MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: f(t(),x,y) -> f(g(x,y),x,s(y)) g(s(x),0()) -> t() g(s(x),s(y)) -> g(x,y) - Signature: {f/3,g/2} / {0/0,s/1,t/0} - Obligation: innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,s,t} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs f#(t(),x,y) -> c_1(f#(g(x,y),x,s(y)),g#(x,y)) g#(s(x),0()) -> c_2() g#(s(x),s(y)) -> c_3(g#(x,y)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: f#(t(),x,y) -> c_1(f#(g(x,y),x,s(y)),g#(x,y)) g#(s(x),0()) -> c_2() g#(s(x),s(y)) -> c_3(g#(x,y)) - Weak TRS: f(t(),x,y) -> f(g(x,y),x,s(y)) g(s(x),0()) -> t() g(s(x),s(y)) -> g(x,y) - Signature: {f/3,g/2,f#/3,g#/2} / {0/0,s/1,t/0,c_1/2,c_2/0,c_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s,t} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: g(s(x),0()) -> t() g(s(x),s(y)) -> g(x,y) f#(t(),x,y) -> c_1(f#(g(x,y),x,s(y)),g#(x,y)) g#(s(x),0()) -> c_2() g#(s(x),s(y)) -> c_3(g#(x,y)) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: f#(t(),x,y) -> c_1(f#(g(x,y),x,s(y)),g#(x,y)) g#(s(x),0()) -> c_2() g#(s(x),s(y)) -> c_3(g#(x,y)) - Weak TRS: g(s(x),0()) -> t() g(s(x),s(y)) -> g(x,y) - Signature: {f/3,g/2,f#/3,g#/2} / {0/0,s/1,t/0,c_1/2,c_2/0,c_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s,t} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2} by application of Pre({2}) = {1,3}. Here rules are labelled as follows: 1: f#(t(),x,y) -> c_1(f#(g(x,y),x,s(y)),g#(x,y)) 2: g#(s(x),0()) -> c_2() 3: g#(s(x),s(y)) -> c_3(g#(x,y)) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: f#(t(),x,y) -> c_1(f#(g(x,y),x,s(y)),g#(x,y)) g#(s(x),s(y)) -> c_3(g#(x,y)) - Weak DPs: g#(s(x),0()) -> c_2() - Weak TRS: g(s(x),0()) -> t() g(s(x),s(y)) -> g(x,y) - Signature: {f/3,g/2,f#/3,g#/2} / {0/0,s/1,t/0,c_1/2,c_2/0,c_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s,t} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:f#(t(),x,y) -> c_1(f#(g(x,y),x,s(y)),g#(x,y)) -->_2 g#(s(x),s(y)) -> c_3(g#(x,y)):2 -->_2 g#(s(x),0()) -> c_2():3 -->_1 f#(t(),x,y) -> c_1(f#(g(x,y),x,s(y)),g#(x,y)):1 2:S:g#(s(x),s(y)) -> c_3(g#(x,y)) -->_1 g#(s(x),0()) -> c_2():3 -->_1 g#(s(x),s(y)) -> c_3(g#(x,y)):2 3:W:g#(s(x),0()) -> c_2() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: g#(s(x),0()) -> c_2() * Step 5: Failure MAYBE + Considered Problem: - Strict DPs: f#(t(),x,y) -> c_1(f#(g(x,y),x,s(y)),g#(x,y)) g#(s(x),s(y)) -> c_3(g#(x,y)) - Weak TRS: g(s(x),0()) -> t() g(s(x),s(y)) -> g(x,y) - Signature: {f/3,g/2,f#/3,g#/2} / {0/0,s/1,t/0,c_1/2,c_2/0,c_3/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s,t} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE