* Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {plus,times} 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(plus) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(plus) = [1] x1 + [0] p(s) = [1] x1 + [1] p(times) = [12] x1 + [8] x2 + [3] Following rules are strictly oriented: times(x,0()) = [12] x + [3] > [0] = 0() times(x,s(y)) = [12] x + [8] y + [11] > [12] x + [8] y + [3] = plus(times(x,y),x) Following rules are (at-least) weakly oriented: plus(x,0()) = [1] x + [0] >= [1] x + [0] = x plus(x,s(y)) = [1] x + [0] >= [1] x + [1] = s(plus(x,y)) 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: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) - Weak TRS: times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {plus,times} 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(plus) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(plus) = [1] x1 + [2] p(s) = [1] x1 + [1] p(times) = [1] x1 + [4] x2 + [8] Following rules are strictly oriented: plus(x,0()) = [1] x + [2] > [1] x + [0] = x Following rules are (at-least) weakly oriented: plus(x,s(y)) = [1] x + [2] >= [1] x + [3] = s(plus(x,y)) times(x,0()) = [1] x + [16] >= [2] = 0() times(x,s(y)) = [1] x + [4] y + [12] >= [1] x + [4] y + [10] = plus(times(x,y),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: plus(x,s(y)) -> s(plus(x,y)) - Weak TRS: plus(x,0()) -> x times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {plus,times} 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(plus) = {1}, uargs(s) = {1} Following symbols are considered usable: {plus,times} TcT has computed the following interpretation: p(0) = 0 p(plus) = x1 + 3*x2 p(s) = 1 + x1 p(times) = 2*x1 + 3*x1*x2 + x1^2 Following rules are strictly oriented: plus(x,s(y)) = 3 + x + 3*y > 1 + x + 3*y = s(plus(x,y)) Following rules are (at-least) weakly oriented: plus(x,0()) = x >= x = x times(x,0()) = 2*x + x^2 >= 0 = 0() times(x,s(y)) = 5*x + 3*x*y + x^2 >= 5*x + 3*x*y + x^2 = plus(times(x,y),x) * Step 4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {plus,times} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))