2:S:ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))) -->_2 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))):5 -->_3 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))):4 3:S:ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) -->_1 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))):5 4:S:replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) -->_1 ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)):1 5:S:selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L)),min#(cons(N,L))) -->_1 ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)):3 -->_1 ifselsort#(false(),cons(N,L)) -> c_9(min#(cons(N,L)) ,selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L) ,min#(cons(N,L))):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) *** Step 5.b:3.b:3: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/1,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) - Weak DPs: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/1,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} Problem (S) - Strict DPs: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) - Weak DPs: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/1,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} **** Step 5.b:3.b:3.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) - Weak DPs: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/1,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) The strictly oriented rules are moved into the weak component. ***** Step 5.b:3.b:3.a:1.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) - Weak DPs: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/1,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, 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_7) = {1}, uargs(c_9) = {1,2}, uargs(c_10) = {1}, uargs(c_17) = {1}, uargs(c_19) = {1} Following symbols are considered usable: {ifrepl,replace,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} TcT has computed the following interpretation: p(0) = [0] [0] p(cons) = [1 0] x1 + [1 2] x2 + [1] [0 0] [0 1] [1] p(eq) = [0 1] x2 + [0] [1 0] [0] p(false) = [0] [0] p(ifmin) = [0 1] x1 + [0 2] x2 + [0] [1 0] [0 1] [1] p(ifrepl) = [1 0] x3 + [1 0] x4 + [3] [0 0] [0 1] [0] p(ifselsort) = [0 0] x2 + [0] [2 1] [0] p(le) = [1 1] x1 + [0] [2 2] [1] p(min) = [1 0] x1 + [3] [0 1] [3] p(nil) = [3] [0] p(replace) = [1 0] x2 + [1 0] x3 + [3] [0 0] [0 1] [0] p(s) = [0] [0] p(selsort) = [0] [0] p(true) = [0] [0] p(eq#) = [0 2] x1 + [0 0] x2 + [2] [0 2] [2 2] [0] p(ifmin#) = [0] [1] p(ifrepl#) = [0 0] x2 + [0 1] x4 + [0] [3 1] [2 0] [3] p(ifselsort#) = [1 1] x2 + [3] [3 1] [2] p(le#) = [0 0] x2 + [0] [0 1] [2] p(min#) = [0 0] x1 + [0] [0 1] [0] p(replace#) = [0 1] x3 + [0] [1 0] [0] p(selsort#) = [1 2] x1 + [2] [0 0] [2] p(c_1) = [2] [0] p(c_2) = [0] [2] p(c_3) = [0] [0] p(c_4) = [0 1] x1 + [2] [1 0] [0] p(c_5) = [0] [2] p(c_6) = [0] [0] p(c_7) = [1 0] x1 + [0] [0 0] [0] p(c_8) = [1] [1] p(c_9) = [1 0] x1 + [1 0] x2 + [0] [1 0] [2 2] [0] p(c_10) = [1 1] x1 + [0] [0 0] [2] p(c_11) = [0] [0] p(c_12) = [0] [0] p(c_13) = [0 2] x1 + [1] [0 0] [0] p(c_14) = [2 1] x1 + [0] [0 0] [0] p(c_15) = [0] [0] p(c_16) = [0] [0] p(c_17) = [1 0] x1 + [0] [0 0] [1] p(c_18) = [2] [0] p(c_19) = [1 0] x1 + [0] [0 0] [2] p(c_20) = [0] [0] Following rules are strictly oriented: ifrepl#(false(),N,M,cons(K,L)) = [0 0] K + [0 1] L + [0 0] N + [1] [2 0] [2 4] [3 1] [5] > [0 1] L + [0] [0 0] [0] = c_7(replace#(N,M,L)) Following rules are (at-least) weakly oriented: ifselsort#(false(),cons(N,L)) = [1 3] L + [1 0] N + [5] [3 7] [3 0] [6] >= [1 3] L + [1 0] N + [5] [3 4] [1 0] [5] = c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) = [1 3] L + [1 0] N + [5] [3 7] [3 0] [6] >= [1 2] L + [4] [0 0] [2] = c_10(selsort#(L)) replace#(N,M,cons(K,L)) = [0 0] K + [0 1] L + [1] [1 0] [1 2] [1] >= [0 1] L + [1] [0 0] [1] = c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) selsort#(cons(N,L)) = [1 4] L + [1 0] N + [5] [0 0] [0 0] [2] >= [1 3] L + [1 0] N + [5] [0 0] [0 0] [2] = c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) ifrepl(false(),N,M,cons(K,L)) = [1 0] K + [1 2] L + [1 0] M + [4] [0 0] [0 1] [0 0] [1] >= [1 0] K + [1 2] L + [1 0] M + [4] [0 0] [0 1] [0 0] [1] = cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) = [1 0] K + [1 2] L + [1 0] M + [4] [0 0] [0 1] [0 0] [1] >= [1 2] L + [1 0] M + [1] [0 1] [0 0] [1] = cons(M,L) replace(N,M,cons(K,L)) = [1 0] K + [1 2] L + [1 0] M + [4] [0 0] [0 1] [0 0] [1] >= [1 0] K + [1 2] L + [1 0] M + [4] [0 0] [0 1] [0 0] [1] = ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) = [1 0] M + [6] [0 0] [0] >= [3] [0] = nil() ***** Step 5.b:3.b:3.