* Step 1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y))) minus(X,0()) -> X minus(s(X),s(Y)) -> p(minus(X,Y)) p(s(X)) -> X - Signature: {div/2,minus/2,p/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {div,minus,p} 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(div) = {1}, uargs(p) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(div) = [1] x1 + [14] x2 + [0] p(minus) = [1] x1 + [0] p(p) = [1] x1 + [5] p(s) = [1] x1 + [2] Following rules are strictly oriented: div(0(),s(Y)) = [14] Y + [28] > [0] = 0() p(s(X)) = [1] X + [7] > [1] X + [0] = X Following rules are (at-least) weakly oriented: div(s(X),s(Y)) = [1] X + [14] Y + [30] >= [1] X + [14] Y + [30] = s(div(minus(X,Y),s(Y))) minus(X,0()) = [1] X + [0] >= [1] X + [0] = X minus(s(X),s(Y)) = [1] X + [2] >= [1] X + [5] = p(minus(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: div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y))) minus(X,0()) -> X minus(s(X),s(Y)) -> p(minus(X,Y)) - Weak TRS: div(0(),s(Y)) -> 0() p(s(X)) -> X - Signature: {div/2,minus/2,p/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {div,minus,p} 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(div) = {1}, uargs(p) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [8] p(div) = [1] x1 + [1] x2 + [8] p(minus) = [1] x1 + [1] p(p) = [1] x1 + [8] p(s) = [1] x1 + [0] Following rules are strictly oriented: minus(X,0()) = [1] X + [1] > [1] X + [0] = X Following rules are (at-least) weakly oriented: div(0(),s(Y)) = [1] Y + [16] >= [8] = 0() div(s(X),s(Y)) = [1] X + [1] Y + [8] >= [1] X + [1] Y + [9] = s(div(minus(X,Y),s(Y))) minus(s(X),s(Y)) = [1] X + [1] >= [1] X + [9] = p(minus(X,Y)) p(s(X)) = [1] X + [8] >= [1] X + [0] = X Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: MI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y))) minus(s(X),s(Y)) -> p(minus(X,Y)) - Weak TRS: div(0(),s(Y)) -> 0() minus(X,0()) -> X p(s(X)) -> X - Signature: {div/2,minus/2,p/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {div,minus,p} 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(div) = {1}, uargs(p) = {1}, uargs(s) = {1} Following symbols are considered usable: {div,minus,p} TcT has computed the following interpretation: p(0) = [0] p(div) = [2] x_1 + [1] x_2 + [5] p(minus) = [1] x_1 + [3] p(p) = [1] x_1 + [1] p(s) = [1] x_1 + [8] Following rules are strictly oriented: div(s(X),s(Y)) = [2] X + [1] Y + [29] > [2] X + [1] Y + [27] = s(div(minus(X,Y),s(Y))) minus(s(X),s(Y)) = [1] X + [11] > [1] X + [4] = p(minus(X,Y)) Following rules are (at-least) weakly oriented: div(0(),s(Y)) = [1] Y + [13] >= [0] = 0() minus(X,0()) = [1] X + [3] >= [1] X + [0] = X p(s(X)) = [1] X + [9] >= [1] X + [0] = X * Step 4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: div(0(),s(Y)) -> 0() div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y))) minus(X,0()) -> X minus(s(X),s(Y)) -> p(minus(X,Y)) p(s(X)) -> X - Signature: {div/2,minus/2,p/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {div,minus,p} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))