* Step 1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) +(s(x),y) -> s(+(x,y)) double(x) -> +(x,x) double(0()) -> 0() double(s(x)) -> s(s(double(x))) - Signature: {+/2,double/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,double} 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(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(+) = [6] x1 + [0] p(0) = [1] p(double) = [6] x1 + [8] p(s) = [1] x1 + [3] Following rules are strictly oriented: +(s(x),y) = [6] x + [18] > [6] x + [3] = s(+(x,y)) double(x) = [6] x + [8] > [6] x + [0] = +(x,x) double(0()) = [14] > [1] = 0() double(s(x)) = [6] x + [26] > [6] x + [14] = s(s(double(x))) Following rules are (at-least) weakly oriented: +(x,0()) = [6] x + [0] >= [1] x + [0] = x +(x,s(y)) = [6] x + [0] >= [6] x + [3] = s(+(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^1)) + Considered Problem: - Strict TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) - Weak TRS: +(s(x),y) -> s(+(x,y)) double(x) -> +(x,x) double(0()) -> 0() double(s(x)) -> s(s(double(x))) - Signature: {+/2,double/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,double} 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(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(+) = [3] x1 + [2] x2 + [7] p(0) = [1] p(double) = [5] x1 + [9] p(s) = [1] x1 + [1] Following rules are strictly oriented: +(x,0()) = [3] x + [9] > [1] x + [0] = x +(x,s(y)) = [3] x + [2] y + [9] > [3] x + [2] y + [8] = s(+(x,y)) Following rules are (at-least) weakly oriented: +(s(x),y) = [3] x + [2] y + [10] >= [3] x + [2] y + [8] = s(+(x,y)) double(x) = [5] x + [9] >= [5] x + [7] = +(x,x) double(0()) = [14] >= [1] = 0() double(s(x)) = [5] x + [14] >= [5] x + [11] = s(s(double(x))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) +(s(x),y) -> s(+(x,y)) double(x) -> +(x,x) double(0()) -> 0() double(s(x)) -> s(s(double(x))) - Signature: {+/2,double/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,double} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))