* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: a(a(f(x,y))) -> f(a(b(a(b(a(x))))),a(b(a(b(a(y)))))) f(a(x),a(y)) -> a(f(x,y)) f(b(x),b(y)) -> b(f(x,y)) - Signature: {a/1,f/2} / {b/1} - Obligation: innermost runtime complexity wrt. defined symbols {a,f} and constructors {b} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a(a(f(x,y))) -> f(a(b(a(b(a(x))))),a(b(a(b(a(y)))))) f(a(x),a(y)) -> a(f(x,y)) f(b(x),b(y)) -> b(f(x,y)) - Signature: {a/1,f/2} / {b/1} - Obligation: innermost runtime complexity wrt. defined symbols {a,f} and constructors {b} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: f(x,y){x -> b(x),y -> b(y)} = f(b(x),b(y)) ->^+ b(f(x,y)) = C[f(x,y) = f(x,y){}] ** Step 1.b:1: Bounds WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a(a(f(x,y))) -> f(a(b(a(b(a(x))))),a(b(a(b(a(y)))))) f(a(x),a(y)) -> a(f(x,y)) f(b(x),b(y)) -> b(f(x,y)) - Signature: {a/1,f/2} / {b/1} - Obligation: innermost runtime complexity wrt. defined symbols {a,f} and constructors {b} + Applied Processor: Bounds {initialAutomaton = minimal, enrichment = match} + Details: The problem is match-bounded by 1. The enriched problem is compatible with follwoing automaton. a_0(2) -> 1 b_0(2) -> 2 b_1(3) -> 1 b_1(3) -> 3 f_0(2,2) -> 1 f_1(2,2) -> 3 ** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a(a(f(x,y))) -> f(a(b(a(b(a(x))))),a(b(a(b(a(y)))))) f(a(x),a(y)) -> a(f(x,y)) f(b(x),b(y)) -> b(f(x,y)) - Signature: {a/1,f/2} / {b/1} - Obligation: innermost runtime complexity wrt. defined symbols {a,f} and constructors {b} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))