WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum WORST_CASE(Omega(n^1),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: innermost runtime complexity wrt. defined symbols {activate,and,plus,x} and constructors {0,s,tt} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + 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: innermost runtime complexity wrt. defined symbols {activate,and,plus,x} and constructors {0,s,tt} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: plus(x,y){y -> s(y)} = plus(x,s(y)) ->^+ s(plus(x,y)) = C[plus(x,y) = plus(x,y){}] ** Step 1.b:1: NaturalPI 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: innermost 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: {activate,and,plus,x} TcT has computed the following interpretation: p(0) = 0 p(activate) = 4 + 4*x1 p(and) = 4*x1 + 4*x1*x2 + 4*x2 + x2^2 p(plus) = 2*x1 p(s) = x1 p(tt) = 1 p(x) = 0 Following rules are strictly oriented: activate(X) = 4 + 4*X > X = X Following rules are (at-least) weakly oriented: and(tt(),X) = 4 + 8*X + X^2 >= 4 + 4*X = activate(X) plus(N,0()) = 2*N >= N = N plus(N,s(M)) = 2*N >= 2*N = s(plus(N,M)) x(N,0()) = 0 >= 0 = 0() x(N,s(M)) = 0 >= 0 = plus(x(N,M),N) ** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 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) - Weak TRS: activate(X) -> X - Signature: {activate/1,and/2,plus/2,x/2} / {0/0,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,and,plus,x} and constructors {0,s,tt} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(plus) = {1}, uargs(s) = {1} Following symbols are considered usable: {activate,and,plus,x} TcT has computed the following interpretation: p(0) = 3 p(activate) = 8*x1 p(and) = 2 + 8*x2 p(plus) = x1 p(s) = x1 p(tt) = 0 p(x) = 3 Following rules are strictly oriented: and(tt(),X) = 2 + 8*X > 8*X = activate(X) Following rules are (at-least) weakly oriented: activate(X) = 8*X >= X = X plus(N,0()) = N >= N = N plus(N,s(M)) = N >= N = s(plus(N,M)) x(N,0()) = 3 >= 3 = 0() x(N,s(M)) = 3 >= 3 = plus(x(N,M),N) ** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 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) - Weak TRS: activate(X) -> X and(tt(),X) -> activate(X) - Signature: {activate/1,and/2,plus/2,x/2} / {0/0,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,and,plus,x} and constructors {0,s,tt} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(plus) = {1}, uargs(s) = {1} Following symbols are considered usable: {activate,and,plus,x} TcT has computed the following interpretation: p(0) = 4 p(activate) = 7 + 8*x1 p(and) = 9 + 12*x2 p(plus) = x1 p(s) = x1 p(tt) = 0 p(x) = 1 + 4*x1 + 4*x2 Following rules are strictly oriented: x(N,0()) = 17 + 4*N > 4 = 0() Following rules are (at-least) weakly oriented: activate(X) = 7 + 8*X >= X = X and(tt(),X) = 9 + 12*X >= 7 + 8*X = activate(X) plus(N,0()) = N >= N = N plus(N,s(M)) = N >= N = s(plus(N,M)) x(N,s(M)) = 1 + 4*M + 4*N >= 1 + 4*M + 4*N = plus(x(N,M),N) ** Step 1.b:4: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: plus(N,0()) -> N plus(N,s(M)) -> s(plus(N,M)) x(N,s(M)) -> plus(x(N,M),N) - Weak TRS: activate(X) -> X and(tt(),X) -> activate(X) x(N,0()) -> 0() - Signature: {activate/1,and/2,plus/2,x/2} / {0/0,s/1,tt/0} - Obligation: innermost 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: {activate,and,plus,x} TcT has computed the following interpretation: p(0) = 0 p(activate) = 5*x1 p(and) = 2 + x1 + 2*x1*x2 + 7*x1^2 + 6*x2 + 3*x2^2 p(plus) = x1 + 2*x2 p(s) = 2 + x1 p(tt) = 1 p(x) = x1*x2 + 2*x2 + x2^2 Following rules are strictly oriented: plus(N,s(M)) = 4 + 2*M + N > 2 + 2*M + N = s(plus(N,M)) x(N,s(M)) = 8 + 6*M + M*N + M^2 + 2*N > 2*M + M*N + M^2 + 2*N = plus(x(N,M),N) Following rules are (at-least) weakly oriented: activate(X) = 5*X >= X = X and(tt(),X) = 10 + 8*X + 3*X^2 >= 5*X = activate(X) plus(N,0()) = N >= N = N x(N,0()) = 0 >= 0 = 0() ** Step 1.b:5: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: plus(N,0()) -> N - Weak TRS: activate(X) -> X and(tt(),X) -> activate(X) 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: innermost 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: {activate,and,plus,x} TcT has computed the following interpretation: p(0) = 3 p(activate) = 2*x1 p(and) = 1 + 4*x1 + 3*x1*x2 + 7*x2 p(plus) = x1 + 3*x2 p(s) = 3 + x1 p(tt) = 1 p(x) = 3 + x1 + x1*x2 + 2*x1^2 Following rules are strictly oriented: plus(N,0()) = 9 + N > N = N Following rules are (at-least) weakly oriented: activate(X) = 2*X >= X = X and(tt(),X) = 5 + 10*X >= 2*X = activate(X) plus(N,s(M)) = 9 + 3*M + N >= 3 + 3*M + N = s(plus(N,M)) x(N,0()) = 3 + 4*N + 2*N^2 >= 3 = 0() x(N,s(M)) = 3 + M*N + 4*N + 2*N^2 >= 3 + M*N + 4*N + 2*N^2 = plus(x(N,M),N) ** Step 1.b:6: 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: innermost 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(Omega(n^1),O(n^2))