* Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) sum(0()) -> 0() sum(s(x)) -> +(sum(x),s(x)) - Signature: {+/2,sum/1} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {+,sum} and constructors {0,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(+) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(+) = [1] x1 + [5] p(0) = [0] p(s) = [1] x1 + [8] p(sum) = [2] x1 + [0] Following rules are strictly oriented: +(x,0()) = [1] x + [5] > [1] x + [0] = x sum(s(x)) = [2] x + [16] > [2] x + [5] = +(sum(x),s(x)) Following rules are (at-least) weakly oriented: +(x,s(y)) = [1] x + [5] >= [1] x + [13] = s(+(x,y)) sum(0()) = [0] >= [0] = 0() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: +(x,s(y)) -> s(+(x,y)) sum(0()) -> 0() - Weak TRS: +(x,0()) -> x sum(s(x)) -> +(sum(x),s(x)) - Signature: {+/2,sum/1} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {+,sum} and constructors {0,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(+) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(+) = [1] x1 + [0] p(0) = [0] p(s) = [1] x1 + [0] p(sum) = [7] Following rules are strictly oriented: sum(0()) = [7] > [0] = 0() Following rules are (at-least) weakly oriented: +(x,0()) = [1] x + [0] >= [1] x + [0] = x +(x,s(y)) = [1] x + [0] >= [1] x + [0] = s(+(x,y)) sum(s(x)) = [7] >= [7] = +(sum(x),s(x)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: +(x,s(y)) -> s(+(x,y)) - Weak TRS: +(x,0()) -> x sum(0()) -> 0() sum(s(x)) -> +(sum(x),s(x)) - Signature: {+/2,sum/1} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {+,sum} 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}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(+) = x1 + 4*x2 p(0) = 0 p(s) = 1 + x1 p(sum) = 2 + x1 + 3*x1^2 Following rules are strictly oriented: +(x,s(y)) = 4 + x + 4*y > 1 + x + 4*y = s(+(x,y)) Following rules are (at-least) weakly oriented: +(x,0()) = x >= x = x sum(0()) = 2 >= 0 = 0() sum(s(x)) = 6 + 7*x + 3*x^2 >= 6 + 5*x + 3*x^2 = +(sum(x),s(x)) * Step 4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) sum(0()) -> 0() sum(s(x)) -> +(sum(x),s(x)) - Signature: {+/2,sum/1} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {+,sum} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))