WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^2)) + Considered Problem: - Strict TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) double(0()) -> 0() double(s(x)) -> s(s(double(x))) sqr(0()) -> 0() sqr(s(x)) -> +(sqr(x),s(double(x))) sqr(s(x)) -> s(+(sqr(x),double(x))) - Signature: {+/2,double/1,sqr/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s} + 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)) double(0()) -> 0() double(s(x)) -> s(s(double(x))) sqr(0()) -> 0() sqr(s(x)) -> +(sqr(x),s(double(x))) sqr(s(x)) -> s(+(sqr(x),double(x))) - Signature: {+/2,double/1,sqr/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s} + 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: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) double(0()) -> 0() double(s(x)) -> s(s(double(x))) sqr(0()) -> 0() sqr(s(x)) -> +(sqr(x),s(double(x))) sqr(s(x)) -> s(+(sqr(x),double(x))) - Signature: {+/2,double/1,sqr/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s} + 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(+) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {+,double,sqr} TcT has computed the following interpretation: p(+) = x1 + 4*x2 p(0) = 0 p(double) = 0 p(s) = x1 p(sqr) = 1 Following rules are strictly oriented: sqr(0()) = 1 > 0 = 0() Following rules are (at-least) weakly oriented: +(x,0()) = x >= x = x +(x,s(y)) = x + 4*y >= x + 4*y = s(+(x,y)) double(0()) = 0 >= 0 = 0() double(s(x)) = 0 >= 0 = s(s(double(x))) sqr(s(x)) = 1 >= 1 = +(sqr(x),s(double(x))) sqr(s(x)) = 1 >= 1 = s(+(sqr(x),double(x))) ** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) double(0()) -> 0() double(s(x)) -> s(s(double(x))) sqr(s(x)) -> +(sqr(x),s(double(x))) sqr(s(x)) -> s(+(sqr(x),double(x))) - Weak TRS: sqr(0()) -> 0() - Signature: {+/2,double/1,sqr/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s} + 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(+) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {+,double,sqr} TcT has computed the following interpretation: p(+) = x1 + 4*x2 p(0) = 0 p(double) = 2*x1 p(s) = 2 + x1 p(sqr) = 2*x1^2 Following rules are strictly oriented: +(x,s(y)) = 8 + x + 4*y > 2 + x + 4*y = s(+(x,y)) sqr(s(x)) = 8 + 8*x + 2*x^2 > 2 + 8*x + 2*x^2 = s(+(sqr(x),double(x))) Following rules are (at-least) weakly oriented: +(x,0()) = x >= x = x double(0()) = 0 >= 0 = 0() double(s(x)) = 4 + 2*x >= 4 + 2*x = s(s(double(x))) sqr(0()) = 0 >= 0 = 0() sqr(s(x)) = 8 + 8*x + 2*x^2 >= 8 + 8*x + 2*x^2 = +(sqr(x),s(double(x))) ** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: +(x,0()) -> x double(0()) -> 0() double(s(x)) -> s(s(double(x))) sqr(s(x)) -> +(sqr(x),s(double(x))) - Weak TRS: +(x,s(y)) -> s(+(x,y)) sqr(0()) -> 0() sqr(s(x)) -> s(+(sqr(x),double(x))) - Signature: {+/2,double/1,sqr/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s} + 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(+) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {+,double,sqr} TcT has computed the following interpretation: p(+) = x1 + 4*x2 p(0) = 1 p(double) = 1 + 2*x1 p(s) = 2 + x1 p(sqr) = 2*x1 + 2*x1^2 Following rules are strictly oriented: +(x,0()) = 4 + x > x = x double(0()) = 3 > 1 = 0() Following rules are (at-least) weakly oriented: +(x,s(y)) = 8 + x + 4*y >= 2 + x + 4*y = s(+(x,y)) double(s(x)) = 5 + 2*x >= 5 + 2*x = s(s(double(x))) sqr(0()) = 4 >= 1 = 0() sqr(s(x)) = 12 + 10*x + 2*x^2 >= 12 + 10*x + 2*x^2 = +(sqr(x),s(double(x))) sqr(s(x)) = 12 + 10*x + 2*x^2 >= 6 + 10*x + 2*x^2 = s(+(sqr(x),double(x))) ** Step 1.b:4: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: double(s(x)) -> s(s(double(x))) sqr(s(x)) -> +(sqr(x),s(double(x))) - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) double(0()) -> 0() sqr(0()) -> 0() sqr(s(x)) -> s(+(sqr(x),double(x))) - Signature: {+/2,double/1,sqr/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s} + 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(+) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {+,double,sqr} TcT has computed the following interpretation: p(+) = 1 + x1 + x2 p(0) = 0 p(double) = 3*x1 p(s) = 1 + x1 p(sqr) = 2*x1^2 Following rules are strictly oriented: double(s(x)) = 3 + 3*x > 2 + 3*x = s(s(double(x))) Following rules are (at-least) weakly oriented: +(x,0()) = 1 + x >= x = x +(x,s(y)) = 2 + x + y >= 2 + x + y = s(+(x,y)) double(0()) = 0 >= 0 = 0() sqr(0()) = 0 >= 0 = 0() sqr(s(x)) = 2 + 4*x + 2*x^2 >= 2 + 3*x + 2*x^2 = +(sqr(x),s(double(x))) sqr(s(x)) = 2 + 4*x + 2*x^2 >= 2 + 3*x + 2*x^2 = s(+(sqr(x),double(x))) ** Step 1.b:5: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: sqr(s(x)) -> +(sqr(x),s(double(x))) - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) double(0()) -> 0() double(s(x)) -> s(s(double(x))) sqr(0()) -> 0() sqr(s(x)) -> s(+(sqr(x),double(x))) - Signature: {+/2,double/1,sqr/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s} + 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(+) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {+,double,sqr} TcT has computed the following interpretation: p(+) = 3 + x1 + x2 p(0) = 0 p(double) = 2*x1 p(s) = 1 + x1 p(sqr) = 3*x1 + 2*x1^2 Following rules are strictly oriented: sqr(s(x)) = 5 + 7*x + 2*x^2 > 4 + 5*x + 2*x^2 = +(sqr(x),s(double(x))) Following rules are (at-least) weakly oriented: +(x,0()) = 3 + x >= x = x +(x,s(y)) = 4 + x + y >= 4 + x + y = s(+(x,y)) double(0()) = 0 >= 0 = 0() double(s(x)) = 2 + 2*x >= 2 + 2*x = s(s(double(x))) sqr(0()) = 0 >= 0 = 0() sqr(s(x)) = 5 + 7*x + 2*x^2 >= 4 + 5*x + 2*x^2 = s(+(sqr(x),double(x))) ** Step 1.b:6: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) double(0()) -> 0() double(s(x)) -> s(s(double(x))) sqr(0()) -> 0() sqr(s(x)) -> +(sqr(x),s(double(x))) sqr(s(x)) -> s(+(sqr(x),double(x))) - Signature: {+/2,double/1,sqr/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^2))