WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__d(X)) -> d(X) activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(X) c(X) -> d(activate(X)) d(X) -> n__d(X) f(X) -> n__f(X) f(f(X)) -> c(n__f(n__g(n__f(X)))) g(X) -> n__g(X) h(X) -> c(n__d(X)) - Signature: {activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,n__f,n__g} + 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 activate(n__d(X)) -> d(X) activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(X) c(X) -> d(activate(X)) d(X) -> n__d(X) f(X) -> n__f(X) f(f(X)) -> c(n__f(n__g(n__f(X)))) g(X) -> n__g(X) h(X) -> c(n__d(X)) - Signature: {activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,n__f,n__g} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: activate(x){x -> n__f(x)} = activate(n__f(x)) ->^+ f(activate(x)) = C[activate(x) = activate(x){}] ** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__d(X)) -> d(X) activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(X) c(X) -> d(activate(X)) d(X) -> n__d(X) f(X) -> n__f(X) f(f(X)) -> c(n__f(n__g(n__f(X)))) g(X) -> n__g(X) h(X) -> c(n__d(X)) - Signature: {activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,n__f,n__g} + 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(d) = {1}, uargs(f) = {1} Following symbols are considered usable: {activate,c,d,f,g,h} TcT has computed the following interpretation: p(activate) = 2*x1 p(c) = 8*x1 p(d) = 2*x1 p(f) = x1 p(g) = 0 p(h) = 10 + 14*x1 p(n__d) = x1 p(n__f) = x1 p(n__g) = 0 Following rules are strictly oriented: h(X) = 10 + 14*X > 8*X = c(n__d(X)) Following rules are (at-least) weakly oriented: activate(X) = 2*X >= X = X activate(n__d(X)) = 2*X >= 2*X = d(X) activate(n__f(X)) = 2*X >= 2*X = f(activate(X)) activate(n__g(X)) = 0 >= 0 = g(X) c(X) = 8*X >= 4*X = d(activate(X)) d(X) = 2*X >= X = n__d(X) f(X) = X >= X = n__f(X) f(f(X)) = X >= 0 = c(n__f(n__g(n__f(X)))) g(X) = 0 >= 0 = n__g(X) ** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__d(X)) -> d(X) activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(X) c(X) -> d(activate(X)) d(X) -> n__d(X) f(X) -> n__f(X) f(f(X)) -> c(n__f(n__g(n__f(X)))) g(X) -> n__g(X) - Weak TRS: h(X) -> c(n__d(X)) - Signature: {activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,n__f,n__g} + 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(d) = {1}, uargs(f) = {1} Following symbols are considered usable: {activate,c,d,f,g,h} TcT has computed the following interpretation: p(activate) = 3*x1 p(c) = 3*x1 p(d) = x1 p(f) = 3 + x1 p(g) = 0 p(h) = 1 + 4*x1 p(n__d) = x1 p(n__f) = 2 + x1 p(n__g) = 0 Following rules are strictly oriented: activate(n__f(X)) = 6 + 3*X > 3 + 3*X = f(activate(X)) f(X) = 3 + X > 2 + X = n__f(X) Following rules are (at-least) weakly oriented: activate(X) = 3*X >= X = X activate(n__d(X)) = 3*X >= X = d(X) activate(n__g(X)) = 0 >= 0 = g(X) c(X) = 3*X >= 3*X = d(activate(X)) d(X) = X >= X = n__d(X) f(f(X)) = 6 + X >= 6 = c(n__f(n__g(n__f(X)))) g(X) = 0 >= 0 = n__g(X) h(X) = 1 + 4*X >= 3*X = c(n__d(X)) ** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__d(X)) -> d(X) activate(n__g(X)) -> g(X) c(X) -> d(activate(X)) d(X) -> n__d(X) f(f(X)) -> c(n__f(n__g(n__f(X)))) g(X) -> n__g(X) - Weak TRS: activate(n__f(X)) -> f(activate(X)) f(X) -> n__f(X) h(X) -> c(n__d(X)) - Signature: {activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,n__f,n__g} + 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(d) = {1}, uargs(f) = {1} Following symbols are considered usable: {activate,c,d,f,g,h} TcT has computed the following interpretation: p(activate) = 2*x1 p(c) = 2 + 2*x1 p(d) = 2 + x1 p(f) = 3 + x1 p(g) = 0 p(h) = 9 + 2*x1 p(n__d) = 2 + x1 p(n__f) = 2 + x1 p(n__g) = 0 Following rules are strictly oriented: activate(n__d(X)) = 4 + 2*X > 2 + X = d(X) Following rules are (at-least) weakly oriented: activate(X) = 2*X >= X = X activate(n__f(X)) = 4 + 2*X >= 3 + 2*X = f(activate(X)) activate(n__g(X)) = 0 >= 0 = g(X) c(X) = 2 + 2*X >= 2 + 2*X = d(activate(X)) d(X) = 2 + X >= 2 + X = n__d(X) f(X) = 3 + X >= 2 + X = n__f(X) f(f(X)) = 6 + X >= 6 = c(n__f(n__g(n__f(X)))) g(X) = 0 >= 0 = n__g(X) h(X) = 9 + 2*X >= 6 + 2*X = c(n__d(X)) ** Step 1.b:4: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__g(X)) -> g(X) c(X) -> d(activate(X)) d(X) -> n__d(X) f(f(X)) -> c(n__f(n__g(n__f(X)))) g(X) -> n__g(X) - Weak TRS: activate(n__d(X)) -> d(X) activate(n__f(X)) -> f(activate(X)) f(X) -> n__f(X) h(X) -> c(n__d(X)) - Signature: {activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,n__f,n__g} + 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(d) = {1}, uargs(f) = {1} Following symbols are considered usable: {activate,c,d,f,g,h} TcT has computed the following interpretation: p(activate) = 1 + 8*x1 p(c) = 2 + 12*x1 p(d) = x1 p(f) = 8 + x1 p(g) = 1 p(h) = 5 + 12*x1 p(n__d) = x1 p(n__f) = 1 + x1 p(n__g) = 0 Following rules are strictly oriented: activate(X) = 1 + 8*X > X = X c(X) = 2 + 12*X > 1 + 8*X = d(activate(X)) f(f(X)) = 16 + X > 14 = c(n__f(n__g(n__f(X)))) g(X) = 1 > 0 = n__g(X) Following rules are (at-least) weakly oriented: activate(n__d(X)) = 1 + 8*X >= X = d(X) activate(n__f(X)) = 9 + 8*X >= 9 + 8*X = f(activate(X)) activate(n__g(X)) = 1 >= 1 = g(X) d(X) = X >= X = n__d(X) f(X) = 8 + X >= 1 + X = n__f(X) h(X) = 5 + 12*X >= 2 + 12*X = c(n__d(X)) ** Step 1.b:5: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(n__g(X)) -> g(X) d(X) -> n__d(X) - Weak TRS: activate(X) -> X activate(n__d(X)) -> d(X) activate(n__f(X)) -> f(activate(X)) c(X) -> d(activate(X)) f(X) -> n__f(X) f(f(X)) -> c(n__f(n__g(n__f(X)))) g(X) -> n__g(X) h(X) -> c(n__d(X)) - Signature: {activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,n__f,n__g} + 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(d) = {1}, uargs(f) = {1} Following symbols are considered usable: {activate,c,d,f,g,h} TcT has computed the following interpretation: p(activate) = 8*x1 p(c) = 1 + 8*x1 p(d) = x1 p(f) = 13 + x1 p(g) = 4 p(h) = 8 + 11*x1 p(n__d) = x1 p(n__f) = 2 + x1 p(n__g) = 1 Following rules are strictly oriented: activate(n__g(X)) = 8 > 4 = g(X) Following rules are (at-least) weakly oriented: activate(X) = 8*X >= X = X activate(n__d(X)) = 8*X >= X = d(X) activate(n__f(X)) = 16 + 8*X >= 13 + 8*X = f(activate(X)) c(X) = 1 + 8*X >= 8*X = d(activate(X)) d(X) = X >= X = n__d(X) f(X) = 13 + X >= 2 + X = n__f(X) f(f(X)) = 26 + X >= 25 = c(n__f(n__g(n__f(X)))) g(X) = 4 >= 1 = n__g(X) h(X) = 8 + 11*X >= 1 + 8*X = c(n__d(X)) ** Step 1.b:6: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: d(X) -> n__d(X) - Weak TRS: activate(X) -> X activate(n__d(X)) -> d(X) activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(X) c(X) -> d(activate(X)) f(X) -> n__f(X) f(f(X)) -> c(n__f(n__g(n__f(X)))) g(X) -> n__g(X) h(X) -> c(n__d(X)) - Signature: {activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,n__f,n__g} + 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(d) = {1}, uargs(f) = {1} Following symbols are considered usable: {activate,c,d,f,g,h} TcT has computed the following interpretation: p(activate) = 2 + 8*x1 p(c) = 5 + 8*x1 p(d) = 3 + x1 p(f) = 15 + x1 p(g) = 0 p(h) = 13 + 10*x1 p(n__d) = 1 + x1 p(n__f) = 2 + x1 p(n__g) = 0 Following rules are strictly oriented: d(X) = 3 + X > 1 + X = n__d(X) Following rules are (at-least) weakly oriented: activate(X) = 2 + 8*X >= X = X activate(n__d(X)) = 10 + 8*X >= 3 + X = d(X) activate(n__f(X)) = 18 + 8*X >= 17 + 8*X = f(activate(X)) activate(n__g(X)) = 2 >= 0 = g(X) c(X) = 5 + 8*X >= 5 + 8*X = d(activate(X)) f(X) = 15 + X >= 2 + X = n__f(X) f(f(X)) = 30 + X >= 21 = c(n__f(n__g(n__f(X)))) g(X) = 0 >= 0 = n__g(X) h(X) = 13 + 10*X >= 13 + 8*X = c(n__d(X)) ** Step 1.b:7: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__d(X)) -> d(X) activate(n__f(X)) -> f(activate(X)) activate(n__g(X)) -> g(X) c(X) -> d(activate(X)) d(X) -> n__d(X) f(X) -> n__f(X) f(f(X)) -> c(n__f(n__g(n__f(X)))) g(X) -> n__g(X) h(X) -> c(n__d(X)) - Signature: {activate/1,c/1,d/1,f/1,g/1,h/1} / {n__d/1,n__f/1,n__g/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,c,d,f,g,h} and constructors {n__d,n__f,n__g} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))