* Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: log(s(0())) -> 0() log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {log,min,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(log) = {1}, uargs(min) = {1}, uargs(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [7] p(log) = [1] x1 + [0] p(min) = [1] x1 + [2] p(quot) = [1] x1 + [9] p(s) = [1] x1 + [0] Following rules are strictly oriented: min(X,0()) = [1] X + [2] > [1] X + [0] = X quot(0(),s(Y)) = [16] > [7] = 0() Following rules are (at-least) weakly oriented: log(s(0())) = [7] >= [7] = 0() log(s(s(X))) = [1] X + [0] >= [1] X + [9] = s(log(s(quot(X,s(s(0())))))) min(s(X),s(Y)) = [1] X + [2] >= [1] X + [2] = min(X,Y) quot(s(X),s(Y)) = [1] X + [9] >= [1] X + [11] = s(quot(min(X,Y),s(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: log(s(0())) -> 0() log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) min(s(X),s(Y)) -> min(X,Y) quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Weak TRS: min(X,0()) -> X quot(0(),s(Y)) -> 0() - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {log,min,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(log) = {1}, uargs(min) = {1}, uargs(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(log) = [1] x1 + [0] p(min) = [1] x1 + [7] p(quot) = [1] x1 + [8] p(s) = [1] x1 + [1] Following rules are strictly oriented: log(s(0())) = [1] > [0] = 0() min(s(X),s(Y)) = [1] X + [8] > [1] X + [7] = min(X,Y) Following rules are (at-least) weakly oriented: log(s(s(X))) = [1] X + [2] >= [1] X + [10] = s(log(s(quot(X,s(s(0())))))) min(X,0()) = [1] X + [7] >= [1] X + [0] = X quot(0(),s(Y)) = [8] >= [0] = 0() quot(s(X),s(Y)) = [1] X + [9] >= [1] X + [16] = s(quot(min(X,Y),s(Y))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: MI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Weak TRS: log(s(0())) -> 0() min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {log,min,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(log) = {1}, uargs(min) = {1}, uargs(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(log) = [8] x_1 + [6] p(min) = [1] x_1 + [0] p(quot) = [1] x_1 + [0] p(s) = [1] x_1 + [1] Following rules are strictly oriented: log(s(s(X))) = [8] X + [22] > [8] X + [15] = s(log(s(quot(X,s(s(0())))))) Following rules are (at-least) weakly oriented: log(s(0())) = [30] >= [2] = 0() min(X,0()) = [1] X + [0] >= [1] X + [0] = X min(s(X),s(Y)) = [1] X + [1] >= [1] X + [0] = min(X,Y) quot(0(),s(Y)) = [2] >= [2] = 0() quot(s(X),s(Y)) = [1] X + [1] >= [1] X + [1] = s(quot(min(X,Y),s(Y))) * Step 4: MI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Weak TRS: log(s(0())) -> 0() log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s} + Applied Processor: MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 2, 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(log) = {1}, uargs(min) = {1}, uargs(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] [1] p(log) = [2 0] x_1 + [4] [0 1] [1] p(min) = [1 0] x_1 + [0] [0 1] [0] p(quot) = [1 1] x_1 + [0] [0 1] [0] p(s) = [1 2] x_1 + [0] [0 1] [1] Following rules are strictly oriented: quot(s(X),s(Y)) = [1 3] X + [1] [0 1] [1] > [1 3] X + [0] [0 1] [1] = s(quot(min(X,Y),s(Y))) Following rules are (at-least) weakly oriented: log(s(0())) = [12] [3] >= [2] [1] = 0() log(s(s(X))) = [2 8] X + [8] [0 1] [3] >= [2 8] X + [8] [0 1] [3] = s(log(s(quot(X,s(s(0())))))) min(X,0()) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X min(s(X),s(Y)) = [1 2] X + [0] [0 1] [1] >= [1 0] X + [0] [0 1] [0] = min(X,Y) quot(0(),s(Y)) = [3] [1] >= [2] [1] = 0() * Step 5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: log(s(0())) -> 0() log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))