* Step 1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {minus/2,quot/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {minus,quot} 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(minus) = {1}, uargs(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(minus) = [1] x1 + [0] p(quot) = [1] x1 + [4] x2 + [1] p(s) = [1] x1 + [2] Following rules are strictly oriented: minus(s(x),s(y)) = [1] x + [2] > [1] x + [0] = minus(x,y) quot(0(),s(y)) = [4] y + [9] > [0] = 0() Following rules are (at-least) weakly oriented: minus(x,0()) = [1] x + [0] >= [1] x + [0] = x quot(s(x),s(y)) = [1] x + [4] y + [11] >= [1] x + [4] y + [11] = s(quot(minus(x,y),s(y))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: MI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: minus(x,0()) -> x quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Weak TRS: minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() - Signature: {minus/2,quot/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {minus,quot} and constructors {0,s} + Applied Processor: MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)): The following argument positions are considered usable: uargs(minus) = {1}, uargs(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [5] p(minus) = [1] x_1 + [1] p(quot) = [4] x_1 + [2] x_2 + [0] p(s) = [1] x_1 + [4] Following rules are strictly oriented: minus(x,0()) = [1] x + [1] > [1] x + [0] = x quot(s(x),s(y)) = [4] x + [2] y + [24] > [4] x + [2] y + [16] = s(quot(minus(x,y),s(y))) Following rules are (at-least) weakly oriented: minus(s(x),s(y)) = [1] x + [5] >= [1] x + [1] = minus(x,y) quot(0(),s(y)) = [2] y + [28] >= [5] = 0() * Step 3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {minus/2,quot/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {minus,quot} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))