* Step 1: MI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(x,y,s(z)) -> s(f(0(),1(),z)) f(0(),1(),x) -> f(s(x),x,x) g(x,y) -> x g(x,y) -> y - Signature: {f/3,g/2} / {0/0,1/0,s/1} - Obligation: runtime complexity wrt. defined symbols {f,g} and constructors {0,1,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(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(1) = [0] p(f) = [9] p(g) = [2] x_1 + [2] x_2 + [5] p(s) = [1] x_1 + [0] Following rules are strictly oriented: g(x,y) = [2] x + [2] y + [5] > [1] x + [0] = x g(x,y) = [2] x + [2] y + [5] > [1] y + [0] = y Following rules are (at-least) weakly oriented: f(x,y,s(z)) = [9] >= [9] = s(f(0(),1(),z)) f(0(),1(),x) = [9] >= [9] = f(s(x),x,x) * Step 2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(x,y,s(z)) -> s(f(0(),1(),z)) f(0(),1(),x) -> f(s(x),x,x) - Weak TRS: g(x,y) -> x g(x,y) -> y - Signature: {f/3,g/2} / {0/0,1/0,s/1} - Obligation: runtime complexity wrt. defined symbols {f,g} and constructors {0,1,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(0) = [1] p(1) = [1] p(f) = [8] x3 + [0] p(g) = [1] x1 + [2] x2 + [8] p(s) = [1] x1 + [2] Following rules are strictly oriented: f(x,y,s(z)) = [8] z + [16] > [8] z + [2] = s(f(0(),1(),z)) Following rules are (at-least) weakly oriented: f(0(),1(),x) = [8] x + [0] >= [8] x + [0] = f(s(x),x,x) g(x,y) = [1] x + [2] y + [8] >= [1] x + [0] = x g(x,y) = [1] x + [2] y + [8] >= [1] y + [0] = y 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: f(0(),1(),x) -> f(s(x),x,x) - Weak TRS: f(x,y,s(z)) -> s(f(0(),1(),z)) g(x,y) -> x g(x,y) -> y - Signature: {f/3,g/2} / {0/0,1/0,s/1} - Obligation: runtime complexity wrt. defined symbols {f,g} and constructors {0,1,s} + Applied Processor: MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 1))), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 1))): The following argument positions are considered usable: uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [4] [1] p(1) = [0] [0] p(f) = [0 2] x_1 + [5 9] x_3 + [1] [0 0] [0 2] [4] p(g) = [1 4] x_1 + [2 1] x_2 + [0] [1 2] [8 8] [0] p(s) = [1 3] x_1 + [4] [0 0] [0] Following rules are strictly oriented: f(0(),1(),x) = [5 9] x + [3] [0 2] [4] > [5 9] x + [1] [0 2] [4] = f(s(x),x,x) Following rules are (at-least) weakly oriented: f(x,y,s(z)) = [0 2] x + [5 15] z + [21] [0 0] [0 0] [4] >= [5 15] z + [19] [0 0] [0] = s(f(0(),1(),z)) g(x,y) = [1 4] x + [2 1] y + [0] [1 2] [8 8] [0] >= [1 0] x + [0] [0 1] [0] = x g(x,y) = [1 4] x + [2 1] y + [0] [1 2] [8 8] [0] >= [1 0] y + [0] [0 1] [0] = y * Step 4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: f(x,y,s(z)) -> s(f(0(),1(),z)) f(0(),1(),x) -> f(s(x),x,x) g(x,y) -> x g(x,y) -> y - Signature: {f/3,g/2} / {0/0,1/0,s/1} - Obligation: runtime complexity wrt. defined symbols {f,g} and constructors {0,1,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))