* Step 1: ToInnermost WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: half(0()) -> 0() half(s(s(x))) -> s(half(x)) log(s(0())) -> 0() log(s(s(x))) -> s(log(s(half(x)))) - Signature: {half/1,log/1} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {half,log} and constructors {0,s} + Applied Processor: ToInnermost + Details: switch to innermost, as the system is overlay and right linear and does not contain weak rules * Step 2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: half(0()) -> 0() half(s(s(x))) -> s(half(x)) log(s(0())) -> 0() log(s(s(x))) -> s(log(s(half(x)))) - Signature: {half/1,log/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {half,log} and constructors {0,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(log) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(half) = [8] p(log) = [1] x1 + [3] p(s) = [1] x1 + [0] Following rules are strictly oriented: half(0()) = [8] > [1] = 0() log(s(0())) = [4] > [1] = 0() Following rules are (at-least) weakly oriented: half(s(s(x))) = [8] >= [8] = s(half(x)) log(s(s(x))) = [1] x + [3] >= [11] = s(log(s(half(x)))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: MI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: half(s(s(x))) -> s(half(x)) log(s(s(x))) -> s(log(s(half(x)))) - Weak TRS: half(0()) -> 0() log(s(0())) -> 0() - Signature: {half/1,log/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {half,log} and constructors {0,s} + Applied Processor: MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)): The following argument positions are considered usable: uargs(log) = {1}, uargs(s) = {1} Following symbols are considered usable: {half,log} TcT has computed the following interpretation: p(0) = [0] p(half) = [1] x_1 + [0] p(log) = [1] x_1 + [6] p(s) = [1] x_1 + [8] Following rules are strictly oriented: half(s(s(x))) = [1] x + [16] > [1] x + [8] = s(half(x)) Following rules are (at-least) weakly oriented: half(0()) = [0] >= [0] = 0() log(s(0())) = [14] >= [0] = 0() log(s(s(x))) = [1] x + [22] >= [1] x + [22] = s(log(s(half(x)))) * Step 4: MI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: log(s(s(x))) -> s(log(s(half(x)))) - Weak TRS: half(0()) -> 0() half(s(s(x))) -> s(half(x)) log(s(0())) -> 0() - Signature: {half/1,log/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {half,log} and constructors {0,s} + Applied Processor: MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)): The following argument positions are considered usable: uargs(log) = {1}, uargs(s) = {1} Following symbols are considered usable: {half,log} TcT has computed the following interpretation: p(0) = [2] p(half) = [1] x_1 + [0] p(log) = [4] x_1 + [13] p(s) = [1] x_1 + [2] Following rules are strictly oriented: log(s(s(x))) = [4] x + [29] > [4] x + [23] = s(log(s(half(x)))) Following rules are (at-least) weakly oriented: half(0()) = [2] >= [2] = 0() half(s(s(x))) = [1] x + [4] >= [1] x + [2] = s(half(x)) log(s(0())) = [29] >= [2] = 0() * Step 5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: half(0()) -> 0() half(s(s(x))) -> s(half(x)) log(s(0())) -> 0() log(s(s(x))) -> s(log(s(half(x)))) - Signature: {half/1,log/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {half,log} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))