WORST_CASE(?,O(n^2)) * 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: innermost 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(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(log) = [1] x1 + [1] p(min) = [1] x1 + [8] p(quot) = [1] x1 + [8] x2 + [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: log(s(0())) = [2] > [1] = 0() min(X,0()) = [1] X + [8] > [1] X + [0] = X Following rules are (at-least) weakly oriented: log(s(s(X))) = [1] X + [1] >= [1] X + [9] = s(log(s(quot(X,s(s(0())))))) min(s(X),s(Y)) = [1] X + [8] >= [1] X + [8] = min(X,Y) quot(0(),s(Y)) = [8] Y + [1] >= [1] = 0() quot(s(X),s(Y)) = [1] X + [8] Y + [0] >= [1] X + [8] Y + [8] = 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(s(X))) -> s(log(s(quot(X,s(s(0())))))) 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))) - Weak TRS: log(s(0())) -> 0() min(X,0()) -> X - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: innermost 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(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(log) = [1] x1 + [0] p(min) = [1] x1 + [1] p(quot) = [1] x1 + [3] p(s) = [1] x1 + [1] Following rules are strictly oriented: min(s(X),s(Y)) = [1] X + [2] > [1] X + [1] = min(X,Y) quot(0(),s(Y)) = [4] > [1] = 0() Following rules are (at-least) weakly oriented: log(s(0())) = [2] >= [1] = 0() log(s(s(X))) = [1] X + [2] >= [1] X + [5] = s(log(s(quot(X,s(s(0())))))) min(X,0()) = [1] X + [1] >= [1] X + [0] = X quot(s(X),s(Y)) = [1] X + [4] >= [1] X + [5] = 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: NaturalMI 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: innermost runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(log) = {1}, uargs(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: {log,min,quot} TcT has computed the following interpretation: p(0) = [0] p(log) = [2] x1 + [12] p(min) = [1] x1 + [0] p(quot) = [1] x1 + [0] p(s) = [1] x1 + [4] Following rules are strictly oriented: log(s(s(X))) = [2] X + [28] > [2] X + [24] = s(log(s(quot(X,s(s(0())))))) Following rules are (at-least) weakly oriented: log(s(0())) = [20] >= [0] = 0() min(X,0()) = [1] X + [0] >= [1] X + [0] = X min(s(X),s(Y)) = [1] X + [4] >= [1] X + [0] = min(X,Y) quot(0(),s(Y)) = [0] >= [0] = 0() quot(s(X),s(Y)) = [1] X + [4] >= [1] X + [4] = s(quot(min(X,Y),s(Y))) * Step 4: NaturalMI 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: innermost runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(log) = {1}, uargs(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: {log,min,quot} TcT has computed the following interpretation: p(0) = [0] [2] p(log) = [2 0] x1 + [0] [0 1] [0] p(min) = [1 0] x1 + [1] [0 1] [0] p(quot) = [1 1] x1 + [0] [0 1] [0] p(s) = [1 2] x1 + [0] [0 1] [2] Following rules are strictly oriented: quot(s(X),s(Y)) = [1 3] X + [2] [0 1] [2] > [1 3] X + [1] [0 1] [2] = s(quot(min(X,Y),s(Y))) Following rules are (at-least) weakly oriented: log(s(0())) = [8] [4] >= [0] [2] = 0() log(s(s(X))) = [2 8] X + [8] [0 1] [4] >= [2 8] X + [4] [0 1] [4] = s(log(s(quot(X,s(s(0())))))) min(X,0()) = [1 0] X + [1] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X min(s(X),s(Y)) = [1 2] X + [1] [0 1] [2] >= [1 0] X + [1] [0 1] [0] = min(X,Y) quot(0(),s(Y)) = [2] [2] >= [0] [2] = 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: innermost 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))