MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: main(0()) -> 0() main(S(x1)) -> sum#1(map#2(Cons(S(x1),unfoldr#2(S(x1))))) map#2(Cons(x2,x5)) -> Cons(mult#2(x2,x2),map#2(x5)) map#2(Nil()) -> Nil() mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> plus#2(x2,mult#2(x4,x2)) plus#2(0(),x8) -> x8 plus#2(S(x4),x2) -> S(plus#2(x4,x2)) sum#1(Cons(x2,x1)) -> plus#2(x2,sum#1(x1)) sum#1(Nil()) -> 0() unfoldr#2(0()) -> Nil() unfoldr#2(S(x2)) -> Cons(x2,unfoldr#2(x2)) - Signature: {main/1,map#2/1,mult#2/2,plus#2/2,sum#1/1,unfoldr#2/1} / {0/0,Cons/2,Nil/0,S/1} - Obligation: innermost runtime complexity wrt. defined symbols {main,map#2,mult#2,plus#2,sum#1 ,unfoldr#2} and constructors {0,Cons,Nil,S} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs main#(0()) -> c_1() main#(S(x1)) -> c_2(sum#1#(map#2(Cons(S(x1),unfoldr#2(S(x1))))) ,map#2#(Cons(S(x1),unfoldr#2(S(x1)))) ,unfoldr#2#(S(x1))) map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5)) map#2#(Nil()) -> c_4() mult#2#(0(),x2) -> c_5() mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2)) plus#2#(0(),x8) -> c_7() plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2)) sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1)) sum#1#(Nil()) -> c_10() unfoldr#2#(0()) -> c_11() unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: main#(0()) -> c_1() main#(S(x1)) -> c_2(sum#1#(map#2(Cons(S(x1),unfoldr#2(S(x1))))) ,map#2#(Cons(S(x1),unfoldr#2(S(x1)))) ,unfoldr#2#(S(x1))) map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5)) map#2#(Nil()) -> c_4() mult#2#(0(),x2) -> c_5() mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2)) plus#2#(0(),x8) -> c_7() plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2)) sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1)) sum#1#(Nil()) -> c_10() unfoldr#2#(0()) -> c_11() unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2)) - Weak TRS: main(0()) -> 0() main(S(x1)) -> sum#1(map#2(Cons(S(x1),unfoldr#2(S(x1))))) map#2(Cons(x2,x5)) -> Cons(mult#2(x2,x2),map#2(x5)) map#2(Nil()) -> Nil() mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> plus#2(x2,mult#2(x4,x2)) plus#2(0(),x8) -> x8 plus#2(S(x4),x2) -> S(plus#2(x4,x2)) sum#1(Cons(x2,x1)) -> plus#2(x2,sum#1(x1)) sum#1(Nil()) -> 0() unfoldr#2(0()) -> Nil() unfoldr#2(S(x2)) -> Cons(x2,unfoldr#2(x2)) - Signature: {main/1,map#2/1,mult#2/2,plus#2/2,sum#1/1,unfoldr#2/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2,sum#1#/1 ,unfoldr#2#/1} / {0/0,Cons/2,Nil/0,S/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/2,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {main#,map#2#,mult#2#,plus#2#,sum#1# ,unfoldr#2#} and constructors {0,Cons,Nil,S} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: map#2(Cons(x2,x5)) -> Cons(mult#2(x2,x2),map#2(x5)) map#2(Nil()) -> Nil() mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> plus#2(x2,mult#2(x4,x2)) plus#2(0(),x8) -> x8 plus#2(S(x4),x2) -> S(plus#2(x4,x2)) sum#1(Cons(x2,x1)) -> plus#2(x2,sum#1(x1)) sum#1(Nil()) -> 0() unfoldr#2(0()) -> Nil() unfoldr#2(S(x2)) -> Cons(x2,unfoldr#2(x2)) main#(0()) -> c_1() main#(S(x1)) -> c_2(sum#1#(map#2(Cons(S(x1),unfoldr#2(S(x1))))) ,map#2#(Cons(S(x1),unfoldr#2(S(x1)))) ,unfoldr#2#(S(x1))) map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5)) map#2#(Nil()) -> c_4() mult#2#(0(),x2) -> c_5() mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2)) plus#2#(0(),x8) -> c_7() plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2)) sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1)) sum#1#(Nil()) -> c_10() unfoldr#2#(0()) -> c_11() unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2)) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: main#(0()) -> c_1() main#(S(x1)) -> c_2(sum#1#(map#2(Cons(S(x1),unfoldr#2(S(x1))))) ,map#2#(Cons(S(x1),unfoldr#2(S(x1)))) ,unfoldr#2#(S(x1))) map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5)) map#2#(Nil()) -> c_4() mult#2#(0(),x2) -> c_5() mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2)) plus#2#(0(),x8) -> c_7() plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2)) sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1)) sum#1#(Nil()) -> c_10() unfoldr#2#(0()) -> c_11() unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2)) - Weak TRS: map#2(Cons(x2,x5)) -> Cons(mult#2(x2,x2),map#2(x5)) map#2(Nil()) -> Nil() mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> plus#2(x2,mult#2(x4,x2)) plus#2(0(),x8) -> x8 plus#2(S(x4),x2) -> S(plus#2(x4,x2)) sum#1(Cons(x2,x1)) -> plus#2(x2,sum#1(x1)) sum#1(Nil()) -> 0() unfoldr#2(0()) -> Nil() unfoldr#2(S(x2)) -> Cons(x2,unfoldr#2(x2)) - Signature: {main/1,map#2/1,mult#2/2,plus#2/2,sum#1/1,unfoldr#2/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2,sum#1#/1 ,unfoldr#2#/1} / {0/0,Cons/2,Nil/0,S/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/2,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {main#,map#2#,mult#2#,plus#2#,sum#1# ,unfoldr#2#} and constructors {0,Cons,Nil,S} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,4,5,7,10,11} by application of Pre({1,4,5,7,10,11}) = {2,3,6,8,9,12}. Here rules are labelled as follows: 1: main#(0()) -> c_1() 2: main#(S(x1)) -> c_2(sum#1#(map#2(Cons(S(x1),unfoldr#2(S(x1))))) ,map#2#(Cons(S(x1),unfoldr#2(S(x1)))) ,unfoldr#2#(S(x1))) 3: map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5)) 4: map#2#(Nil()) -> c_4() 5: mult#2#(0(),x2) -> c_5() 6: mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2)) 7: plus#2#(0(),x8) -> c_7() 8: plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2)) 9: sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1)) 10: sum#1#(Nil()) -> c_10() 11: unfoldr#2#(0()) -> c_11() 12: unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2)) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: main#(S(x1)) -> c_2(sum#1#(map#2(Cons(S(x1),unfoldr#2(S(x1))))) ,map#2#(Cons(S(x1),unfoldr#2(S(x1)))) ,unfoldr#2#(S(x1))) map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5)) mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2)) plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2)) sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1)) unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2)) - Weak DPs: main#(0()) -> c_1() map#2#(Nil()) -> c_4() mult#2#(0(),x2) -> c_5() plus#2#(0(),x8) -> c_7() sum#1#(Nil()) -> c_10() unfoldr#2#(0()) -> c_11() - Weak TRS: map#2(Cons(x2,x5)) -> Cons(mult#2(x2,x2),map#2(x5)) map#2(Nil()) -> Nil() mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> plus#2(x2,mult#2(x4,x2)) plus#2(0(),x8) -> x8 plus#2(S(x4),x2) -> S(plus#2(x4,x2)) sum#1(Cons(x2,x1)) -> plus#2(x2,sum#1(x1)) sum#1(Nil()) -> 0() unfoldr#2(0()) -> Nil() unfoldr#2(S(x2)) -> Cons(x2,unfoldr#2(x2)) - Signature: {main/1,map#2/1,mult#2/2,plus#2/2,sum#1/1,unfoldr#2/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2,sum#1#/1 ,unfoldr#2#/1} / {0/0,Cons/2,Nil/0,S/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/2,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {main#,map#2#,mult#2#,plus#2#,sum#1# ,unfoldr#2#} and constructors {0,Cons,Nil,S} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:main#(S(x1)) -> c_2(sum#1#(map#2(Cons(S(x1),unfoldr#2(S(x1))))) ,map#2#(Cons(S(x1),unfoldr#2(S(x1)))) ,unfoldr#2#(S(x1))) -->_3 unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2)):6 -->_1 sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1)):5 -->_2 map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5)):2 -->_1 sum#1#(Nil()) -> c_10():11 2:S:map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5)) -->_1 mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2)):3 -->_1 mult#2#(0(),x2) -> c_5():9 -->_2 map#2#(Nil()) -> c_4():8 -->_2 map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5)):2 3:S:mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2)) -->_1 plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2)):4 -->_1 plus#2#(0(),x8) -> c_7():10 -->_2 mult#2#(0(),x2) -> c_5():9 -->_2 mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2)):3 4:S:plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2)) -->_1 plus#2#(0(),x8) -> c_7():10 -->_1 plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2)):4 5:S:sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1)) -->_2 sum#1#(Nil()) -> c_10():11 -->_1 plus#2#(0(),x8) -> c_7():10 -->_2 sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1)):5 -->_1 plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2)):4 6:S:unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2)) -->_1 unfoldr#2#(0()) -> c_11():12 -->_1 unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2)):6 7:W:main#(0()) -> c_1() 8:W:map#2#(Nil()) -> c_4() 9:W:mult#2#(0(),x2) -> c_5() 10:W:plus#2#(0(),x8) -> c_7() 11:W:sum#1#(Nil()) -> c_10() 12:W:unfoldr#2#(0()) -> c_11() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: main#(0()) -> c_1() 8: map#2#(Nil()) -> c_4() 9: mult#2#(0(),x2) -> c_5() 10: plus#2#(0(),x8) -> c_7() 11: sum#1#(Nil()) -> c_10() 12: unfoldr#2#(0()) -> c_11() * Step 5: NaturalMI MAYBE + Considered Problem: - Strict DPs: main#(S(x1)) -> c_2(sum#1#(map#2(Cons(S(x1),unfoldr#2(S(x1))))) ,map#2#(Cons(S(x1),unfoldr#2(S(x1)))) ,unfoldr#2#(S(x1))) map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5)) mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2)) plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2)) sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1)) unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2)) - Weak TRS: map#2(Cons(x2,x5)) -> Cons(mult#2(x2,x2),map#2(x5)) map#2(Nil()) -> Nil() mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> plus#2(x2,mult#2(x4,x2)) plus#2(0(),x8) -> x8 