* Step 1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: *(x,+(y,z)) -> +(*(x,y),*(x,z)) *(x,1()) -> x *(+(x,y),z) -> +(*(x,z),*(y,z)) *(1(),y) -> y - Signature: {*/2} / {+/2,1/0} - Obligation: runtime complexity wrt. defined symbols {*} and constructors {+,1} + 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} Following symbols are considered usable: all TcT has computed the following interpretation: p(*) = x1 + 4*x1*x2 + 2*x2 p(+) = 1 + x1 + x2 p(1) = 0 Following rules are strictly oriented: *(x,+(y,z)) = 2 + 5*x + 4*x*y + 4*x*z + 2*y + 2*z > 1 + 2*x + 4*x*y + 4*x*z + 2*y + 2*z = +(*(x,y),*(x,z)) Following rules are (at-least) weakly oriented: *(x,1()) = x >= x = x *(+(x,y),z) = 1 + x + 4*x*z + y + 4*y*z + 6*z >= 1 + x + 4*x*z + y + 4*y*z + 4*z = +(*(x,z),*(y,z)) *(1(),y) = 2*y >= y = y * Step 2: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: *(x,1()) -> x *(+(x,y),z) -> +(*(x,z),*(y,z)) *(1(),y) -> y - Weak TRS: *(x,+(y,z)) -> +(*(x,y),*(x,z)) - Signature: {*/2} / {+/2,1/0} - Obligation: runtime complexity wrt. defined symbols {*} and constructors {+,1} + 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} Following symbols are considered usable: all TcT has computed the following interpretation: p(*) = 2*x1 + 2*x1*x2 + x2 p(+) = 1 + x1 + x2 p(1) = 0 Following rules are strictly oriented: *(+(x,y),z) = 2 + 2*x + 2*x*z + 2*y + 2*y*z + 3*z > 1 + 2*x + 2*x*z + 2*y + 2*y*z + 2*z = +(*(x,z),*(y,z)) Following rules are (at-least) weakly oriented: *(x,+(y,z)) = 1 + 4*x + 2*x*y + 2*x*z + y + z >= 1 + 4*x + 2*x*y + 2*x*z + y + z = +(*(x,y),*(x,z)) *(x,1()) = 2*x >= x = x *(1(),y) = y >= y = y * Step 3: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: *(x,1()) -> x *(1(),y) -> y - Weak TRS: *(x,+(y,z)) -> +(*(x,y),*(x,z)) *(+(x,y),z) -> +(*(x,z),*(y,z)) - Signature: {*/2} / {+/2,1/0} - Obligation: runtime complexity wrt. defined symbols {*} and constructors {+,1} + 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} Following symbols are considered usable: all TcT has computed the following interpretation: p(*) = 2 + 7*x1 + 7*x1*x2 + 6*x2 p(+) = 1 + x1 + x2 p(1) = 1 Following rules are strictly oriented: *(x,1()) = 8 + 14*x > x = x *(1(),y) = 9 + 13*y > y = y Following rules are (at-least) weakly oriented: *(x,+(y,z)) = 8 + 14*x + 7*x*y + 7*x*z + 6*y + 6*z >= 5 + 14*x + 7*x*y + 7*x*z + 6*y + 6*z = +(*(x,y),*(x,z)) *(+(x,y),z) = 9 + 7*x + 7*x*z + 7*y + 7*y*z + 13*z >= 5 + 7*x + 7*x*z + 7*y + 7*y*z + 12*z = +(*(x,z),*(y,z)) * Step 4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: *(x,+(y,z)) -> +(*(x,y),*(x,z)) *(x,1()) -> x *(+(x,y),z) -> +(*(x,z),*(y,z)) *(1(),y) -> y - Signature: {*/2} / {+/2,1/0} - Obligation: runtime complexity wrt. defined symbols {*} and constructors {+,1} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))