WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: append#2(Cons(x6,x4),Cons(x2,Nil())) -> Cons(x6,append#2(x4,Cons(x2,Nil()))) append#2(Nil(),Cons(x2,Nil())) -> Cons(x2,Nil()) main(x0) -> rev#1(x0) rev#1(Cons(x2,x1)) -> append#2(rev#1(x1),Cons(x2,Nil())) rev#1(Nil()) -> Nil() - Signature: {append#2/2,main/1,rev#1/1} / {Cons/2,Nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#2,main,rev#1} and constructors {Cons,Nil} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs append#2#(Cons(x6,x4),Cons(x2,Nil())) -> c_1(append#2#(x4,Cons(x2,Nil()))) append#2#(Nil(),Cons(x2,Nil())) -> c_2() main#(x0) -> c_3(rev#1#(x0)) rev#1#(Cons(x2,x1)) -> c_4(append#2#(rev#1(x1),Cons(x2,Nil())),rev#1#(x1)) rev#1#(Nil()) -> c_5() Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#2#(Cons(x6,x4),Cons(x2,Nil())) -> c_1(append#2#(x4,Cons(x2,Nil()))) append#2#(Nil(),Cons(x2,Nil())) -> c_2() main#(x0) -> c_3(rev#1#(x0)) rev#1#(Cons(x2,x1)) -> c_4(append#2#(rev#1(x1),Cons(x2,Nil())),rev#1#(x1)) rev#1#(Nil()) -> c_5() - Weak TRS: append#2(Cons(x6,x4),Cons(x2,Nil())) -> Cons(x6,append#2(x4,Cons(x2,Nil()))) append#2(Nil(),Cons(x2,Nil())) -> Cons(x2,Nil()) main(x0) -> rev#1(x0) rev#1(Cons(x2,x1)) -> append#2(rev#1(x1),Cons(x2,Nil())) rev#1(Nil()) -> Nil() - Signature: {append#2/2,main/1,rev#1/1,append#2#/2,main#/1,rev#1#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#2#,main#,rev#1#} and constructors {Cons,Nil} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,5} by application of Pre({2,5}) = {1,3,4}. Here rules are labelled as follows: 1: append#2#(Cons(x6,x4),Cons(x2,Nil())) -> c_1(append#2#(x4,Cons(x2,Nil()))) 2: append#2#(Nil(),Cons(x2,Nil())) -> c_2() 3: main#(x0) -> c_3(rev#1#(x0)) 4: rev#1#(Cons(x2,x1)) -> c_4(append#2#(rev#1(x1),Cons(x2,Nil())),rev#1#(x1)) 5: rev#1#(Nil()) -> c_5() * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#2#(Cons(x6,x4),Cons(x2,Nil())) -> c_1(append#2#(x4,Cons(x2,Nil()))) main#(x0) -> c_3(rev#1#(x0)) rev#1#(Cons(x2,x1)) -> c_4(append#2#(rev#1(x1),Cons(x2,Nil())),rev#1#(x1)) - Weak DPs: append#2#(Nil(),Cons(x2,Nil())) -> c_2() rev#1#(Nil()) -> c_5() - Weak TRS: append#2(Cons(x6,x4),Cons(x2,Nil())) -> Cons(x6,append#2(x4,Cons(x2,Nil()))) append#2(Nil(),Cons(x2,Nil())) -> Cons(x2,Nil()) main(x0) -> rev#1(x0) rev#1(Cons(x2,x1)) -> append#2(rev#1(x1),Cons(x2,Nil())) rev#1(Nil()) -> Nil() - Signature: {append#2/2,main/1,rev#1/1,append#2#/2,main#/1,rev#1#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#2#,main#,rev#1#} and constructors {Cons,Nil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:append#2#(Cons(x6,x4),Cons(x2,Nil())) -> c_1(append#2#(x4,Cons(x2,Nil()))) -->_1 append#2#(Nil(),Cons(x2,Nil())) -> c_2():4 -->_1 append#2#(Cons(x6,x4),Cons(x2,Nil())) -> c_1(append#2#(x4,Cons(x2,Nil()))):1 2:S:main#(x0) -> c_3(rev#1#(x0)) -->_1 rev#1#(Cons(x2,x1)) -> c_4(append#2#(rev#1(x1),Cons(x2,Nil())),rev#1#(x1)):3 -->_1 rev#1#(Nil()) -> c_5():5 3:S:rev#1#(Cons(x2,x1)) -> c_4(append#2#(rev#1(x1),Cons(x2,Nil())),rev#1#(x1)) -->_2 rev#1#(Nil()) -> c_5():5 -->_1 append#2#(Nil(),Cons(x2,Nil())) -> c_2():4 -->_2 rev#1#(Cons(x2,x1)) -> c_4(append#2#(rev#1(x1),Cons(x2,Nil())),rev#1#(x1)):3 -->_1 append#2#(Cons(x6,x4),Cons(x2,Nil())) -> c_1(append#2#(x4,Cons(x2,Nil()))):1 4:W:append#2#(Nil(),Cons(x2,Nil())) -> c_2() 5:W:rev#1#(Nil()) -> c_5() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: rev#1#(Nil()) -> c_5() 4: append#2#(Nil(),Cons(x2,Nil())) -> c_2() * Step 4: RemoveHeads WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#2#(Cons(x6,x4),Cons(x2,Nil())) -> c_1(append#2#(x4,Cons(x2,Nil()))) main#(x0) -> c_3(rev#1#(x0)) rev#1#(Cons(x2,x1)) -> c_4(append#2#(rev#1(x1),Cons(x2,Nil())),rev#1#(x1)) - Weak TRS: append#2(Cons(x6,x4),Cons(x2,Nil())) -> Cons(x6,append#2(x4,Cons(x2,Nil()))) append#2(Nil(),Cons(x2,Nil())) -> Cons(x2,Nil()) main(x0) -> rev#1(x0) rev#1(Cons(x2,x1)) -> append#2(rev#1(x1),Cons(x2,Nil())) rev#1(Nil()) -> Nil() - Signature: {append#2/2,main/1,rev#1/1,append#2#/2,main#/1,rev#1#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#2#,main#,rev#1#} and constructors {Cons,Nil} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:append#2#(Cons(x6,x4),Cons(x2,Nil())) -> c_1(append#2#(x4,Cons(x2,Nil()))) -->_1 append#2#(Cons(x6,x4),Cons(x2,Nil())) -> c_1(append#2#(x4,Cons(x2,Nil()))):1 2:S:main#(x0) -> c_3(rev#1#(x0)) -->_1 rev#1#(Cons(x2,x1)) -> c_4(append#2#(rev#1(x1),Cons(x2,Nil())),rev#1#(x1)):3 3:S:rev#1#(Cons(x2,x1)) -> c_4(append#2#(rev#1(x1),Cons(x2,Nil())),rev#1#(x1)) -->_2 rev#1#(Cons(x2,x1)) -> c_4(append#2#(rev#1(x1),Cons(x2,Nil())),rev#1#(x1)):3 -->_1 append#2#(Cons(x6,x4),Cons(x2,Nil())) -> c_1(append#2#(x4,Cons(x2,Nil()))):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#(x0) -> c_3(rev#1#(x0)))] * Step 5: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#2#(Cons(x6,x4),Cons(x2,Nil())) -> c_1(append#2#(x4,Cons(x2,Nil()))) rev#1#(Cons(x2,x1)) -> c_4(append#2#(rev#1(x1),Cons(x2,Nil())),rev#1#(x1)) - Weak TRS: append#2(Cons(x6,x4),Cons(x2,Nil())) -> Cons(x6,append#2(x4,Cons(x2,Nil()))) append#2(Nil(),Cons(x2,Nil())) -> Cons(x2,Nil()) main(x0) -> rev#1(x0) rev#1(Cons(x2,x1)) -> append#2(rev#1(x1),Cons(x2,Nil())) rev#1(Nil()) -> Nil() - Signature: {append#2/2,main/1,rev#1/1,append#2#/2,main#/1,rev#1#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#2#,main#,rev#1#} and constructors {Cons,Nil} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: append#2(Cons(x6,x4),Cons(x2,Nil())) -> Cons(x6,append#2(x4,Cons(x2,Nil()))) append#2(Nil(),Cons(x2,Nil())) -> Cons(x2,Nil()) rev#1(Cons(x2,x1)) -> append#2(rev#1(x1),Cons(x2,Nil())) rev#1(Nil()) -> Nil() append#2#(Cons(x6,x4),Cons(x2,Nil())) -> c_1(append#2#(x4,Cons(x2,Nil()))) rev#1#(Cons(x2,x1)) -> c_4(append#2#(rev#1(x1),Cons(x2,Nil())),rev#1#(x1)) * Step 6: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: append#2#(Cons(x6,x4),Cons(x2,Nil())) -> c_1(append#2#(x4,Cons(x2,Nil()))) rev#1#(Cons(x2,x1)) -> c_4(append#2#(rev#1(x1),Cons(x2,Nil())),rev#1#(x1)) - Weak TRS: append#2(Cons(x6,x4),Cons(x2,Nil())) -> Cons(x6,append#2(x4,Cons(x2,Nil()))) append#2(Nil(),Cons(x2,Nil())) -> Cons(x2,Nil()) rev#1(Cons(x2,x1)) -> append#2(rev#1(x1),Cons(x2,Nil())) rev#1(Nil()) -> Nil() - Signature: {append#2/2,main/1,rev#1/1,append#2#/2,main#/1,rev#1#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#2#,main#,rev#1#} and constructors {Cons,Nil} + 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 rev#1#(Cons(x2,x1)) -> c_4(append#2#(rev#1(x1),Cons(x2,Nil())),rev#1#(x1)) and a lower component append#2#(Cons(x6,x4),Cons(x2,Nil())) -> c_1(append#2#(x4,Cons(x2,Nil()))) Further, following extension rules are added to the lower component. rev#1#(Cons(x2,x1)) -> append#2#(rev#1(x1),Cons(x2,Nil())) rev#1#(Cons(x2,x1)) -> rev#1#(x1) ** Step 6.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: rev#1#(Cons(x2,x1)) -> c_4(append#2#(rev#1(x1),Cons(x2,Nil())),rev#1#(x1)) - Weak TRS: append#2(Cons(x6,x4),Cons(x2,Nil())) -> Cons(x6,append#2(x4,Cons(x2,Nil()))) append#2(Nil(),Cons(x2,Nil())) -> Cons(x2,Nil()) rev#1(Cons(x2,x1)) -> append#2(rev#1(x1),Cons(x2,Nil())) rev#1(Nil()) -> Nil() - Signature: {append#2/2,main/1,rev#1/1,append#2#/2,main#/1,rev#1#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#2#,main#,rev#1#} and constructors {Cons,Nil} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:rev#1#(Cons(x2,x1)) -> c_4(append#2#(rev#1(x1),Cons(x2,Nil())),rev#1#(x1)) -->_2 rev#1#(Cons(x2,x1)) -> c_4(append#2#(rev#1(x1),Cons(x2,Nil())),rev#1#(x1)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: rev#1#(Cons(x2,x1)) -> c_4(rev#1#(x1)) ** Step 6.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: rev#1#(Cons(x2,x1)) -> c_4(rev#1#(x1)) - Weak TRS: append#2(Cons(x6,x4),Cons(x2,Nil())) -> Cons(x6,append#2(x4,Cons(x2,Nil()))) append#2(Nil(),Cons(x2,Nil())) -> Cons(x2,Nil()) rev#1(Cons(x2,x1)) -> append#2(rev#1(x1),Cons(x2,Nil())) rev#1(Nil()) -> Nil() - Signature: {append#2/2,main/1,rev#1/1,append#2#/2,main#/1,rev#1#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#2#,main#,rev#1#} and constructors {Cons,Nil} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: rev#1#(Cons(x2,x1)) -> c_4(rev#1#(x1)) ** Step 6.a:3: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: rev#1#(Cons(x2,x1)) -> c_4(rev#1#(x1)) - Signature: {append#2/2,main/1,rev#1/1,append#2#/2,main#/1,rev#1#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#2#,main#,rev#1#} and constructors {Cons,Nil} + 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) rev#1# :: ["A"(1)] -(15)-> "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) "c_4_A" :: ["A"(0)] -(0)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: rev#1#(Cons(x2,x1)) -> c_4(rev#1#(x1)) 2. Weak: ** Step 6.b:1: Ara WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: append#2#(Cons(x6,x4),Cons(x2,Nil())) -> c_1(append#2#(x4,Cons(x2,Nil()))) - Weak DPs: rev#1#(Cons(x2,x1)) -> append#2#(rev#1(x1),Cons(x2,Nil())) rev#1#(Cons(x2,x1)) -> rev#1#(x1) - Weak TRS: append#2(Cons(x6,x4),Cons(x2,Nil())) -> Cons(x6,append#2(x4,Cons(x2,Nil()))) append#2(Nil(),Cons(x2,Nil())) -> Cons(x2,Nil()) rev#1(Cons(x2,x1)) -> append#2(rev#1(x1),Cons(x2,Nil())) rev#1(Nil()) -> Nil() - Signature: {append#2/2,main/1,rev#1/1,append#2#/2,main#/1,rev#1#/1} / {Cons/2,Nil/0,c_1/1,c_2/0,c_3/1,c_4/2,c_5/0} - Obligation: innermost runtime complexity wrt. defined symbols {append#2#,main#,rev#1#} and constructors {Cons,Nil} + 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) Cons :: ["A"(0) x "A"(0)] -(0)-> "A"(0) Cons :: ["A"(0) x "A"(9)] -(9)-> "A"(9) Cons :: ["A"(0) x "A"(15)] -(15)-> "A"(15) Cons :: ["A"(0) x "A"(2)] -(2)-> "A"(2) Cons :: ["A"(0) x "A"(3)] -(3)-> "A"(3) Nil :: [] -(0)-> "A"(0) Nil :: [] -(0)-> "A"(9) Nil :: [] -(0)-> "A"(1) Nil :: [] -(0)-> "A"(15) Nil :: [] -(0)-> "A"(7) append#2 :: ["A"(1) x "A"(9)] -(3)-> "A"(1) rev#1 :: ["A"(15)] -(0)-> "A"(1) append#2# :: ["A"(1) x "A"(0)] -(6)-> "A"(8) rev#1# :: ["A"(15)] -(7)-> "A"(3) c_1 :: ["A"(0)] -(0)-> "A"(12) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "Cons_A" :: ["A"(0) x "A"(1)] -(1)-> "A"(1) "Nil_A" :: [] -(0)-> "A"(1) "c_1_A" :: ["A"(0)] -(0)-> "A"(1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: append#2#(Cons(x6,x4),Cons(x2,Nil())) -> c_1(append#2#(x4,Cons(x2,Nil()))) 2. Weak: WORST_CASE(?,O(n^2))