MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: activate(X) -> X activate(n__h(X)) -> h(X) f(X) -> g(n__h(f(X))) h(X) -> n__h(X) - Signature: {activate/1,f/1,h/1} / {g/1,n__h/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,f,h} and constructors {g,n__h} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs activate#(X) -> c_1() activate#(n__h(X)) -> c_2(h#(X)) f#(X) -> c_3(f#(X)) h#(X) -> c_4() Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__h(X)) -> c_2(h#(X)) f#(X) -> c_3(f#(X)) h#(X) -> c_4() - Weak TRS: activate(X) -> X activate(n__h(X)) -> h(X) f(X) -> g(n__h(f(X))) h(X) -> n__h(X) - Signature: {activate/1,f/1,h/1,activate#/1,f#/1,h#/1} / {g/1,n__h/1,c_1/0,c_2/1,c_3/1,c_4/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,h#} and constructors {g,n__h} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: activate#(X) -> c_1() activate#(n__h(X)) -> c_2(h#(X)) f#(X) -> c_3(f#(X)) h#(X) -> c_4() * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: activate#(X) -> c_1() activate#(n__h(X)) -> c_2(h#(X)) f#(X) -> c_3(f#(X)) h#(X) -> c_4() - Signature: {activate/1,f/1,h/1,activate#/1,f#/1,h#/1} / {g/1,n__h/1,c_1/0,c_2/1,c_3/1,c_4/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,h#} and constructors {g,n__h} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,4} by application of Pre({1,4}) = {2}. Here rules are labelled as follows: 1: activate#(X) -> c_1() 2: activate#(n__h(X)) -> c_2(h#(X)) 3: f#(X) -> c_3(f#(X)) 4: h#(X) -> c_4() * Step 4: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: activate#(n__h(X)) -> c_2(h#(X)) f#(X) -> c_3(f#(X)) - Weak DPs: activate#(X) -> c_1() h#(X) -> c_4() - Signature: {activate/1,f/1,h/1,activate#/1,f#/1,h#/1} / {g/1,n__h/1,c_1/0,c_2/1,c_3/1,c_4/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,h#} and constructors {g,n__h} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1} by application of Pre({1}) = {}. Here rules are labelled as follows: 1: activate#(n__h(X)) -> c_2(h#(X)) 2: f#(X) -> c_3(f#(X)) 3: activate#(X) -> c_1() 4: h#(X) -> c_4() * Step 5: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: f#(X) -> c_3(f#(X)) - Weak DPs: activate#(X) -> c_1() activate#(n__h(X)) -> c_2(h#(X)) h#(X) -> c_4() - Signature: {activate/1,f/1,h/1,activate#/1,f#/1,h#/1} / {g/1,n__h/1,c_1/0,c_2/1,c_3/1,c_4/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,h#} and constructors {g,n__h} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:f#(X) -> c_3(f#(X)) -->_1 f#(X) -> c_3(f#(X)):1 2:W:activate#(X) -> c_1() 3:W:activate#(n__h(X)) -> c_2(h#(X)) -->_1 h#(X) -> c_4():4 4:W:h#(X) -> c_4() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: activate#(n__h(X)) -> c_2(h#(X)) 4: h#(X) -> c_4() 2: activate#(X) -> c_1() * Step 6: Failure MAYBE + Considered Problem: - Strict DPs: f#(X) -> c_3(f#(X)) - Signature: {activate/1,f/1,h/1,activate#/1,f#/1,h#/1} / {g/1,n__h/1,c_1/0,c_2/1,c_3/1,c_4/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate#,f#,h#} and constructors {g,n__h} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE