* Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: activate(X) -> X and(tt(),X) -> activate(X) plus(N,0()) -> N plus(N,s(M)) -> s(plus(N,M)) x(N,0()) -> 0() x(N,s(M)) -> plus(x(N,M),N) - Signature: {activate/1,and/2,plus/2,x/2} / {0/0,s/1,tt/0} - Obligation: runtime complexity wrt. defined symbols {activate,and,plus,x} 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(plus) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [3] p(activate) = [2] x1 + [4] p(and) = [7] x1 + [4] x2 + [0] p(plus) = [1] x1 + [1] p(s) = [1] x1 + [7] p(tt) = [1] p(x) = [1] x1 + [3] x2 + [0] Following rules are strictly oriented: activate(X) = [2] X + [4] > [1] X + [0] = X and(tt(),X) = [4] X + [7] > [2] X + [4] = activate(X) plus(N,0()) = [1] N + [1] > [1] N + [0] = N x(N,0()) = [1] N + [9] > [3] = 0() x(N,s(M)) = [3] M + [1] N + [21] > [3] M + [1] N + [1] = plus(x(N,M),N) Following rules are (at-least) weakly oriented: plus(N,s(M)) = [1] N + [1] >= [1] N + [8] = s(plus(N,M)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: plus(N,s(M)) -> s(plus(N,M)) - Weak TRS: activate(X) -> X and(tt(),X) -> activate(X) plus(N,0()) -> N x(N,0()) -> 0() x(N,s(M)) -> plus(x(N,M),N) - Signature: {activate/1,and/2,plus/2,x/2} / {0/0,s/1,tt/0} - Obligation: runtime complexity wrt. defined symbols {activate,and,plus,x} and constructors {0,s,tt} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(plus) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = 0 p(activate) = 4*x1 p(and) = 4 + x1^2 + 7*x2 + 5*x2^2 p(plus) = x1 + 2*x2 p(s) = 1 + x1 p(tt) = 0 p(x) = 2*x1*x2 Following rules are strictly oriented: plus(N,s(M)) = 2 + 2*M + N > 1 + 2*M + N = s(plus(N,M)) Following rules are (at-least) weakly oriented: activate(X) = 4*X >= X = X and(tt(),X) = 4 + 7*X + 5*X^2 >= 4*X = activate(X) plus(N,0()) = N >= N = N x(N,0()) = 0 >= 0 = 0() x(N,s(M)) = 2*M*N + 2*N >= 2*M*N + 2*N = plus(x(N,M),N) * Step 3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X and(tt(),X) -> activate(X) plus(N,0()) -> N plus(N,s(M)) -> s(plus(N,M)) x(N,0()) -> 0() x(N,s(M)) -> plus(x(N,M),N) - Signature: {activate/1,and/2,plus/2,x/2} / {0/0,s/1,tt/0} - Obligation: runtime complexity wrt. defined symbols {activate,and,plus,x} and constructors {0,s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))