WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x6,x4),Cons(Zero(),x2)) -> Cons(x6,add#2(x4,x2)) add#2(Cons(One(),x4),Cons(One(),x2)) -> Cons(Zero(),inc#1(add#2(x4,x2))) add#2(Cons(Zero(),x4),Cons(One(),x2)) -> Cons(One(),add#2(x4,x2)) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) inc#1(Cons(One(),x8)) -> Cons(Zero(),inc#1(x8)) inc#1(Cons(Zero(),x8)) -> Cons(One(),x8) inc#1(Nil()) -> Cons(One(),Nil()) main(x2,x1) -> add#2(x2,x1) - Signature: {add#2/2,inc#1/1,main/2} / {Cons/2,Nil/0,One/0,Zero/0} - Obligation: innermost runtime complexity wrt. defined symbols {add#2,inc#1,main} and constructors {Cons,Nil,One,Zero} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs add#2#(x2,Nil()) -> c_1() add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)) add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)) add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)) add#2#(Nil(),Cons(x4,x2)) -> c_5() inc#1#(Cons(One(),x8)) -> c_6(inc#1#(x8)) inc#1#(Cons(Zero(),x8)) -> c_7() inc#1#(Nil()) -> c_8() main#(x2,x1) -> c_9(add#2#(x2,x1)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: add#2#(x2,Nil()) -> c_1() add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)) add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)) add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)) add#2#(Nil(),Cons(x4,x2)) -> c_5() inc#1#(Cons(One(),x8)) -> c_6(inc#1#(x8)) inc#1#(Cons(Zero(),x8)) -> c_7() inc#1#(Nil()) -> c_8() main#(x2,x1) -> c_9(add#2#(x2,x1)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x6,x4),Cons(Zero(),x2)) -> Cons(x6,add#2(x4,x2)) add#2(Cons(One(),x4),Cons(One(),x2)) -> Cons(Zero(),inc#1(add#2(x4,x2))) add#2(Cons(Zero(),x4),Cons(One(),x2)) -> Cons(One(),add#2(x4,x2)) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) inc#1(Cons(One(),x8)) -> Cons(Zero(),inc#1(x8)) inc#1(Cons(Zero(),x8)) -> Cons(One(),x8) inc#1(Nil()) -> Cons(One(),Nil()) main(x2,x1) -> add#2(x2,x1) - Signature: {add#2/2,inc#1/1,main/2,add#2#/2,inc#1#/1,main#/2} / {Cons/2,Nil/0,One/0,Zero/0,c_1/0,c_2/1,c_3/2,c_4/1 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,inc#1#,main#} and constructors {Cons,Nil,One,Zero} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,5,7,8} by application of Pre({1,5,7,8}) = {2,3,4,6,9}. Here rules are labelled as follows: 1: add#2#(x2,Nil()) -> c_1() 2: add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)) 3: add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)) 4: add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)) 5: add#2#(Nil(),Cons(x4,x2)) -> c_5() 6: inc#1#(Cons(One(),x8)) -> c_6(inc#1#(x8)) 7: inc#1#(Cons(Zero(),x8)) -> c_7() 8: inc#1#(Nil()) -> c_8() 9: main#(x2,x1) -> c_9(add#2#(x2,x1)) * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)) add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)) add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)) inc#1#(Cons(One(),x8)) -> c_6(inc#1#(x8)) main#(x2,x1) -> c_9(add#2#(x2,x1)) - Weak DPs: add#2#(x2,Nil()) -> c_1() add#2#(Nil(),Cons(x4,x2)) -> c_5() inc#1#(Cons(Zero(),x8)) -> c_7() inc#1#(Nil()) -> c_8() - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x6,x4),Cons(Zero(),x2)) -> Cons(x6,add#2(x4,x2)) add#2(Cons(One(),x4),Cons(One(),x2)) -> Cons(Zero(),inc#1(add#2(x4,x2))) add#2(Cons(Zero(),x4),Cons(One(),x2)) -> Cons(One(),add#2(x4,x2)) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) inc#1(Cons(One(),x8)) -> Cons(Zero(),inc#1(x8)) inc#1(Cons(Zero(),x8)) -> Cons(One(),x8) inc#1(Nil()) -> Cons(One(),Nil()) main(x2,x1) -> add#2(x2,x1) - Signature: {add#2/2,inc#1/1,main/2,add#2#/2,inc#1#/1,main#/2} / {Cons/2,Nil/0,One/0,Zero/0,c_1/0,c_2/1,c_3/2,c_4/1 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,inc#1#,main#} and constructors {Cons,Nil,One,Zero} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)) -->_1 add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)):3 -->_1 add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)):2 -->_1 add#2#(Nil(),Cons(x4,x2)) -> c_5():7 -->_1 add#2#(x2,Nil()) -> c_1():6 -->_1 add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)):1 2:S:add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)) -->_1 inc#1#(Cons(One(),x8)) -> c_6(inc#1#(x8)):4 -->_2 add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)):3 -->_1 inc#1#(Nil()) -> c_8():9 -->_1 inc#1#(Cons(Zero(),x8)) -> c_7():8 -->_2 add#2#(Nil(),Cons(x4,x2)) -> c_5():7 -->_2 add#2#(x2,Nil()) -> c_1():6 -->_2 add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)):2 -->_2 add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)):1 3:S:add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)) -->_1 add#2#(Nil(),Cons(x4,x2)) -> c_5():7 -->_1 add#2#(x2,Nil()) -> c_1():6 -->_1 add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)):3 -->_1 add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)):2 -->_1 add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)):1 4:S:inc#1#(Cons(One(),x8)) -> c_6(inc#1#(x8)) -->_1 inc#1#(Nil()) -> c_8():9 -->_1 inc#1#(Cons(Zero(),x8)) -> c_7():8 -->_1 inc#1#(Cons(One(),x8)) -> c_6(inc#1#(x8)):4 5:S:main#(x2,x1) -> c_9(add#2#(x2,x1)) -->_1 add#2#(Nil(),Cons(x4,x2)) -> c_5():7 -->_1 add#2#(x2,Nil()) -> c_1():6 -->_1 add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)):3 -->_1 add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)):2 -->_1 add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)):1 6:W:add#2#(x2,Nil()) -> c_1() 7:W:add#2#(Nil(),Cons(x4,x2)) -> c_5() 8:W:inc#1#(Cons(Zero(),x8)) -> c_7() 9:W:inc#1#(Nil()) -> c_8() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: inc#1#(Cons(Zero(),x8)) -> c_7() 9: inc#1#(Nil()) -> c_8() 6: add#2#(x2,Nil()) -> c_1() 7: add#2#(Nil(),Cons(x4,x2)) -> c_5() * Step 4: RemoveHeads WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)) add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)) add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)) inc#1#(Cons(One(),x8)) -> c_6(inc#1#(x8)) main#(x2,x1) -> c_9(add#2#(x2,x1)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x6,x4),Cons(Zero(),x2)) -> Cons(x6,add#2(x4,x2)) add#2(Cons(One(),x4),Cons(One(),x2)) -> Cons(Zero(),inc#1(add#2(x4,x2))) add#2(Cons(Zero(),x4),Cons(One(),x2)) -> Cons(One(),add#2(x4,x2)) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) inc#1(Cons(One(),x8)) -> Cons(Zero(),inc#1(x8)) inc#1(Cons(Zero(),x8)) -> Cons(One(),x8) inc#1(Nil()) -> Cons(One(),Nil()) main(x2,x1) -> add#2(x2,x1) - Signature: {add#2/2,inc#1/1,main/2,add#2#/2,inc#1#/1,main#/2} / {Cons/2,Nil/0,One/0,Zero/0,c_1/0,c_2/1,c_3/2,c_4/1 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,inc#1#,main#} and constructors {Cons,Nil,One,Zero} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)) -->_1 add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)):3 -->_1 add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)):2 -->_1 add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)):1 2:S:add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)) -->_1 inc#1#(Cons(One(),x8)) -> c_6(inc#1#(x8)):4 -->_2 add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)):3 -->_2 add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)):2 -->_2 add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)):1 3:S:add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)) -->_1 add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)):3 -->_1 add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)):2 -->_1 add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)):1 4:S:inc#1#(Cons(One(),x8)) -> c_6(inc#1#(x8)) -->_1 inc#1#(Cons(One(),x8)) -> c_6(inc#1#(x8)):4 5:S:main#(x2,x1) -> c_9(add#2#(x2,x1)) -->_1 add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)):3 -->_1 add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)):2 -->_1 add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)):1 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(5,main#(x2,x1) -> c_9(add#2#(x2,x1)))] * Step 5: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)) add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)) add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)) inc#1#(Cons(One(),x8)) -> c_6(inc#1#(x8)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x6,x4),Cons(Zero(),x2)) -> Cons(x6,add#2(x4,x2)) add#2(Cons(One(),x4),Cons(One(),x2)) -> Cons(Zero(),inc#1(add#2(x4,x2))) add#2(Cons(Zero(),x4),Cons(One(),x2)) -> Cons(One(),add#2(x4,x2)) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) inc#1(Cons(One(),x8)) -> Cons(Zero(),inc#1(x8)) inc#1(Cons(Zero(),x8)) -> Cons(One(),x8) inc#1(Nil()) -> Cons(One(),Nil()) main(x2,x1) -> add#2(x2,x1) - Signature: {add#2/2,inc#1/1,main/2,add#2#/2,inc#1#/1,main#/2} / {Cons/2,Nil/0,One/0,Zero/0,c_1/0,c_2/1,c_3/2,c_4/1 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,inc#1#,main#} and constructors {Cons,Nil,One,Zero} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: add#2(x2,Nil()) -> x2 add#2(Cons(x6,x4),Cons(Zero(),x2)) -> Cons(x6,add#2(x4,x2)) add#2(Cons(One(),x4),Cons(One(),x2)) -> Cons(Zero(),inc#1(add#2(x4,x2))) add#2(Cons(Zero(),x4),Cons(One(),x2)) -> Cons(One(),add#2(x4,x2)) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) inc#1(Cons(One(),x8)) -> Cons(Zero(),inc#1(x8)) inc#1(Cons(Zero(),x8)) -> Cons(One(),x8) inc#1(Nil()) -> Cons(One(),Nil()) add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)) add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)) add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)) inc#1#(Cons(One(),x8)) -> c_6(inc#1#(x8)) * Step 6: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)) add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)) add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)) inc#1#(Cons(One(),x8)) -> c_6(inc#1#(x8)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x6,x4),Cons(Zero(),x2)) -> Cons(x6,add#2(x4,x2)) add#2(Cons(One(),x4),Cons(One(),x2)) -> Cons(Zero(),inc#1(add#2(x4,x2))) add#2(Cons(Zero(),x4),Cons(One(),x2)) -> Cons(One(),add#2(x4,x2)) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) inc#1(Cons(One(),x8)) -> Cons(Zero(),inc#1(x8)) inc#1(Cons(Zero(),x8)) -> Cons(One(),x8) inc#1(Nil()) -> Cons(One(),Nil()) - Signature: {add#2/2,inc#1/1,main/2,add#2#/2,inc#1#/1,main#/2} / {Cons/2,Nil/0,One/0,Zero/0,c_1/0,c_2/1,c_3/2,c_4/1 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,inc#1#,main#} and constructors {Cons,Nil,One,Zero} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)) add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)) add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)) and a lower component inc#1#(Cons(One(),x8)) -> c_6(inc#1#(x8)) Further, following extension rules are added to the lower component. add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> add#2#(x4,x2) add#2#(Cons(One(),x4),Cons(One(),x2)) -> add#2#(x4,x2) add#2#(Cons(One(),x4),Cons(One(),x2)) -> inc#1#(add#2(x4,x2)) add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> add#2#(x4,x2) ** Step 6.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)) add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)) add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x6,x4),Cons(Zero(),x2)) -> Cons(x6,add#2(x4,x2)) add#2(Cons(One(),x4),Cons(One(),x2)) -> Cons(Zero(),inc#1(add#2(x4,x2))) add#2(Cons(Zero(),x4),Cons(One(),x2)) -> Cons(One(),add#2(x4,x2)) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) inc#1(Cons(One(),x8)) -> Cons(Zero(),inc#1(x8)) inc#1(Cons(Zero(),x8)) -> Cons(One(),x8) inc#1(Nil()) -> Cons(One(),Nil()) - Signature: {add#2/2,inc#1/1,main/2,add#2#/2,inc#1#/1,main#/2} / {Cons/2,Nil/0,One/0,Zero/0,c_1/0,c_2/1,c_3/2,c_4/1 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,inc#1#,main#} and constructors {Cons,Nil,One,Zero} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)) -->_1 add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)):3 -->_1 add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)):2 -->_1 add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)):1 2:S:add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)) -->_2 add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)):3 -->_2 add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)):2 -->_2 add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)):1 3:S:add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)) -->_1 add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)):3 -->_1 add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(inc#1#(add#2(x4,x2)),add#2#(x4,x2)):2 -->_1 add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(add#2#(x4,x2)) ** Step 6.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)) add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(add#2#(x4,x2)) add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x6,x4),Cons(Zero(),x2)) -> Cons(x6,add#2(x4,x2)) add#2(Cons(One(),x4),Cons(One(),x2)) -> Cons(Zero(),inc#1(add#2(x4,x2))) add#2(Cons(Zero(),x4),Cons(One(),x2)) -> Cons(One(),add#2(x4,x2)) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) inc#1(Cons(One(),x8)) -> Cons(Zero(),inc#1(x8)) inc#1(Cons(Zero(),x8)) -> Cons(One(),x8) inc#1(Nil()) -> Cons(One(),Nil()) - Signature: {add#2/2,inc#1/1,main/2,add#2#/2,inc#1#/1,main#/2} / {Cons/2,Nil/0,One/0,Zero/0,c_1/0,c_2/1,c_3/1,c_4/1 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,inc#1#,main#} and constructors {Cons,Nil,One,Zero} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)) add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(add#2#(x4,x2)) add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)) ** Step 6.a:3: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)) add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(add#2#(x4,x2)) add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)) - Signature: {add#2/2,inc#1/1,main/2,add#2#/2,inc#1#/1,main#/2} / {Cons/2,Nil/0,One/0,Zero/0,c_1/0,c_2/1,c_3/1,c_4/1 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,inc#1#,main#} and constructors {Cons,Nil,One,Zero} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- Cons :: ["A"(0) x "A"(1)] -(1)-> "A"(1) One :: [] -(0)-> "A"(0) Zero :: [] -(0)-> "A"(0) add#2# :: ["A"(1) x "A"(1)] -(14)-> "A"(0) c_2 :: ["A"(0)] -(0)-> "A"(14) c_3 :: ["A"(0)] -(0)-> "A"(14) c_4 :: ["A"(0)] -(0)-> "A"(14) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "Cons_A" :: ["A"(0) x "A"(1)] -(1)-> "A"(1) "One_A" :: [] -(0)-> "A"(1) "Zero_A" :: [] -(0)-> "A"(1) "c_2_A" :: ["A"(0)] -(0)-> "A"(1) "c_3_A" :: ["A"(0)] -(0)-> "A"(1) "c_4_A" :: ["A"(0)] -(0)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)) 2. Weak: add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)) add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(add#2#(x4,x2)) ** Step 6.a:4: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)) add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(add#2#(x4,x2)) - Weak DPs: add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> c_4(add#2#(x4,x2)) - Signature: {add#2/2,inc#1/1,main/2,add#2#/2,inc#1#/1,main#/2} / {Cons/2,Nil/0,One/0,Zero/0,c_1/0,c_2/1,c_3/1,c_4/1 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,inc#1#,main#} and constructors {Cons,Nil,One,Zero} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- Cons :: ["A"(0) x "A"(12)] -(12)-> "A"(12) Cons :: ["A"(0) x "A"(2)] -(2)-> "A"(2) One :: [] -(0)-> "A"(0) Zero :: [] -(0)-> "A"(0) add#2# :: ["A"(12) x "A"(2)] -(0)-> "A"(0) c_2 :: ["A"(0)] -(0)-> "A"(14) c_3 :: ["A"(0)] -(0)-> "A"(0) c_4 :: ["A"(0)] -(0)-> "A"(14) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "Cons_A" :: ["A"(0) x "A"(1)] -(1)-> "A"(1) "One_A" :: [] -(0)-> "A"(1) "Zero_A" :: [] -(0)-> "A"(1) "c_2_A" :: ["A"(0)] -(0)-> "A"(1) "c_3_A" :: ["A"(0)] -(0)-> "A"(1) "c_4_A" :: ["A"(0)] -(0)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)) add#2#(Cons(One(),x4),Cons(One(),x2)) -> c_3(add#2#(x4,x2)) 2. Weak: ** Step 6.b:1: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: inc#1#(Cons(One(),x8)) -> c_6(inc#1#(x8)) - Weak DPs: add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> add#2#(x4,x2) add#2#(Cons(One(),x4),Cons(One(),x2)) -> add#2#(x4,x2) add#2#(Cons(One(),x4),Cons(One(),x2)) -> inc#1#(add#2(x4,x2)) add#2#(Cons(Zero(),x4),Cons(One(),x2)) -> add#2#(x4,x2) - Weak TRS: add#2(x2,Nil()) -> x2 add#2(Cons(x6,x4),Cons(Zero(),x2)) -> Cons(x6,add#2(x4,x2)) add#2(Cons(One(),x4),Cons(One(),x2)) -> Cons(Zero(),inc#1(add#2(x4,x2))) add#2(Cons(Zero(),x4),Cons(One(),x2)) -> Cons(One(),add#2(x4,x2)) add#2(Nil(),Cons(x4,x2)) -> Cons(x4,x2) inc#1(Cons(One(),x8)) -> Cons(Zero(),inc#1(x8)) inc#1(Cons(Zero(),x8)) -> Cons(One(),x8) inc#1(Nil()) -> Cons(One(),Nil()) - Signature: {add#2/2,inc#1/1,main/2,add#2#/2,inc#1#/1,main#/2} / {Cons/2,Nil/0,One/0,Zero/0,c_1/0,c_2/1,c_3/2,c_4/1 ,c_5/0,c_6/1,c_7/0,c_8/0,c_9/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#2#,inc#1#,main#} and constructors {Cons,Nil,One,Zero} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- Cons :: ["A"(8) x "A"(8)] -(8)-> "A"(8) Cons :: ["A"(15) x "A"(15)] -(15)-> "A"(15) Cons :: ["A"(9) x "A"(9)] -(9)-> "A"(9) Cons :: ["A"(10) x "A"(10)] -(10)-> "A"(10) Nil :: [] -(0)-> "A"(15) Nil :: [] -(0)-> "A"(9) One :: [] -(0)-> "A"(8) One :: [] -(0)-> "A"(15) One :: [] -(0)-> "A"(9) Zero :: [] -(0)-> "A"(15) Zero :: [] -(0)-> "A"(9) add#2 :: ["A"(15) x "A"(15)] -(3)-> "A"(9) inc#1 :: ["A"(9)] -(9)-> "A"(9) add#2# :: ["A"(15) x "A"(15)] -(4)-> "A"(0) inc#1# :: ["A"(8)] -(0)-> "A"(0) c_6 :: ["A"(0)] -(0)-> "A"(12) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "Cons_A" :: ["A"(1) x "A"(1)] -(1)-> "A"(1) "Nil_A" :: [] -(0)-> "A"(1) "One_A" :: [] -(0)-> "A"(1) "Zero_A" :: [] -(0)-> "A"(1) "c_6_A" :: ["A"(0)] -(0)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: inc#1#(Cons(One(),x8)) -> c_6(inc#1#(x8)) 2. Weak: WORST_CASE(?,O(n^2))