WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: append_xs#1(Cons(x7,x12),x14) -> Cons(x7,append_xs#1(x12,x14)) append_xs#1(Nil(),x8) -> x8 main(x12,x3) -> map#2(append_xs(x12),x3) map#2(append_xs(x2),Nil()) -> Nil() map#2(append_xs(x6),Cons(x4,x2)) -> Cons(append_xs#1(x6,x4),map#2(append_xs(x6),x2)) - Signature: {append_xs#1/2,main/2,map#2/2} / {Cons/2,Nil/0,append_xs/1} - Obligation: innermost runtime complexity wrt. defined symbols {append_xs#1,main,map#2} and constructors {Cons,Nil ,append_xs} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs append_xs#1#(Cons(x7,x12),x14) -> c_1(append_xs#1#(x12,x14)) append_xs#1#(Nil(),x8) -> c_2() main#(x12,x3) -> c_3(map#2#(append_xs(x12),x3)) map#2#(append_xs(x2),Nil()) -> c_4() map#2#(append_xs(x6),Cons(x4,x2)) -> c_5(append_xs#1#(x6,x4),map#2#(append_xs(x6),x2)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append_xs#1#(Cons(x7,x12),x14) -> c_1(append_xs#1#(x12,x14)) append_xs#1#(Nil(),x8) -> c_2() main#(x12,x3) -> c_3(map#2#(append_xs(x12),x3)) map#2#(append_xs(x2),Nil()) -> c_4() map#2#(append_xs(x6),Cons(x4,x2)) -> c_5(append_xs#1#(x6,x4),map#2#(append_xs(x6),x2)) - Weak TRS: append_xs#1(Cons(x7,x12),x14) -> Cons(x7,append_xs#1(x12,x14)) append_xs#1(Nil(),x8) -> x8 main(x12,x3) -> map#2(append_xs(x12),x3) map#2(append_xs(x2),Nil()) -> Nil() map#2(append_xs(x6),Cons(x4,x2)) -> Cons(append_xs#1(x6,x4),map#2(append_xs(x6),x2)) - Signature: {append_xs#1/2,main/2,map#2/2,append_xs#1#/2,main#/2,map#2#/2} / {Cons/2,Nil/0,append_xs/1,c_1/1,c_2/0,c_3/1 ,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {append_xs#1#,main#,map#2#} and constructors {Cons,Nil ,append_xs} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,4} by application of Pre({2,4}) = {1,3,5}. Here rules are labelled as follows: 1: append_xs#1#(Cons(x7,x12),x14) -> c_1(append_xs#1#(x12,x14)) 2: append_xs#1#(Nil(),x8) -> c_2() 3: main#(x12,x3) -> c_3(map#2#(append_xs(x12),x3)) 4: map#2#(append_xs(x2),Nil()) -> c_4() 5: map#2#(append_xs(x6),Cons(x4,x2)) -> c_5(append_xs#1#(x6,x4),map#2#(append_xs(x6),x2)) * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append_xs#1#(Cons(x7,x12),x14) -> c_1(append_xs#1#(x12,x14)) main#(x12,x3) -> c_3(map#2#(append_xs(x12),x3)) map#2#(append_xs(x6),Cons(x4,x2)) -> c_5(append_xs#1#(x6,x4),map#2#(append_xs(x6),x2)) - Weak DPs: append_xs#1#(Nil(),x8) -> c_2() map#2#(append_xs(x2),Nil()) -> c_4() - Weak TRS: append_xs#1(Cons(x7,x12),x14) -> Cons(x7,append_xs#1(x12,x14)) append_xs#1(Nil(),x8) -> x8 main(x12,x3) -> map#2(append_xs(x12),x3) map#2(append_xs(x2),Nil()) -> Nil() map#2(append_xs(x6),Cons(x4,x2)) -> Cons(append_xs#1(x6,x4),map#2(append_xs(x6),x2)) - Signature: {append_xs#1/2,main/2,map#2/2,append_xs#1#/2,main#/2,map#2#/2} / {Cons/2,Nil/0,append_xs/1,c_1/1,c_2/0,c_3/1 ,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {append_xs#1#,main#,map#2#} and constructors {Cons,Nil ,append_xs} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:append_xs#1#(Cons(x7,x12),x14) -> c_1(append_xs#1#(x12,x14)) -->_1 append_xs#1#(Nil(),x8) -> c_2():4 -->_1 append_xs#1#(Cons(x7,x12),x14) -> c_1(append_xs#1#(x12,x14)):1 2:S:main#(x12,x3) -> c_3(map#2#(append_xs(x12),x3)) -->_1 map#2#(append_xs(x6),Cons(x4,x2)) -> c_5(append_xs#1#(x6,x4),map#2#(append_xs(x6),x2)):3 -->_1 map#2#(append_xs(x2),Nil()) -> c_4():5 3:S:map#2#(append_xs(x6),Cons(x4,x2)) -> c_5(append_xs#1#(x6,x4),map#2#(append_xs(x6),x2)) -->_2 map#2#(append_xs(x2),Nil()) -> c_4():5 -->_1 append_xs#1#(Nil(),x8) -> c_2():4 -->_2 map#2#(append_xs(x6),Cons(x4,x2)) -> c_5(append_xs#1#(x6,x4),map#2#(append_xs(x6),x2)):3 -->_1 append_xs#1#(Cons(x7,x12),x14) -> c_1(append_xs#1#(x12,x14)):1 4:W:append_xs#1#(Nil(),x8) -> c_2() 5:W:map#2#(append_xs(x2),Nil()) -> c_4() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: map#2#(append_xs(x2),Nil()) -> c_4() 4: append_xs#1#(Nil(),x8) -> c_2() * Step 4: RemoveHeads WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append_xs#1#(Cons(x7,x12),x14) -> c_1(append_xs#1#(x12,x14)) main#(x12,x3) -> c_3(map#2#(append_xs(x12),x3)) map#2#(append_xs(x6),Cons(x4,x2)) -> c_5(append_xs#1#(x6,x4),map#2#(append_xs(x6),x2)) - Weak TRS: append_xs#1(Cons(x7,x12),x14) -> Cons(x7,append_xs#1(x12,x14)) append_xs#1(Nil(),x8) -> x8 main(x12,x3) -> map#2(append_xs(x12),x3) map#2(append_xs(x2),Nil()) -> Nil() map#2(append_xs(x6),Cons(x4,x2)) -> Cons(append_xs#1(x6,x4),map#2(append_xs(x6),x2)) - Signature: {append_xs#1/2,main/2,map#2/2,append_xs#1#/2,main#/2,map#2#/2} / {Cons/2,Nil/0,append_xs/1,c_1/1,c_2/0,c_3/1 ,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {append_xs#1#,main#,map#2#} and constructors {Cons,Nil ,append_xs} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:append_xs#1#(Cons(x7,x12),x14) -> c_1(append_xs#1#(x12,x14)) -->_1 append_xs#1#(Cons(x7,x12),x14) -> c_1(append_xs#1#(x12,x14)):1 2:S:main#(x12,x3) -> c_3(map#2#(append_xs(x12),x3)) -->_1 map#2#(append_xs(x6),Cons(x4,x2)) -> c_5(append_xs#1#(x6,x4),map#2#(append_xs(x6),x2)):3 3:S:map#2#(append_xs(x6),Cons(x4,x2)) -> c_5(append_xs#1#(x6,x4),map#2#(append_xs(x6),x2)) -->_2 map#2#(append_xs(x6),Cons(x4,x2)) -> c_5(append_xs#1#(x6,x4),map#2#(append_xs(x6),x2)):3 -->_1 append_xs#1#(Cons(x7,x12),x14) -> c_1(append_xs#1#(x12,x14)):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). [(2,main#(x12,x3) -> c_3(map#2#(append_xs(x12),x3)))] * Step 5: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append_xs#1#(Cons(x7,x12),x14) -> c_1(append_xs#1#(x12,x14)) map#2#(append_xs(x6),Cons(x4,x2)) -> c_5(append_xs#1#(x6,x4),map#2#(append_xs(x6),x2)) - Weak TRS: append_xs#1(Cons(x7,x12),x14) -> Cons(x7,append_xs#1(x12,x14)) append_xs#1(Nil(),x8) -> x8 main(x12,x3) -> map#2(append_xs(x12),x3) map#2(append_xs(x2),Nil()) -> Nil() map#2(append_xs(x6),Cons(x4,x2)) -> Cons(append_xs#1(x6,x4),map#2(append_xs(x6),x2)) - Signature: {append_xs#1/2,main/2,map#2/2,append_xs#1#/2,main#/2,map#2#/2} / {Cons/2,Nil/0,append_xs/1,c_1/1,c_2/0,c_3/1 ,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {append_xs#1#,main#,map#2#} and constructors {Cons,Nil ,append_xs} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: append_xs#1#(Cons(x7,x12),x14) -> c_1(append_xs#1#(x12,x14)) map#2#(append_xs(x6),Cons(x4,x2)) -> c_5(append_xs#1#(x6,x4),map#2#(append_xs(x6),x2)) * Step 6: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append_xs#1#(Cons(x7,x12),x14) -> c_1(append_xs#1#(x12,x14)) map#2#(append_xs(x6),Cons(x4,x2)) -> c_5(append_xs#1#(x6,x4),map#2#(append_xs(x6),x2)) - Signature: {append_xs#1/2,main/2,map#2/2,append_xs#1#/2,main#/2,map#2#/2} / {Cons/2,Nil/0,append_xs/1,c_1/1,c_2/0,c_3/1 ,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {append_xs#1#,main#,map#2#} and constructors {Cons,Nil ,append_xs} + 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 map#2#(append_xs(x6),Cons(x4,x2)) -> c_5(append_xs#1#(x6,x4),map#2#(append_xs(x6),x2)) and a lower component append_xs#1#(Cons(x7,x12),x14) -> c_1(append_xs#1#(x12,x14)) Further, following extension rules are added to the lower component. map#2#(append_xs(x6),Cons(x4,x2)) -> append_xs#1#(x6,x4) map#2#(append_xs(x6),Cons(x4,x2)) -> map#2#(append_xs(x6),x2) ** Step 6.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: map#2#(append_xs(x6),Cons(x4,x2)) -> c_5(append_xs#1#(x6,x4),map#2#(append_xs(x6),x2)) - Signature: {append_xs#1/2,main/2,map#2/2,append_xs#1#/2,main#/2,map#2#/2} / {Cons/2,Nil/0,append_xs/1,c_1/1,c_2/0,c_3/1 ,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {append_xs#1#,main#,map#2#} and constructors {Cons,Nil ,append_xs} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:map#2#(append_xs(x6),Cons(x4,x2)) -> c_5(append_xs#1#(x6,x4),map#2#(append_xs(x6),x2)) -->_2 map#2#(append_xs(x6),Cons(x4,x2)) -> c_5(append_xs#1#(x6,x4),map#2#(append_xs(x6),x2)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: map#2#(append_xs(x6),Cons(x4,x2)) -> c_5(map#2#(append_xs(x6),x2)) ** Step 6.a:2: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: map#2#(append_xs(x6),Cons(x4,x2)) -> c_5(map#2#(append_xs(x6),x2)) - Signature: {append_xs#1/2,main/2,map#2/2,append_xs#1#/2,main#/2,map#2#/2} / {Cons/2,Nil/0,append_xs/1,c_1/1,c_2/0,c_3/1 ,c_4/0,c_5/1} - Obligation: innermost runtime complexity wrt. defined symbols {append_xs#1#,main#,map#2#} and constructors {Cons,Nil ,append_xs} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- Cons :: ["A"(0) x "A"(8)] -(8)-> "A"(8) append_xs :: ["A"(0)] -(1)-> "A"(1) append_xs :: ["A"(0)] -(4)-> "A"(4) map#2# :: ["A"(1) x "A"(8)] -(5)-> "A"(14) c_5 :: ["A"(0)] -(0)-> "A"(14) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "Cons_A" :: ["A"(0) x "A"(1)] -(1)-> "A"(1) "append_xs_A" :: ["A"(0)] -(1)-> "A"(1) "c_5_A" :: ["A"(0)] -(0)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: map#2#(append_xs(x6),Cons(x4,x2)) -> c_5(map#2#(append_xs(x6),x2)) 2. Weak: ** Step 6.b:1: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: append_xs#1#(Cons(x7,x12),x14) -> c_1(append_xs#1#(x12,x14)) - Weak DPs: map#2#(append_xs(x6),Cons(x4,x2)) -> append_xs#1#(x6,x4) map#2#(append_xs(x6),Cons(x4,x2)) -> map#2#(append_xs(x6),x2) - Signature: {append_xs#1/2,main/2,map#2/2,append_xs#1#/2,main#/2,map#2#/2} / {Cons/2,Nil/0,append_xs/1,c_1/1,c_2/0,c_3/1 ,c_4/0,c_5/2} - Obligation: innermost runtime complexity wrt. defined symbols {append_xs#1#,main#,map#2#} and constructors {Cons,Nil ,append_xs} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- Cons :: ["A"(0) x "A"(15)] -(15)-> "A"(15) Cons :: ["A"(0) x "A"(7)] -(7)-> "A"(7) append_xs :: ["A"(15)] -(0)-> "A"(15) append_xs#1# :: ["A"(15) x "A"(0)] -(11)-> "A"(4) map#2# :: ["A"(15) x "A"(7)] -(5)-> "A"(0) c_1 :: ["A"(0)] -(0)-> "A"(14) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "Cons_A" :: ["A"(0) x "A"(1)] -(1)-> "A"(1) "append_xs_A" :: ["A"(0)] -(0)-> "A"(1) "c_1_A" :: ["A"(0)] -(0)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: append_xs#1#(Cons(x7,x12),x14) -> c_1(append_xs#1#(x12,x14)) 2. Weak: WORST_CASE(?,O(n^2))