* Step 1: ToInnermost WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: +(0(),y) -> y +(s(x),0()) -> s(x) +(s(x),s(y)) -> s(+(s(x),+(y,0()))) - Signature: {+/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {+} 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: +(0(),y) -> y +(s(x),0()) -> s(x) +(s(x),s(y)) -> s(+(s(x),+(y,0()))) - Signature: {+/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+} 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(+) = {2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(+) = [1] x1 + [1] x2 + [0] p(0) = [1] p(s) = [1] x1 + [0] Following rules are strictly oriented: +(0(),y) = [1] y + [1] > [1] y + [0] = y +(s(x),0()) = [1] x + [1] > [1] x + [0] = s(x) Following rules are (at-least) weakly oriented: +(s(x),s(y)) = [1] x + [1] y + [0] >= [1] x + [1] y + [1] = s(+(s(x),+(y,0()))) 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: +(s(x),s(y)) -> s(+(s(x),+(y,0()))) - Weak TRS: +(0(),y) -> y +(s(x),0()) -> s(x) - Signature: {+/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+} 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(+) = {2}, uargs(s) = {1} Following symbols are considered usable: {+} TcT has computed the following interpretation: p(+) = [1] x_1 + [4] x_2 + [0] p(0) = [0] p(s) = [1] x_1 + [1] Following rules are strictly oriented: +(s(x),s(y)) = [1] x + [4] y + [5] > [1] x + [4] y + [2] = s(+(s(x),+(y,0()))) Following rules are (at-least) weakly oriented: +(0(),y) = [4] y + [0] >= [1] y + [0] = y +(s(x),0()) = [1] x + [1] >= [1] x + [1] = s(x) * Step 4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(0(),y) -> y +(s(x),0()) -> s(x) +(s(x),s(y)) -> s(+(s(x),+(y,0()))) - Signature: {+/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))