MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: cond(false(),n,l) -> tail(nthtail(s(n),l)) cond(true(),n,l) -> l ge(u,0()) -> true() ge(0(),s(v)) -> false() ge(s(u),s(v)) -> ge(u,v) length(cons(x,l)) -> s(length(l)) length(nil()) -> 0() nthtail(n,l) -> cond(ge(n,length(l)),n,l) tail(cons(x,l)) -> l tail(nil()) -> nil() - Signature: {cond/3,ge/2,length/1,nthtail/2,tail/1} / {0/0,cons/2,false/0,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond,ge,length,nthtail,tail} and constructors {0,cons ,false,nil,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs cond#(false(),n,l) -> c_1(tail#(nthtail(s(n),l)),nthtail#(s(n),l)) cond#(true(),n,l) -> c_2() ge#(u,0()) -> c_3() ge#(0(),s(v)) -> c_4() ge#(s(u),s(v)) -> c_5(ge#(u,v)) length#(cons(x,l)) -> c_6(length#(l)) length#(nil()) -> c_7() nthtail#(n,l) -> c_8(cond#(ge(n,length(l)),n,l),ge#(n,length(l)),length#(l)) tail#(cons(x,l)) -> c_9() tail#(nil()) -> c_10() Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: cond#(false(),n,l) -> c_1(tail#(nthtail(s(n),l)),nthtail#(s(n),l)) cond#(true(),n,l) -> c_2() ge#(u,0()) -> c_3() ge#(0(),s(v)) -> c_4() ge#(s(u),s(v)) -> c_5(ge#(u,v)) length#(cons(x,l)) -> c_6(length#(l)) length#(nil()) -> c_7() nthtail#(n,l) -> c_8(cond#(ge(n,length(l)),n,l),ge#(n,length(l)),length#(l)) tail#(cons(x,l)) -> c_9() tail#(nil()) -> c_10() - Weak TRS: cond(false(),n,l) -> tail(nthtail(s(n),l)) cond(true(),n,l) -> l ge(u,0()) -> true() ge(0(),s(v)) -> false() ge(s(u),s(v)) -> ge(u,v) length(cons(x,l)) -> s(length(l)) length(nil()) -> 0() nthtail(n,l) -> cond(ge(n,length(l)),n,l) tail(cons(x,l)) -> l tail(nil()) -> nil() - Signature: {cond/3,ge/2,length/1,nthtail/2,tail/1,cond#/3,ge#/2,length#/1,nthtail#/2,tail#/1} / {0/0,cons/2,false/0 ,nil/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0,c_8/3,c_9/0,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond#,ge#,length#,nthtail#,tail#} 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 {2,3,4,7,9,10} by application of Pre({2,3,4,7,9,10}) = {1,5,6,8}. Here rules are labelled as follows: 1: cond#(false(),n,l) -> c_1(tail#(nthtail(s(n),l)),nthtail#(s(n),l)) 2: cond#(true(),n,l) -> c_2() 3: ge#(u,0()) -> c_3() 4: ge#(0(),s(v)) -> c_4() 5: ge#(s(u),s(v)) -> c_5(ge#(u,v)) 6: length#(cons(x,l)) -> c_6(length#(l)) 7: length#(nil()) -> c_7() 8: nthtail#(n,l) -> c_8(cond#(ge(n,length(l)),n,l),ge#(n,length(l)),length#(l)) 9: tail#(cons(x,l)) -> c_9() 10: tail#(nil()) -> c_10() * Step 3: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: cond#(false(),n,l) -> c_1(tail#(nthtail(s(n),l)),nthtail#(s(n),l)) ge#(s(u),s(v)) -> c_5(ge#(u,v)) length#(cons(x,l)) -> c_6(length#(l)) nthtail#(n,l) -> c_8(cond#(ge(n,length(l)),n,l),ge#(n,length(l)),length#(l)) - Weak DPs: cond#(true(),n,l) -> c_2() ge#(u,0()) -> c_3() ge#(0(),s(v)) -> c_4() length#(nil()) -> c_7() tail#(cons(x,l)) -> c_9() tail#(nil()) -> c_10() - Weak TRS: cond(false(),n,l) -> tail(nthtail(s(n),l)) cond(true(),n,l) -> l ge(u,0()) -> true() ge(0(),s(v)) -> false() ge(s(u),s(v)) -> ge(u,v) length(cons(x,l)) -> s(length(l)) length(nil()) -> 0() nthtail(n,l) -> cond(ge(n,length(l)),n,l) tail(cons(x,l)) -> l tail(nil()) -> nil() - Signature: {cond/3,ge/2,length/1,nthtail/2,tail/1,cond#/3,ge#/2,length#/1,nthtail#/2,tail#/1} / {0/0,cons/2,false/0 ,nil/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0,c_8/3,c_9/0,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond#,ge#,length#,nthtail#,tail#} and constructors {0 ,cons,false,nil,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:cond#(false(),n,l) -> c_1(tail#(nthtail(s(n),l)),nthtail#(s(n),l)) -->_2 nthtail#(n,l) -> c_8(cond#(ge(n,length(l)),n,l),ge#(n,length(l)),length#(l)):4 -->_1 tail#(nil()) -> c_10():10 -->_1 tail#(cons(x,l)) -> c_9():9 2:S:ge#(s(u),s(v)) -> c_5(ge#(u,v)) -->_1 ge#(0(),s(v)) -> c_4():7 -->_1 ge#(u,0()) -> c_3():6 -->_1 ge#(s(u),s(v)) -> c_5(ge#(u,v)):2 3:S:length#(cons(x,l)) -> c_6(length#(l)) -->_1 length#(nil()) -> c_7():8 -->_1 length#(cons(x,l)) -> c_6(length#(l)):3 4:S:nthtail#(n,l) -> c_8(cond#(ge(n,length(l)),n,l),ge#(n,length(l)),length#(l)) -->_3 length#(nil()) -> c_7():8 -->_2 ge#(0(),s(v)) -> c_4():7 -->_2 ge#(u,0()) -> c_3():6 -->_1 cond#(true(),n,l) -> c_2():5 -->_3 length#(cons(x,l)) -> c_6(length#(l)):3 -->_2 ge#(s(u),s(v)) -> c_5(ge#(u,v)):2 -->_1 cond#(false(),n,l) -> c_1(tail#(nthtail(s(n),l)),nthtail#(s(n),l)):1 5:W:cond#(true(),n,l) -> c_2() 6:W:ge#(u,0()) -> c_3() 7:W:ge#(0(),s(v)) -> c_4() 8:W:length#(nil()) -> c_7() 9:W:tail#(cons(x,l)) -> c_9() 10:W:tail#(nil()) -> c_10() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: tail#(cons(x,l)) -> c_9() 10: tail#(nil()) -> c_10() 5: cond#(true(),n,l) -> c_2() 6: ge#(u,0()) -> c_3() 7: ge#(0(),s(v)) -> c_4() 8: length#(nil()) -> c_7() * Step 4: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: cond#(false(),n,l) -> c_1(tail#(nthtail(s(n),l)),nthtail#(s(n),l)) ge#(s(u),s(v)) -> c_5(ge#(u,v)) length#(cons(x,l)) -> c_6(length#(l)) nthtail#(n,l) -> c_8(cond#(ge(n,length(l)),n,l),ge#(n,length(l)),length#(l)) - Weak TRS: cond(false(),n,l) -> tail(nthtail(s(n),l)) cond(true(),n,l) -> l ge(u,0()) -> true() ge(0(),s(v)) -> false() ge(s(u),s(v)) -> ge(u,v) length(cons(x,l)) -> s(length(l)) length(nil()) -> 0() nthtail(n,l) -> cond(ge(n,length(l)),n,l) tail(cons(x,l)) -> l tail(nil()) -> nil() - Signature: {cond/3,ge/2,length/1,nthtail/2,tail/1,cond#/3,ge#/2,length#/1,nthtail#/2,tail#/1} / {0/0,cons/2,false/0 ,nil/0,s/1,true/0,c_1/2,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0,c_8/3,c_9/0,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond#,ge#,length#,nthtail#,tail#} and