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