/export/starexec/sandbox2/solver/bin/starexec_run_tct_rc /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^2)) + Considered Problem: - Strict TRS: dx(X) -> one() dx(a()) -> zero() dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two())))) dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))) ,times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA)))) dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA) dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA)) dx(neg(ALPHA)) -> neg(dx(ALPHA)) dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA)) dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA))) - Signature: {dx/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0} - Obligation: runtime complexity wrt. defined symbols {dx} and constructors {a,div,exp,ln,minus,neg,one,plus,times,two ,zero} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: dx(X) -> one() dx(a()) -> zero() dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two())))) dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))) ,times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA)))) dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA) dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA)) dx(neg(ALPHA)) -> neg(dx(ALPHA)) dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA)) dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA))) - Signature: {dx/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0} - Obligation: runtime complexity wrt. defined symbols {dx} and constructors {a,div,exp,ln,minus,neg,one,plus,times,two ,zero} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: dx(X) -> one() dx(a()) -> zero() dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two())))) dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))) ,times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA)))) dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA) dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA)) dx(neg(ALPHA)) -> neg(dx(ALPHA)) dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA)) dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA))) - Signature: {dx/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0} - Obligation: runtime complexity wrt. defined symbols {dx} and constructors {a,div,exp,ln,minus,neg,one,plus,times,two ,zero} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: dx(x){x -> div(x,y)} = dx(div(x,y)) ->^+ minus(div(dx(x),y),times(x,div(dx(y),exp(y,two())))) = C[dx(x) = dx(x){}] ** Step 1.b:1: NaturalPI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: dx(X) -> one() dx(a()) -> zero() dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two())))) dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))) ,times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA)))) dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA) dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA)) dx(neg(ALPHA)) -> neg(dx(ALPHA)) dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA)) dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA))) - Signature: {dx/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0} - Obligation: runtime complexity wrt. defined symbols {dx} and constructors {a,div,exp,ln,minus,neg,one,plus,times,two ,zero} + 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(div) = {1}, uargs(minus) = {1,2}, uargs(neg) = {1}, uargs(plus) = {1,2}, uargs(times) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = 0 p(div) = 2 + x1 + x2 p(dx) = 1 + 5*x1 + x1^2 p(exp) = 2 + x1 + x2 p(ln) = 2 + x1 p(minus) = 2 + x1 + x2 p(neg) = x1 p(one) = 0 p(plus) = 1 + x1 + x2 p(times) = 1 + x1 + x2 p(two) = 2 p(zero) = 1 Following rules are strictly oriented: dx(X) = 1 + 5*X + X^2 > 0 = one() dx(div(ALPHA,BETA)) = 15 + 9*ALPHA + 2*ALPHA*BETA + ALPHA^2 + 9*BETA + BETA^2 > 13 + 6*ALPHA + ALPHA^2 + 7*BETA + BETA^2 = minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two())))) dx(ln(ALPHA)) = 15 + 9*ALPHA + ALPHA^2 > 3 + 6*ALPHA + ALPHA^2 = div(dx(ALPHA),ALPHA) dx(minus(ALPHA,BETA)) = 15 + 9*ALPHA + 2*ALPHA*BETA + ALPHA^2 + 9*BETA + BETA^2 > 4 + 5*ALPHA + ALPHA^2 + 5*BETA + BETA^2 = minus(dx(ALPHA),dx(BETA)) dx(plus(ALPHA,BETA)) = 7 + 7*ALPHA + 2*ALPHA*BETA + ALPHA^2 + 7*BETA + BETA^2 > 3 + 5*ALPHA + ALPHA^2 + 5*BETA + BETA^2 = plus(dx(ALPHA),dx(BETA)) dx(times(ALPHA,BETA)) = 7 + 7*ALPHA + 2*ALPHA*BETA + ALPHA^2 + 7*BETA + BETA^2 > 5 + 6*ALPHA + ALPHA^2 + 6*BETA + BETA^2 = plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA))) Following rules are (at-least) weakly oriented: dx(a()) = 1 >= 1 = zero() dx(exp(ALPHA,BETA)) = 15 + 9*ALPHA + 2*ALPHA*BETA + ALPHA^2 + 9*BETA + BETA^2 >= 15 + 8*ALPHA + ALPHA^2 + 8*BETA + BETA^2 = plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))) ,times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA)))) dx(neg(ALPHA)) = 1 + 5*ALPHA + ALPHA^2 >= 1 + 5*ALPHA + ALPHA^2 = neg(dx(ALPHA)) ** Step 1.b:2: NaturalPI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: dx(a()) -> zero() dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))) ,times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA)))) dx(neg(ALPHA)) -> neg(dx(ALPHA)) - Weak TRS: dx(X) -> one() dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two())))) dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA) dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA)) dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA)) dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA))) - Signature: {dx/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0} - Obligation: runtime complexity wrt. defined symbols {dx} and constructors {a,div,exp,ln,minus,neg,one,plus,times,two ,zero} + 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(div) = {1}, uargs(minus) = {1,2}, uargs(neg) = {1}, uargs(plus) = {1,2}, uargs(times) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = 0 p(div) = 1 + x1 + x2 p(dx) = 4*x1 + 4*x1^2 p(exp) = 1 + x1 + x2 p(ln) = 1 + x1 p(minus) = 1 + x1 + x2 p(neg) = 1 + x1 p(one) = 0 p(plus) = x1 + x2 p(times) = 1 + x1 + x2 p(two) = 2 p(zero) = 0 Following rules are strictly oriented: dx(neg(ALPHA)) = 8 + 12*ALPHA + 4*ALPHA^2 > 1 + 4*ALPHA + 4*ALPHA^2 = neg(dx(ALPHA)) Following rules are (at-least) weakly oriented: dx(X) = 4*X + 4*X^2 >= 0 = one() dx(a()) = 0 >= 0 = zero() dx(div(ALPHA,BETA)) = 8 + 12*ALPHA + 8*ALPHA*BETA + 4*ALPHA^2 + 12*BETA + 4*BETA^2 >= 7 + 5*ALPHA + 4*ALPHA^2 + 6*BETA + 4*BETA^2 = minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two())))) dx(exp(ALPHA,BETA)) = 8 + 12*ALPHA + 8*ALPHA*BETA + 4*ALPHA^2 + 12*BETA + 4*BETA^2 >= 8 + 7*ALPHA + 4*ALPHA^2 + 7*BETA + 4*BETA^2 = plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))) ,times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA)))) dx(ln(ALPHA)) = 8 + 12*ALPHA + 4*ALPHA^2 >= 1 + 5*ALPHA + 4*ALPHA^2 = div(dx(ALPHA),ALPHA) dx(minus(ALPHA,BETA)) = 8 + 12*ALPHA + 8*ALPHA*BETA + 4*ALPHA^2 + 12*BETA + 4*BETA^2 >= 1 + 4*ALPHA + 4*ALPHA^2 + 4*BETA + 4*BETA^2 = minus(dx(ALPHA),dx(BETA)) dx(plus(ALPHA,BETA)) = 4*ALPHA + 8*ALPHA*BETA + 4*ALPHA^2 + 4*BETA + 4*BETA^2 >= 4*ALPHA + 4*ALPHA^2 + 4*BETA + 4*BETA^2 = plus(dx(ALPHA),dx(BETA)) dx(times(ALPHA,BETA)) = 8 + 12*ALPHA + 8*ALPHA*BETA + 4*ALPHA^2 + 12*BETA + 4*BETA^2 >= 2 + 5*ALPHA + 4*ALPHA^2 + 5*BETA + 4*BETA^2 = plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA))) ** Step 1.b:3: NaturalPI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: dx(a()) -> zero() dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))) ,times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA)))) - Weak TRS: dx(X) -> one() dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two())))) dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA) dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA)) dx(neg(ALPHA)) -> neg(dx(ALPHA)) dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA)) dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA))) - Signature: {dx/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0} - Obligation: runtime complexity wrt. defined symbols {dx} and constructors {a,div,exp,ln,minus,neg,one,plus,times,two ,zero} + 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(div) = {1}, uargs(minus) = {1,2}, uargs(neg) = {1}, uargs(plus) = {1,2}, uargs(times) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = 1 p(div) = 2 + x1 + x2 p(dx) = 3*x1 + 2*x1^2 p(exp) = 2 + x1 + x2 p(ln) = 2 + x1 p(minus) = x1 + x2 p(neg) = 1 + x1 p(one) = 0 p(plus) = x1 + x2 p(times) = 2 + x1 + x2 p(two) = 5 p(zero) = 3 Following rules are strictly oriented: dx(a()) = 5 > 3 = zero() Following rules are (at-least) weakly oriented: dx(X) = 3*X + 2*X^2 >= 0 = one() dx(div(ALPHA,BETA)) = 14 + 11*ALPHA + 4*ALPHA*BETA + 2*ALPHA^2 + 11*BETA + 2*BETA^2 >= 13 + 4*ALPHA + 2*ALPHA^2 + 5*BETA + 2*BETA^2 = minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two())))) dx(exp(ALPHA,BETA)) = 14 + 11*ALPHA + 4*ALPHA*BETA + 2*ALPHA^2 + 11*BETA + 2*BETA^2 >= 14 + 6*ALPHA + 2*ALPHA^2 + 6*BETA + 2*BETA^2 = plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))) ,times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA)))) dx(ln(ALPHA)) = 14 + 11*ALPHA + 2*ALPHA^2 >= 2 + 4*ALPHA + 2*ALPHA^2 = div(dx(ALPHA),ALPHA) dx(minus(ALPHA,BETA)) = 3*ALPHA + 4*ALPHA*BETA + 2*ALPHA^2 + 3*BETA + 2*BETA^2 >= 3*ALPHA + 2*ALPHA^2 + 3*BETA + 2*BETA^2 = minus(dx(ALPHA),dx(BETA)) dx(neg(ALPHA)) = 5 + 7*ALPHA + 2*ALPHA^2 >= 1 + 3*ALPHA + 2*ALPHA^2 = neg(dx(ALPHA)) dx(plus(ALPHA,BETA)) = 3*ALPHA + 4*ALPHA*BETA + 2*ALPHA^2 + 3*BETA + 2*BETA^2 >= 3*ALPHA + 2*ALPHA^2 + 3*BETA + 2*BETA^2 = plus(dx(ALPHA),dx(BETA)) dx(times(ALPHA,BETA)) = 14 + 11*ALPHA + 4*ALPHA*BETA + 2*ALPHA^2 + 11*BETA + 2*BETA^2 >= 4 + 4*ALPHA + 2*ALPHA^2 + 4*BETA + 2*BETA^2 = plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA))) ** Step 1.b:4: NaturalPI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))) ,times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA)))) - Weak TRS: dx(X) -> one() dx(a()) -> zero() dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two())))) dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA) dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA)) dx(neg(ALPHA)) -> neg(dx(ALPHA)) dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA)) dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA))) - Signature: {dx/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0} - Obligation: runtime complexity wrt. defined symbols {dx} and constructors {a,div,exp,ln,minus,neg,one,plus,times,two ,zero} + 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(div) = {1}, uargs(minus) = {1,2}, uargs(neg) = {1}, uargs(plus) = {1,2}, uargs(times) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = 0 p(div) = 1 + x1 + x2 p(dx) = 2*x1 + 6*x1^2 p(exp) = 1 + x1 + x2 p(ln) = 1 + x1 p(minus) = x1 + x2 p(neg) = x1 p(one) = 0 p(plus) = x1 + x2 p(times) = 1 + x1 + x2 p(two) = 2 p(zero) = 0 Following rules are strictly oriented: dx(exp(ALPHA,BETA)) = 8 + 14*ALPHA + 12*ALPHA*BETA + 6*ALPHA^2 + 14*BETA + 6*BETA^2 > 7 + 5*ALPHA + 6*ALPHA^2 + 5*BETA + 6*BETA^2 = plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))) ,times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA)))) Following rules are (at-least) weakly oriented: dx(X) = 2*X + 6*X^2 >= 0 = one() dx(a()) = 0 >= 0 = zero() dx(div(ALPHA,BETA)) = 8 + 14*ALPHA + 12*ALPHA*BETA + 6*ALPHA^2 + 14*BETA + 6*BETA^2 >= 6 + 3*ALPHA + 6*ALPHA^2 + 4*BETA + 6*BETA^2 = minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two())))) dx(ln(ALPHA)) = 8 + 14*ALPHA + 6*ALPHA^2 >= 1 + 3*ALPHA + 6*ALPHA^2 = div(dx(ALPHA),ALPHA) dx(minus(ALPHA,BETA)) = 2*ALPHA + 12*ALPHA*BETA + 6*ALPHA^2 + 2*BETA + 6*BETA^2 >= 2*ALPHA + 6*ALPHA^2 + 2*BETA + 6*BETA^2 = minus(dx(ALPHA),dx(BETA)) dx(neg(ALPHA)) = 2*ALPHA + 6*ALPHA^2 >= 2*ALPHA + 6*ALPHA^2 = neg(dx(ALPHA)) dx(plus(ALPHA,BETA)) = 2*ALPHA + 12*ALPHA*BETA + 6*ALPHA^2 + 2*BETA + 6*BETA^2 >= 2*ALPHA + 6*ALPHA^2 + 2*BETA + 6*BETA^2 = plus(dx(ALPHA),dx(BETA)) dx(times(ALPHA,BETA)) = 8 + 14*ALPHA + 12*ALPHA*BETA + 6*ALPHA^2 + 14*BETA + 6*BETA^2 >= 2 + 3*ALPHA + 6*ALPHA^2 + 3*BETA + 6*BETA^2 = plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA))) ** Step 1.b:5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: dx(X) -> one() dx(a()) -> zero() dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two())))) dx(exp(ALPHA,BETA)) -> plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))) ,times(exp(ALPHA,BETA),times(ln(ALPHA),dx(BETA)))) dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA) dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA)) dx(neg(ALPHA)) -> neg(dx(ALPHA)) dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA)) dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA))) - Signature: {dx/1} / {a/0,div/2,exp/2,ln/1,minus/2,neg/1,one/0,plus/2,times/2,two/0,zero/0} - Obligation: runtime complexity wrt. defined symbols {dx} and constructors {a,div,exp,ln,minus,neg,one,plus,times,two ,zero} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^2))