/export/starexec/sandbox/solver/bin/starexec_run_tct_rc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/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: +(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: runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. 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: runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: 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: 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: NaturalMI. 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: runtime complexity wrt. defined symbols {+,double,sqr} and constructors {0,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(+) = {1,2}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(+) = [1] x1 + [2] x2 + [0] p(0) = [0] p(double) = [0] p(s) = [1] x1 + [0] p(sqr) = [2] Following rules are strictly oriented: sqr(0()) = [2] > [0] = 0() Following rules are (at-least) weakly oriented: +(x,0()) = [1] x + [0] >= [1] x + [0] = x +(x,s(y)) = [1] x + [2] y + [0] >= [1] x + [2] y + [0] = s(+(x,y)) double(0()) = [0] >= [0] = 0() double(s(x)) = [0] >= [0] = s(s(double(x))) sqr(s(x)) = [2] >= [2] = +(sqr(x),s(double(x))) sqr(s(x)) = [2] >= [2] = 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: 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: all TcT has computed the following interpretation: p(+) = 6 + x1 + 2*x2 p(0) = 2 p(double) = 2*x1 p(s) = 2 + x1 p(sqr) = x1 + 2*x1^2 Following rules are strictly oriented: +(x,0()) = 10 + x > x = x +(x,s(y)) = 10 + x + 2*y > 8 + x + 2*y = s(+(x,y)) double(0()) = 4 > 2 = 0() sqr(s(x)) = 10 + 9*x + 2*x^2 > 8 + 5*x + 2*x^2 = s(+(sqr(x),double(x))) Following rules are (at-least) weakly oriented: double(s(x)) = 4 + 2*x >= 4 + 2*x = s(s(double(x))) sqr(0()) = 10 >= 2 = 0() sqr(s(x)) = 10 + 9*x + 2*x^2 >= 10 + 5*x + 2*x^2 = +(sqr(x),s(double(x))) ** Step 1.b:3: 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: 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: all TcT has computed the following interpretation: p(+) = 4 + x1 + 2*x2 p(0) = 2 p(double) = 2*x1 p(s) = 2 + x1 p(sqr) = x1 + 2*x1^2 Following rules are strictly oriented: sqr(s(x)) = 10 + 9*x + 2*x^2 > 8 + 5*x + 2*x^2 = +(sqr(x),s(double(x))) Following rules are (at-least) weakly oriented: +(x,0()) = 8 + x >= x = x +(x,s(y)) = 8 + x + 2*y >= 6 + x + 2*y = s(+(x,y)) double(0()) = 4 >= 2 = 0() double(s(x)) = 4 + 2*x >= 4 + 2*x = s(s(double(x))) sqr(0()) = 10 >= 2 = 0() sqr(s(x)) = 10 + 9*x + 2*x^2 >= 6 + 5*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))) - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) double(0()) -> 0() 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: 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: all TcT has computed the following interpretation: p(+) = 4 + x1 + x2 p(0) = 2 p(double) = 3*x1 p(s) = 2 + x1 p(sqr) = 2 + 2*x1 + x1^2 Following rules are strictly oriented: double(s(x)) = 6 + 3*x > 4 + 3*x = s(s(double(x))) Following rules are (at-least) weakly oriented: +(x,0()) = 6 + x >= x = x +(x,s(y)) = 6 + x + y >= 6 + x + y = s(+(x,y)) double(0()) = 6 >= 2 = 0() sqr(0()) = 10 >= 2 = 0() sqr(s(x)) = 10 + 6*x + x^2 >= 8 + 5*x + x^2 = +(sqr(x),s(double(x))) sqr(s(x)) = 10 + 6*x + x^2 >= 8 + 5*x + x^2 = s(+(sqr(x),double(x))) ** Step 1.b:5: 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: 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))