plus#2(S(x4),x2) -> S(plus#2(x4,x2)) sum#1(Cons(x2,x1)) -> plus#2(x2,sum#1(x1)) sum#1(Nil()) -> 0() unfoldr#2(0()) -> Nil() unfoldr#2(S(x2)) -> Cons(x2,unfoldr#2(x2)) - Signature: {main/1,map#2/1,mult#2/2,plus#2/2,sum#1/1,unfoldr#2/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2,sum#1#/1 ,unfoldr#2#/1} / {0/0,Cons/2,Nil/0,S/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/2,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {main#,map#2#,mult#2#,plus#2#,sum#1# ,unfoldr#2#} and constructors {0,Cons,Nil,S} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_2) = {1,2,3}, uargs(c_3) = {1,2}, uargs(c_6) = {1,2}, uargs(c_8) = {1}, uargs(c_9) = {1,2}, uargs(c_12) = {1} Following symbols are considered usable: {main#,map#2#,mult#2#,plus#2#,sum#1#,unfoldr#2#} TcT has computed the following interpretation: p(0) = [0] p(Cons) = [0] p(Nil) = [0] p(S) = [9] p(main) = [0] p(map#2) = [0] p(mult#2) = [0] p(plus#2) = [0] p(sum#1) = [0] p(unfoldr#2) = [1] x1 + [15] p(main#) = [13] p(map#2#) = [0] p(mult#2#) = [0] p(plus#2#) = [0] p(sum#1#) = [0] p(unfoldr#2#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [8] x3 + [0] p(c_3) = [1] x1 + [2] x2 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [4] x1 + [4] x2 + [0] p(c_7) = [1] p(c_8) = [8] x1 + [0] p(c_9) = [2] x1 + [1] x2 + [0] p(c_10) = [2] p(c_11) = [0] p(c_12) = [1] x1 + [0] Following rules are strictly oriented: main#(S(x1)) = [13] > [0] = c_2(sum#1#(map#2(Cons(S(x1),unfoldr#2(S(x1))))),map#2#(Cons(S(x1),unfoldr#2(S(x1)))),unfoldr#2#(S(x1))) Following rules are (at-least) weakly oriented: map#2#(Cons(x2,x5)) = [0] >= [0] = c_3(mult#2#(x2,x2),map#2#(x5)) mult#2#(S(x4),x2) = [0] >= [0] = c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2)) plus#2#(S(x4),x2) = [0] >= [0] = c_8(plus#2#(x4,x2)) sum#1#(Cons(x2,x1)) = [0] >= [0] = c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1)) unfoldr#2#(S(x2)) = [0] >= [0] = c_12(unfoldr#2#(x2)) * Step 6: Failure MAYBE + Considered Problem: - Strict DPs: map#2#(Cons(x2,x5)) -> c_3(mult#2#(x2,x2),map#2#(x5)) mult#2#(S(x4),x2) -> c_6(plus#2#(x2,mult#2(x4,x2)),mult#2#(x4,x2)) plus#2#(S(x4),x2) -> c_8(plus#2#(x4,x2)) sum#1#(Cons(x2,x1)) -> c_9(plus#2#(x2,sum#1(x1)),sum#1#(x1)) unfoldr#2#(S(x2)) -> c_12(unfoldr#2#(x2)) - Weak DPs: main#(S(x1)) -> c_2(sum#1#(map#2(Cons(S(x1),unfoldr#2(S(x1))))) ,map#2#(Cons(S(x1),unfoldr#2(S(x1)))) ,unfoldr#2#(S(x1))) - Weak TRS: map#2(Cons(x2,x5)) -> Cons(mult#2(x2,x2),map#2(x5)) map#2(Nil()) -> Nil() mult#2(0(),x2) -> 0() mult#2(S(x4),x2) -> plus#2(x2,mult#2(x4,x2)) plus#2(0(),x8) -> x8 plus#2(S(x4),x2) -> S(plus#2(x4,x2)) sum#1(Cons(x2,x1)) -> plus#2(x2,sum#1(x1)) sum#1(Nil()) -> 0() unfoldr#2(0()) -> Nil() unfoldr#2(S(x2)) -> Cons(x2,unfoldr#2(x2)) - Signature: {main/1,map#2/1,mult#2/2,plus#2/2,sum#1/1,unfoldr#2/1,main#/1,map#2#/1,mult#2#/2,plus#2#/2,sum#1#/1 ,unfoldr#2#/1} / {0/0,Cons/2,Nil/0,S/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/0,c_6/2,c_7/0,c_8/1,c_9/2,c_10/0,c_11/0 ,c_12/1} - Obligation: innermost runtime complexity wrt. defined symbols {main#,map#2#,mult#2#,plus#2#,sum#1# ,unfoldr#2#} and constructors {0,Cons,Nil,S} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE