WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + Considered Problem: - Strict TRS: a__b() -> a() a__b() -> b() a__f(X1,X2,X3) -> f(X1,X2,X3) a__f(a(),X,X) -> a__f(X,a__b(),b()) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Signature: {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: a__b() -> a() a__b() -> b() a__f(X1,X2,X3) -> f(X1,X2,X3) a__f(a(),X,X) -> a__f(X,a__b(),b()) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Signature: {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: mark(y){y -> f(x,y,z)} = mark(f(x,y,z)) ->^+ a__f(x,mark(y),z) = C[mark(y) = mark(y){}] ** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__b() -> a() a__b() -> b() a__f(X1,X2,X3) -> f(X1,X2,X3) a__f(a(),X,X) -> a__f(X,a__b(),b()) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Signature: {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f} + 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(a__f) = {2} Following symbols are considered usable: {a__b,a__f,mark} TcT has computed the following interpretation: p(a) = 0 p(a__b) = 0 p(a__f) = 2 + x2 p(b) = 0 p(f) = 2 + x2 p(mark) = 1 + x1 Following rules are strictly oriented: mark(a()) = 1 > 0 = a() mark(b()) = 1 > 0 = a__b() Following rules are (at-least) weakly oriented: a__b() = 0 >= 0 = a() a__b() = 0 >= 0 = b() a__f(X1,X2,X3) = 2 + X2 >= 2 + X2 = f(X1,X2,X3) a__f(a(),X,X) = 2 + X >= 2 = a__f(X,a__b(),b()) mark(f(X1,X2,X3)) = 3 + X2 >= 3 + X2 = a__f(X1,mark(X2),X3) ** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__b() -> a() a__b() -> b() a__f(X1,X2,X3) -> f(X1,X2,X3) a__f(a(),X,X) -> a__f(X,a__b(),b()) mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Weak TRS: mark(a()) -> a() mark(b()) -> a__b() - Signature: {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f} + 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(a__f) = {2} Following symbols are considered usable: {a__b,a__f,mark} TcT has computed the following interpretation: p(a) = 0 p(a__b) = 0 p(a__f) = 4 + x2 p(b) = 0 p(f) = 2 + x2 p(mark) = 8 + 8*x1 Following rules are strictly oriented: a__f(X1,X2,X3) = 4 + X2 > 2 + X2 = f(X1,X2,X3) mark(f(X1,X2,X3)) = 24 + 8*X2 > 12 + 8*X2 = a__f(X1,mark(X2),X3) Following rules are (at-least) weakly oriented: a__b() = 0 >= 0 = a() a__b() = 0 >= 0 = b() a__f(a(),X,X) = 4 + X >= 4 = a__f(X,a__b(),b()) mark(a()) = 8 >= 0 = a() mark(b()) = 8 >= 0 = a__b() ** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__b() -> a() a__b() -> b() a__f(a(),X,X) -> a__f(X,a__b(),b()) - Weak TRS: a__f(X1,X2,X3) -> f(X1,X2,X3) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Signature: {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f} + 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(a__f) = {2} Following symbols are considered usable: {a__b,a__f,mark} TcT has computed the following interpretation: p(a) = 2 p(a__b) = 4 p(a__f) = 10 + 4*x1 + x2 + 4*x3 p(b) = 1 p(f) = 2 + x1 + x2 + x3 p(mark) = 9 + 8*x1 Following rules are strictly oriented: a__b() = 4 > 2 = a() a__b() = 4 > 1 = b() Following rules are (at-least) weakly oriented: a__f(X1,X2,X3) = 10 + 4*X1 + X2 + 4*X3 >= 2 + X1 + X2 + X3 = f(X1,X2,X3) a__f(a(),X,X) = 18 + 5*X >= 18 + 4*X = a__f(X,a__b(),b()) mark(a()) = 25 >= 2 = a() mark(b()) = 17 >= 4 = a__b() mark(f(X1,X2,X3)) = 25 + 8*X1 + 8*X2 + 8*X3 >= 19 + 4*X1 + 8*X2 + 4*X3 = a__f(X1,mark(X2),X3) ** Step 1.b:4: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__f(a(),X,X) -> a__f(X,a__b(),b()) - Weak TRS: a__b() -> a() a__b() -> b() a__f(X1,X2,X3) -> f(X1,X2,X3) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Signature: {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f} + 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(a__f) = {2} Following symbols are considered usable: {a__b,a__f,mark} TcT has computed the following interpretation: p(a) = 2 p(a__b) = 9 p(a__f) = 8*x1 + x2 + 8*x3 p(b) = 0 p(f) = x1 + x2 + x3 p(mark) = 10 + 9*x1 Following rules are strictly oriented: a__f(a(),X,X) = 16 + 9*X > 9 + 8*X = a__f(X,a__b(),b()) Following rules are (at-least) weakly oriented: a__b() = 9 >= 2 = a() a__b() = 9 >= 0 = b() a__f(X1,X2,X3) = 8*X1 + X2 + 8*X3 >= X1 + X2 + X3 = f(X1,X2,X3) mark(a()) = 28 >= 2 = a() mark(b()) = 10 >= 9 = a__b() mark(f(X1,X2,X3)) = 10 + 9*X1 + 9*X2 + 9*X3 >= 10 + 8*X1 + 9*X2 + 8*X3 = a__f(X1,mark(X2),X3) ** Step 1.b:5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__b() -> a() a__b() -> b() a__f(X1,X2,X3) -> f(X1,X2,X3) a__f(a(),X,X) -> a__f(X,a__b(),b()) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Signature: {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))