constructors {0 ,cons,false,nil,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:cond#(false(),n,l) -> c_1(tail#(nthtail(s(n),l)),nthtail#(s(n),l)) -->_2 nthtail#(n,l) -> c_8(cond#(ge(n,length(l)),n,l),ge#(n,length(l)),length#(l)):4 2:S:ge#(s(u),s(v)) -> c_5(ge#(u,v)) -->_1 ge#(s(u),s(v)) -> c_5(ge#(u,v)):2 3:S:length#(cons(x,l)) -> c_6(length#(l)) -->_1 length#(cons(x,l)) -> c_6(length#(l)):3 4:S:nthtail#(n,l) -> c_8(cond#(ge(n,length(l)),n,l),ge#(n,length(l)),length#(l)) -->_3 length#(cons(x,l)) -> c_6(length#(l)):3 -->_2 ge#(s(u),s(v)) -> c_5(ge#(u,v)):2 -->_1 cond#(false(),n,l) -> c_1(tail#(nthtail(s(n),l)),nthtail#(s(n),l)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: cond#(false(),n,l) -> c_1(nthtail#(s(n),l)) * Step 5: UsableRules MAYBE + Considered Problem: - Strict DPs: cond#(false(),n,l) -> c_1(nthtail#(s(n),l)) ge#(s(u),s(v)) -> c_5(ge#(u,v)) length#(cons(x,l)) -> c_6(length#(l)) nthtail#(n,l) -> c_8(cond#(ge(n,length(l)),n,l),ge#(n,length(l)),length#(l)) - Weak TRS: cond(false(),n,l) -> tail(nthtail(s(n),l)) cond(true(),n,l) -> l ge(u,0()) -> true() ge(0(),s(v)) -> false() ge(s(u),s(v)) -> ge(u,v) length(cons(x,l)) -> s(length(l)) length(nil()) -> 0() nthtail(n,l) -> cond(ge(n,length(l)),n,l) tail(cons(x,l)) -> l tail(nil()) -> nil() - Signature: {cond/3,ge/2,length/1,nthtail/2,tail/1,cond#/3,ge#/2,length#/1,nthtail#/2,tail#/1} / {0/0,cons/2,false/0 ,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0,c_8/3,c_9/0,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond#,ge#,length#,nthtail#,tail#} and constructors {0 ,cons,false,nil,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: ge(u,0()) -> true() ge(0(),s(v)) -> false() ge(s(u),s(v)) -> ge(u,v) length(cons(x,l)) -> s(length(l)) length(nil()) -> 0() cond#(false(),n,l) -> c_1(nthtail#(s(n),l)) ge#(s(u),s(v)) -> c_5(ge#(u,v)) length#(cons(x,l)) -> c_6(length#(l)) nthtail#(n,l) -> c_8(cond#(ge(n,length(l)),n,l),ge#(n,length(l)),length#(l)) * Step 6: Failure MAYBE + Considered Problem: - Strict DPs: cond#(false(),n,l) -> c_1(nthtail#(s(n),l)) ge#(s(u),s(v)) -> c_5(ge#(u,v)) length#(cons(x,l)) -> c_6(length#(l)) nthtail#(n,l) -> c_8(cond#(ge(n,length(l)),n,l),ge#(n,length(l)),length#(l)) - Weak TRS: ge(u,0()) -> true() ge(0(),s(v)) -> false() ge(s(u),s(v)) -> ge(u,v) length(cons(x,l)) -> s(length(l)) length(nil()) -> 0() - Signature: {cond/3,ge/2,length/1,nthtail/2,tail/1,cond#/3,ge#/2,length#/1,nthtail#/2,tail#/1} / {0/0,cons/2,false/0 ,nil/0,s/1,true/0,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/1,c_7/0,c_8/3,c_9/0,c_10/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond#,ge#,length#,nthtail#,tail#} and constructors {0 ,cons,false,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE