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