MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: f(0(),0(),0(),0(),0()) -> 0() f(0(),0(),0(),0(),s(x5)) -> f(x5,x5,x5,x5,x5) f(0(),0(),0(),s(x4),x5) -> f(x4,x4,x4,x4,x5) f(0(),0(),s(x3),x4,x5) -> f(x3,x3,x3,x4,x5) f(0(),s(x2),x3,x4,x5) -> f(x2,x2,x3,x4,x5) f(s(x1),x2,x3,x4,x5) -> f(x1,x2,x3,x4,x5) - Signature: {f/5} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {f} and constructors {0,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs f#(0(),0(),0(),0(),0()) -> c_1() f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: f#(0(),0(),0(),0(),0()) -> c_1() f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) - Weak TRS: f(0(),0(),0(),0(),0()) -> 0() f(0(),0(),0(),0(),s(x5)) -> f(x5,x5,x5,x5,x5) f(0(),0(),0(),s(x4),x5) -> f(x4,x4,x4,x4,x5) f(0(),0(),s(x3),x4,x5) -> f(x3,x3,x3,x4,x5) f(0(),s(x2),x3,x4,x5) -> f(x2,x2,x3,x4,x5) f(s(x1),x2,x3,x4,x5) -> f(x1,x2,x3,x4,x5) - Signature: {f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: f#(0(),0(),0(),0(),0()) -> c_1() f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: f#(0(),0(),0(),0(),0()) -> c_1() f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) - Signature: {f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1} by application of Pre({1}) = {2,3,4,5,6}. Here rules are labelled as follows: 1: f#(0(),0(),0(),0(),0()) -> c_1() 2: f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) 3: f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) 4: f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) 5: f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) 6: f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) - Weak DPs: f#(0(),0(),0(),0(),0()) -> c_1() - Signature: {f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) -->_1 f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)):5 -->_1 f#(0(),0(),0(),0(),0()) -> c_1():6 2:S:f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) -->_1 f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)):5 -->_1 f#(0(),0(),0(),0(),0()) -> c_1():6 -->_1 f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)):1 3:S:f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) -->_1 f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)):5 -->_1 f#(0(),0(),0(),0(),0()) -> c_1():6 -->_1 f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)):2 -->_1 f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)):1 4:S:f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) -->_1 f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)):5 -->_1 f#(0(),0(),0(),0(),0()) -> c_1():6 -->_1 f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)):3 -->_1 f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)):2 -->_1 f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)):1 5:S:f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) -->_1 f#(0(),0(),0(),0(),0()) -> c_1():6 -->_1 f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)):5 -->_1 f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)):4 -->_1 f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)):3 -->_1 f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)):2 -->_1 f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)):1 6:W:f#(0(),0(),0(),0(),0()) -> c_1() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: f#(0(),0(),0(),0(),0()) -> c_1() * Step 5: Failure MAYBE + Considered Problem: - Strict DPs: f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) - Signature: {f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE