MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: from(X) -> cons(X,from(s(X))) sel(0(),cons(X,Y)) -> X sel(s(X),cons(Y,Z)) -> sel(X,Z) - Signature: {from/1,sel/2} / {0/0,cons/2,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {from,sel} and constructors {0,cons,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs from#(X) -> c_1(from#(s(X))) sel#(0(),cons(X,Y)) -> c_2() sel#(s(X),cons(Y,Z)) -> c_3(sel#(X,Z)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: from#(X) -> c_1(from#(s(X))) sel#(0(),cons(X,Y)) -> c_2() sel#(s(X),cons(Y,Z)) -> c_3(sel#(X,Z)) - Weak TRS: from(X) -> cons(X,from(s(X))) sel(0(),cons(X,Y)) -> X sel(s(X),cons(Y,Z)) -> sel(X,Z) - Signature: {from/1,sel/2,from#/1,sel#/2} / {0/0,cons/2,s/1,c_1/1,c_2/0,c_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {from#,sel#} and constructors {0,cons,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: from#(X) -> c_1(from#(s(X))) sel#(0(),cons(X,Y)) -> c_2() sel#(s(X),cons(Y,Z)) -> c_3(sel#(X,Z)) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: from#(X) -> c_1(from#(s(X))) sel#(0(),cons(X,Y)) -> c_2() sel#(s(X),cons(Y,Z)) -> c_3(sel#(X,Z)) - Signature: {from/1,sel/2,from#/1,sel#/2} / {0/0,cons/2,s/1,c_1/1,c_2/0,c_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {from#,sel#} and constructors {0,cons,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2} by application of Pre({2}) = {3}. Here rules are labelled as follows: 1: from#(X) -> c_1(from#(s(X))) 2: sel#(0(),cons(X,Y)) -> c_2() 3: sel#(s(X),cons(Y,Z)) -> c_3(sel#(X,Z)) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: from#(X) -> c_1(from#(s(X))) sel#(s(X),cons(Y,Z)) -> c_3(sel#(X,Z)) - Weak DPs: sel#(0(),cons(X,Y)) -> c_2() - Signature: {from/1,sel/2,from#/1,sel#/2} / {0/0,cons/2,s/1,c_1/1,c_2/0,c_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {from#,sel#} and constructors {0,cons,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:from#(X) -> c_1(from#(s(X))) -->_1 from#(X) -> c_1(from#(s(X))):1 2:S:sel#(s(X),cons(Y,Z)) -> c_3(sel#(X,Z)) -->_1 sel#(0(),cons(X,Y)) -> c_2():3 -->_1 sel#(s(X),cons(Y,Z)) -> c_3(sel#(X,Z)):2 3:W:sel#(0(),cons(X,Y)) -> c_2() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: sel#(0(),cons(X,Y)) -> c_2() * Step 5: Failure MAYBE + Considered Problem: - Strict DPs: from#(X) -> c_1(from#(s(X))) sel#(s(X),cons(Y,Z)) -> c_3(sel#(X,Z)) - Signature: {from/1,sel/2,from#/1,sel#/2} / {0/0,cons/2,s/1,c_1/1,c_2/0,c_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {from#,sel#} and constructors {0,cons,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE