WORST_CASE(?,O(n^1)) * Step 1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt} + 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(U12) = {2,3}, uargs(plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(U11) = [1] x2 + [1] x3 + [10] p(U12) = [1] x2 + [1] x3 + [0] p(activate) = [1] x1 + [0] p(plus) = [1] x1 + [1] x2 + [13] p(s) = [1] x1 + [8] p(tt) = [0] Following rules are strictly oriented: U11(tt(),M,N) = [1] M + [1] N + [10] > [1] M + [1] N + [0] = U12(tt(),activate(M),activate(N)) plus(N,0()) = [1] N + [13] > [1] N + [0] = N plus(N,s(M)) = [1] M + [1] N + [21] > [1] M + [1] N + [10] = U11(tt(),M,N) Following rules are (at-least) weakly oriented: U12(tt(),M,N) = [1] M + [1] N + [0] >= [1] M + [1] N + [21] = s(plus(activate(N),activate(M))) activate(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 2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt} + 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(U12) = {2,3}, uargs(plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(U11) = [9] x1 + [1] x2 + [1] x3 + [0] p(U12) = [1] x2 + [1] x3 + [1] p(activate) = [1] x1 + [4] p(plus) = [1] x1 + [1] x2 + [12] p(s) = [1] x1 + [4] p(tt) = [1] Following rules are strictly oriented: activate(X) = [1] X + [4] > [1] X + [0] = X Following rules are (at-least) weakly oriented: U11(tt(),M,N) = [1] M + [1] N + [9] >= [1] M + [1] N + [9] = U12(tt(),activate(M),activate(N)) U12(tt(),M,N) = [1] M + [1] N + [1] >= [1] M + [1] N + [24] = s(plus(activate(N),activate(M))) plus(N,0()) = [1] N + [14] >= [1] N + [0] = N plus(N,s(M)) = [1] M + [1] N + [16] >= [1] M + [1] N + [9] = U11(tt(),M,N) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: U12(tt(),M,N) -> s(plus(activate(N),activate(M))) - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(U12) = {2,3}, uargs(plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {U11,U12,activate,plus} TcT has computed the following interpretation: p(0) = [3] p(U11) = [3] x1 + [8] x2 + [2] x3 + [5] p(U12) = [2] x1 + [8] x2 + [2] x3 + [0] p(activate) = [1] x1 + [0] p(plus) = [2] x1 + [8] x2 + [0] p(s) = [1] x1 + [1] p(tt) = [1] Following rules are strictly oriented: U12(tt(),M,N) = [8] M + [2] N + [2] > [8] M + [2] N + [1] = s(plus(activate(N),activate(M))) Following rules are (at-least) weakly oriented: U11(tt(),M,N) = [8] M + [2] N + [8] >= [8] M + [2] N + [2] = U12(tt(),activate(M),activate(N)) activate(X) = [1] X + [0] >= [1] X + [0] = X plus(N,0()) = [2] N + [24] >= [1] N + [0] = N plus(N,s(M)) = [8] M + [2] N + [8] >= [8] M + [2] N + [8] = U11(tt(),M,N) * Step 4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))