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