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: WeightGap 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(Cons) = [1] x1 + [1] x2 + [0] p(Nil) = [1] p(One) = [1] p(Zero) = [0] p(add#2) = [1] x2 + [1] p(inc#1) = [2] x1 + [1] p(main) = [1] x1 + [0] p(add#2#) = [1] x2 + [0] p(inc#1#) = [8] x1 + [0] p(main#) = [1] x1 + [0] p(c_1) = [4] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [1] p(c_8) = [1] p(c_9) = [2] x1 + [0] Following rules are strictly oriented: add#2#(Cons(One(),x4),Cons(One(),x2)) = [1] x2 + [1] > [1] x2 + [0] = c_3(add#2#(x4,x2)) add#2#(Cons(Zero(),x4),Cons(One(),x2)) = [1] x2 + [1] > [1] x2 + [0] = c_4(add#2#(x4,x2)) Following rules are (at-least) weakly oriented: add#2#(Cons(x6,x4),Cons(Zero(),x2)) = [1] x2 + [0] >= [1] x2 + [0] = c_2(add#2#(x4,x2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.a:4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: add#2#(Cons(x6,x4),Cons(Zero(),x2)) -> c_2(add#2#(x4,x2)) - Weak DPs: 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(Cons) = [1] x1 + [1] x2 + [8] p(Nil) = [4] p(One) = [1] p(Zero) = [8] p(add#2) = [1] x2 + [2] p(inc#1) = [1] x1 + [4] p(main) = [1] x1 + [1] x2 + [0] p(add#2#) = [1] x1 + [3] p(inc#1#) = [0] p(main#) = [8] x1 + [1] p(c_1) = [0] p(c_2) = [1] x1 + [4] p(c_3) = [1] x1 + [1] p(c_4) = [1] x1 + [0] p(c_5) = [2] p(c_6) = [1] x1 + [2] p(c_7) = [1] p(c_8) = [1] p(c_9) = [8] x1 + [0] Following rules are strictly oriented: add#2#(Cons(x6,x4),Cons(Zero(),x2)) = [1] x4 + [1] x6 + [11] > [1] x4 + [7] = c_2(add#2#(x4,x2)) Following rules are (at-least) weakly oriented: add#2#(Cons(One(),x4),Cons(One(),x2)) = [1] x4 + [12] >= [1] x4 + [4] = c_3(add#2#(x4,x2)) add#2#(Cons(Zero(),x4),Cons(One(),x2)) = [1] x4 + [19] >= [1] x4 + [3] = c_4(add#2#(x4,x2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.a:5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak 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: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 6.b:1: WeightGap 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: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(Cons) = {2}, uargs(inc#1) = {1}, uargs(inc#1#) = {1}, uargs(c_6) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(Cons) = [1] x2 + [1] p(Nil) = [0] p(One) = [4] p(Zero) = [0] p(add#2) = [1] x1 + [1] x2 + [5] p(inc#1) = [1] x1 + [1] p(main) = [4] x1 + [1] x2 + [1] p(add#2#) = [4] x1 + [4] x2 + [0] p(inc#1#) = [1] x1 + [0] p(main#) = [1] x1 + [1] x2 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x2 + [2] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [2] x1 + [1] Following rules are strictly oriented: inc#1#(Cons(One(),x8)) = [1] x8 + [1] > [1] x8 + [0] = c_6(inc#1#(x8)) Following rules are (at-least) weakly oriented: add#2#(Cons(x6,x4),Cons(Zero(),x2)) = [4] x2 + [4] x4 + [8] >= [4] x2 + [4] x4 + [0] = add#2#(x4,x2) add#2#(Cons(One(),x4),Cons(One(),x2)) = [4] x2 + [4] x4 + [8] >= [4] x2 + [4] x4 + [0] = add#2#(x4,x2) add#2#(Cons(One(),x4),Cons(One(),x2)) = [4] x2 + [4] x4 + [8] >= [1] x2 + [1] x4 + [5] = inc#1#(add#2(x4,x2)) add#2#(Cons(Zero(),x4),Cons(One(),x2)) = [4] x2 + [4] x4 + [8] >= [4] x2 + [4] x4 + [0] = add#2#(x4,x2) add#2(x2,Nil()) = [1] x2 + [5] >= [1] x2 + [0] = x2 add#2(Cons(x6,x4),Cons(Zero(),x2)) = [1] x2 + [1] x4 + [7] >= [1] x2 + [1] x4 + [6] = Cons(x6,add#2(x4,x2)) add#2(Cons(One(),x4),Cons(One(),x2)) = [1] x2 + [1] x4 + [7] >= [1] x2 + [1] x4 + [7] = Cons(Zero(),inc#1(add#2(x4,x2))) add#2(Cons(Zero(),x4),Cons(One(),x2)) = [1] x2 + [1] x4 + [7] >= [1] x2 + [1] x4 + [6] = Cons(One(),add#2(x4,x2)) add#2(Nil(),Cons(x4,x2)) = [1] x2 + [6] >= [1] x2 + [1] = Cons(x4,x2) inc#1(Cons(One(),x8)) = [1] x8 + [2] >= [1] x8 + [2] = Cons(Zero(),inc#1(x8)) inc#1(Cons(Zero(),x8)) = [1] x8 + [2] >= [1] x8 + [1] = Cons(One(),x8) inc#1(Nil()) = [1] >= [1] = Cons(One(),Nil()) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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) 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: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))