* Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: fac(0()) -> s(0()) fac(s(x)) -> times(s(x),fac(p(s(x)))) p(s(x)) -> x - Signature: {fac/1,p/1} / {0/0,s/1,times/2} - Obligation: runtime complexity wrt. defined symbols {fac,p} and constructors {0,s,times} + 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(fac) = {1}, uargs(times) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(fac) = [1] x1 + [0] p(p) = [1] x1 + [3] p(s) = [1] x1 + [0] p(times) = [1] x2 + [0] Following rules are strictly oriented: p(s(x)) = [1] x + [3] > [1] x + [0] = x Following rules are (at-least) weakly oriented: fac(0()) = [0] >= [0] = s(0()) fac(s(x)) = [1] x + [0] >= [1] x + [3] = times(s(x),fac(p(s(x)))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: fac(0()) -> s(0()) fac(s(x)) -> times(s(x),fac(p(s(x)))) - Weak TRS: p(s(x)) -> x - Signature: {fac/1,p/1} / {0/0,s/1,times/2} - Obligation: runtime complexity wrt. defined symbols {fac,p} and constructors {0,s,times} + 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(fac) = {1}, uargs(times) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(fac) = [1] x1 + [2] p(p) = [1] x1 + [0] p(s) = [1] x1 + [0] p(times) = [1] x2 + [13] Following rules are strictly oriented: fac(0()) = [2] > [0] = s(0()) Following rules are (at-least) weakly oriented: fac(s(x)) = [1] x + [2] >= [1] x + [15] = times(s(x),fac(p(s(x)))) p(s(x)) = [1] x + [0] >= [1] x + [0] = x Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: MI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: fac(s(x)) -> times(s(x),fac(p(s(x)))) - Weak TRS: fac(0()) -> s(0()) p(s(x)) -> x - Signature: {fac/1,p/1} / {0/0,s/1,times/2} - Obligation: runtime complexity wrt. defined symbols {fac,p} and constructors {0,s,times} + Applied Processor: MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 2))), miDimension = 3, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 2))): The following argument positions are considered usable: uargs(fac) = {1}, uargs(times) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] [4] [0] p(fac) = [1 0 1] [4] [2 0 0] x_1 + [0] [0 0 0] [4] p(p) = [1 0 0] [2] [2 0 4] x_1 + [3] [0 1 0] [0] p(s) = [1 1 4] [0] [0 0 1] x_1 + [0] [0 0 1] [3] p(times) = [1 0 0] [0] [0 0 0] x_2 + [0] [0 0 0] [4] Following rules are strictly oriented: fac(s(x)) = [1 1 5] [7] [2 2 8] x + [0] [0 0 0] [4] > [1 1 5] [6] [0 0 0] x + [0] [0 0 0] [4] = times(s(x),fac(p(s(x)))) Following rules are (at-least) weakly oriented: fac(0()) = [4] [0] [4] >= [4] [0] [3] = s(0()) p(s(x)) = [1 1 4] [2] [2 2 12] x + [15] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] x + [0] [0 0 1] [0] = x * Step 4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: fac(0()) -> s(0()) fac(s(x)) -> times(s(x),fac(p(s(x)))) p(s(x)) -> x - Signature: {fac/1,p/1} / {0/0,s/1,times/2} - Obligation: runtime complexity wrt. defined symbols {fac,p} and constructors {0,s,times} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))