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) - Weak DPs: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/1,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 5.b:3.b:3.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) - Weak DPs: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/1,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) Consider the set of all dependency pairs 1: replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) 2: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) 3: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) 4: ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) 5: selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) Processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, 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 {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ****** Step 5.b:3.b:3.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) - Weak DPs: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/1,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, 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_7) = {1}, uargs(c_9) = {1,2}, uargs(c_10) = {1}, uargs(c_17) = {1}, uargs(c_19) = {1} Following symbols are considered usable: {ifrepl,replace,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} TcT has computed the following interpretation: p(0) = [0] [0] p(cons) = [0 2] x1 + [1 2] x2 + [0] [0 0] [0 1] [2] p(eq) = [0 0] x1 + [0] [3 0] [0] p(false) = [0] [0] p(ifmin) = [0 0] x1 + [1 0] x2 + [0] [0 1] [0 0] [0] p(ifrepl) = [0 2] x3 + [1 1] x4 + [0] [0 0] [0 1] [0] p(ifselsort) = [0] [1] p(le) = [0 0] x1 + [0] [2 2] [0] p(min) = [0] [0] p(nil) = [2] [2] p(replace) = [0 2] x2 + [1 1] x3 + [0] [0 0] [0 1] [0] p(s) = [0 2] x1 + [0] [0 0] [0] p(selsort) = [0] [0] p(true) = [0] [0] p(eq#) = [2 0] x2 + [0] [1 0] [0] p(ifmin#) = [0] [0] p(ifrepl#) = [0 1] x4 + [0] [0 0] [1] p(ifselsort#) = [2 2] x2 + [0] [2 2] [3] p(le#) = [0 2] x1 + [0 1] x2 + [0] [0 1] [2 2] [1] p(min#) = [0] [2] p(replace#) = [0 1] x3 + [2] [0 1] [0] p(selsort#) = [2 2] x1 + [0] [0 0] [1] p(c_1) = [1] [2] p(c_2) = [0] [0] p(c_3) = [2] [2] p(c_4) = [1 2] x1 + [0] [1 0] [2] p(c_5) = [0] [0] p(c_6) = [1] [1] p(c_7) = [1 0] x1 + [0] [0 0] [0] p(c_8) = [2] [0] p(c_9) = [1 2] x1 + [1 1] x2 + [0] [1 0] [0 0] [0] p(c_10) = [1 2] x1 + [0] [0 1] [1] p(c_11) = [2] [2] p(c_12) = [0] [0] p(c_13) = [1] [0] p(c_14) = [0] [2] p(c_15) = [2] [0] p(c_16) = [2] [1] p(c_17) = [1 1] x1 + [0] [0 2] [0] p(c_18) = [1] [0] p(c_19) = [1 0] x1 + [0] [0 0] [1] p(c_20) = [1] [1] Following rules are strictly oriented: replace#(N,M,cons(K,L)) = [0 1] L + [4] [0 1] [2] > [0 1] L + [3] [0 0] [2] = c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) Following rules are (at-least) weakly oriented: ifrepl#(false(),N,M,cons(K,L)) = [0 1] L + [2] [0 0] [1] >= [0 1] L + [2] [0 0] [0] = c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) = [2 6] L + [0 4] N + [4] [2 6] [0 4] [7] >= [2 6] L + [0 4] N + [4] [2 4] [0 4] [0] = c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) = [2 6] L + [0 4] N + [4] [2 6] [0 4] [7] >= [2 2] L + [2] [0 0] [2] = c_10(selsort#(L)) selsort#(cons(N,L)) = [2 6] L + [0 4] N + [4] [0 0] [0 0] [1] >= [2 6] L + [0 4] N + [4] [0 0] [0 0] [1] = c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) ifrepl(false(),N,M,cons(K,L)) = [0 2] K + [1 3] L + [0 2] M + [2] [0 0] [0 1] [0 0] [2] >= [0 2] K + [1 3] L + [0 2] M + [0] [0 0] [0 1] [0 0] [2] = cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) = [0 2] K + [1 3] L + [0 2] M + [2] [0 0] [0 1] [0 0] [2] >= [1 2] L + [0 2] M + [0] [0 1] [0 0] [2] = cons(M,L) replace(N,M,cons(K,L)) = [0 2] K + [1 3] L + [0 2] M + [2] [0 0] [0 1] [0 0] [2] >= [0 2] K + [1 3] L + [0 2] M + [2] [0 0] [0 1] [0 0] [2] = ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) = [0 2] M + [4] [0 0] [2] >= [2] [2] = nil() ****** Step 5.b:3.b:3.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/1,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 5.b:3.b:3.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/1,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) -->_1 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))):4 2:W:ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) -->_1 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))):5 -->_2 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))):4 3:W:ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) -->_1 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))):5 4:W:replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) -->_1 ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)):1 5:W:selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) -->_1 ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)):3 -->_1 ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L)) 5: selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) 3: ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) 1: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) 4: replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) ****** Step 5.b:3.b:3.a:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/1,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 5.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) - Weak DPs: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/1,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L)) -->_2 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))):5 -->_1 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))):3 2:S:ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) -->_1 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))):3 3:S:selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) -->_1 ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)):2 -->_1 ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L)):1 4:W:ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) -->_1 replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))):5 5:W:replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) -->_1 ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: replace#(N,M,cons(K,L)) -> c_17(ifrepl#(eq(N,K),N,M,cons(K,L))) 4: ifrepl#(false(),N,M,cons(K,L)) -> c_7(replace#(N,M,L)) **** Step 5.b:3.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)),replace#(min(cons(N,L)),N,L)) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/2,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/1,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L)) -->_1 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))):3 2:S:ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) -->_1 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))):3 3:S:selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) -->_1 ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)):2 -->_1 ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L)) ,replace#(min(cons(N,L)),N,L)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L))) **** Step 5.b:3.b:3.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/1,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,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: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L))) The strictly oriented rules are moved into the weak component. ***** Step 5.b:3.b:3.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/1,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,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_9) = {1}, uargs(c_10) = {1}, uargs(c_19) = {1} Following symbols are considered usable: {ifrepl,replace,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} TcT has computed the following interpretation: p(0) = [0] p(cons) = [1] x2 + [4] p(eq) = [7] x1 + [0] p(false) = [0] p(ifmin) = [4] x1 + [2] p(ifrepl) = [1] x4 + [0] p(ifselsort) = [4] x1 + [0] p(le) = [2] x1 + [1] x2 + [1] p(min) = [1] p(nil) = [4] p(replace) = [1] x3 + [0] p(s) = [0] p(selsort) = [1] p(true) = [0] p(eq#) = [1] x1 + [2] x2 + [2] p(ifmin#) = [2] x1 + [2] p(ifrepl#) = [1] x1 + [2] x3 + [1] x4 + [0] p(ifselsort#) = [1] x2 + [2] p(le#) = [1] x2 + [0] p(min#) = [1] x1 + [1] p(replace#) = [1] x1 + [2] x2 + [1] x3 + [0] p(selsort#) = [1] x1 + [2] p(c_1) = [4] p(c_2) = [1] p(c_3) = [0] p(c_4) = [1] x1 + [1] p(c_5) = [1] p(c_6) = [1] p(c_7) = [2] x1 + [0] p(c_8) = [1] p(c_9) = [1] x1 + [3] p(c_10) = [1] x1 + [4] p(c_11) = [0] p(c_12) = [0] p(c_13) = [4] x1 + [1] p(c_14) = [4] x2 + [4] p(c_15) = [1] p(c_16) = [1] p(c_17) = [1] p(c_18) = [4] p(c_19) = [1] x1 + [0] p(c_20) = [1] Following rules are strictly oriented: ifselsort#(false(),cons(N,L)) = [1] L + [6] > [1] L + [5] = c_9(selsort#(replace(min(cons(N,L)),N,L))) Following rules are (at-least) weakly oriented: ifselsort#(true(),cons(N,L)) = [1] L + [6] >= [1] L + [6] = c_10(selsort#(L)) selsort#(cons(N,L)) = [1] L + [6] >= [1] L + [6] = c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) ifrepl(false(),N,M,cons(K,L)) = [1] L + [4] >= [1] L + [4] = cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) = [1] L + [4] >= [1] L + [4] = cons(M,L) replace(N,M,cons(K,L)) = [1] L + [4] >= [1] L + [4] = ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) = [4] >= [4] = nil() ***** Step 5.b:3.b:3.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) - Weak DPs: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/1,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 5.b:3.b:3.b:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) - Weak DPs: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/1,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,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: ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) 2: selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) Consider the set of all dependency pairs 1: ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) 2: selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) 3: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L))) 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 {1,2} These cover all (indirect) predecessors of dependency pairs {1,2,3} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ****** Step 5.b:3.b:3.b:3.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) - Weak DPs: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/1,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,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_9) = {1}, uargs(c_10) = {1}, uargs(c_19) = {1} Following symbols are considered usable: {ifrepl,replace,eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace#,selsort#} TcT has computed the following interpretation: p(0) = [2] p(cons) = [1] x1 + [1] x2 + [1] p(eq) = [5] x1 + [4] p(false) = [0] p(ifmin) = [2] x1 + [0] p(ifrepl) = [1] x3 + [1] x4 + [0] p(ifselsort) = [0] p(le) = [2] x1 + [4] x2 + [0] p(min) = [4] p(nil) = [3] p(replace) = [1] x2 + [1] x3 + [0] p(s) = [1] p(selsort) = [1] p(true) = [0] p(eq#) = [2] x1 + [4] p(ifmin#) = [4] x2 + [0] p(ifrepl#) = [2] x4 + [1] p(ifselsort#) = [4] x2 + [0] p(le#) = [4] x1 + [0] p(min#) = [1] x1 + [2] p(replace#) = [1] p(selsort#) = [4] x1 + [3] p(c_1) = [0] p(c_2) = [1] p(c_3) = [1] p(c_4) = [2] x1 + [0] p(c_5) = [1] x1 + [2] p(c_6) = [0] p(c_7) = [0] p(c_8) = [1] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [0] p(c_11) = [0] p(c_12) = [1] p(c_13) = [0] p(c_14) = [0] p(c_15) = [4] p(c_16) = [0] p(c_17) = [1] p(c_18) = [0] p(c_19) = [1] x1 + [2] p(c_20) = [1] Following rules are strictly oriented: ifselsort#(true(),cons(N,L)) = [4] L + [4] N + [4] > [4] L + [3] = c_10(selsort#(L)) selsort#(cons(N,L)) = [4] L + [4] N + [7] > [4] L + [4] N + [6] = c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) Following rules are (at-least) weakly oriented: ifselsort#(false(),cons(N,L)) = [4] L + [4] N + [4] >= [4] L + [4] N + [3] = c_9(selsort#(replace(min(cons(N,L)),N,L))) ifrepl(false(),N,M,cons(K,L)) = [1] K + [1] L + [1] M + [1] >= [1] K + [1] L + [1] M + [1] = cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) = [1] K + [1] L + [1] M + [1] >= [1] L + [1] M + [1] = cons(M,L) replace(N,M,cons(K,L)) = [1] K + [1] L + [1] M + [1] >= [1] K + [1] L + [1] M + [1] = ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) = [1] M + [3] >= [3] = nil() ****** Step 5.b:3.b:3.b:3.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/1,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 5.b:3.b:3.b:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L))) ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/1,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L))) -->_1 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))):3 2:W:ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) -->_1 selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))):3 3:W:selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) -->_1 ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)):2 -->_1 ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: ifselsort#(false(),cons(N,L)) -> c_9(selsort#(replace(min(cons(N,L)),N,L))) 3: selsort#(cons(N,L)) -> c_19(ifselsort#(eq(N,min(cons(N,L))),cons(N,L))) 2: ifselsort#(true(),cons(N,L)) -> c_10(selsort#(L)) ****** Step 5.b:3.b:3.b:3.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: eq(0(),0()) -> true() eq(0(),s(Y)) -> false() eq(s(X),0()) -> false() eq(s(X),s(Y)) -> eq(X,Y) ifmin(false(),cons(N,cons(M,L))) -> min(cons(M,L)) ifmin(true(),cons(N,cons(M,L))) -> min(cons(N,L)) ifrepl(false(),N,M,cons(K,L)) -> cons(K,replace(N,M,L)) ifrepl(true(),N,M,cons(K,L)) -> cons(M,L) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) min(cons(N,cons(M,L))) -> ifmin(le(N,M),cons(N,cons(M,L))) min(cons(0(),nil())) -> 0() min(cons(s(N),nil())) -> s(N) replace(N,M,cons(K,L)) -> ifrepl(eq(N,K),N,M,cons(K,L)) replace(N,M,nil()) -> nil() - Signature: {eq/2,ifmin/2,ifrepl/4,ifselsort/2,le/2,min/1,replace/3,selsort/1,eq#/2,ifmin#/2,ifrepl#/4,ifselsort#/2 ,le#/2,min#/1,replace#/3,selsort#/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0,c_1/0,c_2/0,c_3/0,c_4/1,c_5/1 ,c_6/1,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/0,c_13/1,c_14/2,c_15/0,c_16/0,c_17/1,c_18/0,c_19/1,c_20/0} - Obligation: innermost runtime complexity wrt. defined symbols {eq#,ifmin#,ifrepl#,ifselsort#,le#,min#,replace# ,selsort#} and constructors {0,cons,false,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))