* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: f(x,g(y,z)) -> g(f(x,y),z) f(x,nil()) -> g(nil(),x) norm(g(x,y)) -> s(norm(x)) norm(nil()) -> 0() rem(g(x,y),0()) -> g(x,y) rem(g(x,y),s(z)) -> rem(x,z) rem(nil(),y) -> nil() - Signature: {f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {f,norm,rem} and constructors {0,g,nil,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: f(x,g(y,z)) -> g(f(x,y),z) f(x,nil()) -> g(nil(),x) norm(g(x,y)) -> s(norm(x)) norm(nil()) -> 0() rem(g(x,y),0()) -> g(x,y) rem(g(x,y),s(z)) -> rem(x,z) rem(nil(),y) -> nil() - Signature: {f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {f,norm,rem} and constructors {0,g,nil,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: f(x,y){y -> g(y,z)} = f(x,g(y,z)) ->^+ g(f(x,y),z) = C[f(x,y) = f(x,y){}] ** Step 1.b:1: Bounds WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(x,g(y,z)) -> g(f(x,y),z) f(x,nil()) -> g(nil(),x) norm(g(x,y)) -> s(norm(x)) norm(nil()) -> 0() rem(g(x,y),0()) -> g(x,y) rem(g(x,y),s(z)) -> rem(x,z) rem(nil(),y) -> nil() - Signature: {f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {f,norm,rem} and constructors {0,g,nil,s} + Applied Processor: Bounds {initialAutomaton = minimal, enrichment = match} + Details: The problem is match-bounded by 1. The enriched problem is compatible with follwoing automaton. 0_0() -> 2 0_1() -> 1 0_1() -> 4 f_0(2,2) -> 1 f_1(2,2) -> 3 g_0(2,2) -> 2 g_1(2,2) -> 1 g_1(3,2) -> 1 g_1(3,2) -> 3 nil_0() -> 2 nil_1() -> 1 nil_1() -> 3 norm_0(2) -> 1 norm_1(2) -> 4 rem_0(2,2) -> 1 rem_1(2,2) -> 1 s_0(2) -> 2 s_1(4) -> 1 s_1(4) -> 4 ** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: f(x,g(y,z)) -> g(f(x,y),z) f(x,nil()) -> g(nil(),x) norm(g(x,y)) -> s(norm(x)) norm(nil()) -> 0() rem(g(x,y),0()) -> g(x,y) rem(g(x,y),s(z)) -> rem(x,z) rem(nil(),y) -> nil() - Signature: {f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {f,norm,rem} and constructors {0,g,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))