MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: from(X) -> cons(X) length() -> 0() length() -> s(length1()) length1() -> length() - Signature: {from/1,length/0,length1/0} / {0/0,cons/1,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {from,length,length1} and constructors {0,cons,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs from#(X) -> c_1() length#() -> c_2() length#() -> c_3(length1#()) length1#() -> c_4(length#()) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: from#(X) -> c_1() length#() -> c_2() length#() -> c_3(length1#()) length1#() -> c_4(length#()) - Weak TRS: from(X) -> cons(X) length() -> 0() length() -> s(length1()) length1() -> length() - Signature: {from/1,length/0,length1/0,from#/1,length#/0,length1#/0} / {0/0,cons/1,s/1,c_1/0,c_2/0,c_3/1,c_4/1} - Obligation: innermost runtime complexity wrt. defined symbols {from#,length#,length1#} and constructors {0,cons,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: from#(X) -> c_1() length#() -> c_2() length#() -> c_3(length1#()) length1#() -> c_4(length#()) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: from#(X) -> c_1() length#() -> c_2() length#() -> c_3(length1#()) length1#() -> c_4(length#()) - Signature: {from/1,length/0,length1/0,from#/1,length#/0,length1#/0} / {0/0,cons/1,s/1,c_1/0,c_2/0,c_3/1,c_4/1} - Obligation: innermost runtime complexity wrt. defined symbols {from#,length#,length1#} and constructors {0,cons,s} + 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}) = {4}. Here rules are labelled as follows: 1: from#(X) -> c_1() 2: length#() -> c_2() 3: length#() -> c_3(length1#()) 4: length1#() -> c_4(length#()) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: length#() -> c_3(length1#()) length1#() -> c_4(length#()) - Weak DPs: from#(X) -> c_1() length#() -> c_2() - Signature: {from/1,length/0,length1/0,from#/1,length#/0,length1#/0} / {0/0,cons/1,s/1,c_1/0,c_2/0,c_3/1,c_4/1} - Obligation: innermost runtime complexity wrt. defined symbols {from#,length#,length1#} and constructors {0,cons,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:length#() -> c_3(length1#()) -->_1 length1#() -> c_4(length#()):2 2:S:length1#() -> c_4(length#()) -->_1 length#() -> c_2():4 -->_1 length#() -> c_3(length1#()):1 3:W:from#(X) -> c_1() 4:W:length#() -> c_2() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: from#(X) -> c_1() 4: length#() -> c_2() * Step 5: WeightGap MAYBE + Considered Problem: - Strict DPs: length#() -> c_3(length1#()) length1#() -> c_4(length#()) - Signature: {from/1,length/0,length1/0,from#/1,length#/0,length1#/0} / {0/0,cons/1,s/1,c_1/0,c_2/0,c_3/1,c_4/1} - Obligation: innermost runtime complexity wrt. defined symbols {from#,length#,length1#} and constructors {0,cons,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_4) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(cons) = [0] p(from) = [0] p(length) = [0] p(length1) = [0] p(s) = [0] p(from#) = [0] p(length#) = [10] p(length1#) = [11] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] Following rules are strictly oriented: length1#() = [11] > [10] = c_4(length#()) Following rules are (at-least) weakly oriented: length#() = [10] >= [11] = c_3(length1#()) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: Failure MAYBE + Considered Problem: - Strict DPs: length#() -> c_3(length1#()) - Weak DPs: length1#() -> c_4(length#()) - Signature: {from/1,length/0,length1/0,from#/1,length#/0,length1#/0} / {0/0,cons/1,s/1,c_1/0,c_2/0,c_3/1,c_4/1} - Obligation: innermost runtime complexity wrt. defined symbols {from#,length#,length1#} and constructors {0,cons,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE