* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) +(s(x),y) -> s(+(x,y)) not(false()) -> true() not(true()) -> false() odd(0()) -> false() odd(s(x)) -> not(odd(x)) - Signature: {+/2,not/1,odd/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {+,not,odd} and constructors {0,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) +(s(x),y) -> s(+(x,y)) not(false()) -> true() not(true()) -> false() odd(0()) -> false() odd(s(x)) -> not(odd(x)) - Signature: {+/2,not/1,odd/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {+,not,odd} and constructors {0,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: +(x,y){y -> s(y)} = +(x,s(y)) ->^+ s(+(x,y)) = C[+(x,y) = +(x,y){}] ** Step 1.b:1: Bounds WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) +(s(x),y) -> s(+(x,y)) not(false()) -> true() not(true()) -> false() odd(0()) -> false() odd(s(x)) -> not(odd(x)) - Signature: {+/2,not/1,odd/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {+,not,odd} and constructors {0,false,s,true} + Applied Processor: Bounds {initialAutomaton = minimal, enrichment = match} + Details: The problem is match-bounded by 2. The enriched problem is compatible with follwoing automaton. +_0(2,2) -> 1 +_1(2,2) -> 3 0_0() -> 1 0_0() -> 2 0_0() -> 3 false_0() -> 1 false_0() -> 2 false_0() -> 3 false_1() -> 1 false_1() -> 4 false_2() -> 1 false_2() -> 4 not_0(2) -> 1 not_1(4) -> 1 not_1(4) -> 4 odd_0(2) -> 1 odd_1(2) -> 4 s_0(2) -> 1 s_0(2) -> 2 s_0(2) -> 3 s_1(3) -> 1 s_1(3) -> 3 true_0() -> 1 true_0() -> 2 true_0() -> 3 true_1() -> 1 true_2() -> 1 true_2() -> 4 2 -> 1 2 -> 3 ** Step 1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) +(s(x),y) -> s(+(x,y)) not(false()) -> true() not(true()) -> false() odd(0()) -> false() odd(s(x)) -> not(odd(x)) - Signature: {+/2,not/1,odd/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {+,not,odd} and constructors {0,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))