* Step 1: ToInnermost WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__b() -> a() a__b() -> b() a__f(X1,X2,X3) -> f(X1,X2,X3) a__f(a(),X,X) -> a__f(X,a__b(),b()) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Signature: {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3} - Obligation: runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f} + Applied Processor: ToInnermost + Details: switch to innermost, as the system is overlay and right linear and does not contain weak rules * Step 2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__b() -> a() a__b() -> b() a__f(X1,X2,X3) -> f(X1,X2,X3) a__f(a(),X,X) -> a__f(X,a__b(),b()) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Signature: {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f} + 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(a__f) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [4] p(a__b) = [0] p(a__f) = [1] x1 + [1] x2 + [0] p(b) = [10] p(f) = [1] x1 + [1] x2 + [12] p(mark) = [2] x1 + [0] Following rules are strictly oriented: a__f(a(),X,X) = [1] X + [4] > [1] X + [0] = a__f(X,a__b(),b()) mark(a()) = [8] > [4] = a() mark(b()) = [20] > [0] = a__b() mark(f(X1,X2,X3)) = [2] X1 + [2] X2 + [24] > [1] X1 + [2] X2 + [0] = a__f(X1,mark(X2),X3) Following rules are (at-least) weakly oriented: a__b() = [0] >= [4] = a() a__b() = [0] >= [10] = b() a__f(X1,X2,X3) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [12] = f(X1,X2,X3) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__b() -> a() a__b() -> b() a__f(X1,X2,X3) -> f(X1,X2,X3) - Weak TRS: a__f(a(),X,X) -> a__f(X,a__b(),b()) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Signature: {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f} + 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(a__f) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(a) = [1] p(a__b) = [1] p(a__f) = [1] x1 + [1] x2 + [1] x3 + [3] p(b) = [0] p(f) = [1] x1 + [1] x2 + [1] x3 + [2] p(mark) = [9] x1 + [1] Following rules are strictly oriented: a__b() = [1] > [0] = b() a__f(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [3] > [1] X1 + [1] X2 + [1] X3 + [2] = f(X1,X2,X3) Following rules are (at-least) weakly oriented: a__b() = [1] >= [1] = a() a__f(a(),X,X) = [2] X + [4] >= [1] X + [4] = a__f(X,a__b(),b()) mark(a()) = [10] >= [1] = a() mark(b()) = [1] >= [1] = a__b() mark(f(X1,X2,X3)) = [9] X1 + [9] X2 + [9] X3 + [19] >= [1] X1 + [9] X2 + [1] X3 + [4] = a__f(X1,mark(X2),X3) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: MI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__b() -> a() - Weak TRS: a__b() -> b() a__f(X1,X2,X3) -> f(X1,X2,X3) a__f(a(),X,X) -> a__f(X,a__b(),b()) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Signature: {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f} + 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(a__f) = {2} Following symbols are considered usable: {a__b,a__f,mark} TcT has computed the following interpretation: p(a) = [4] p(a__b) = [6] p(a__f) = [4] x_1 + [1] x_2 + [4] x_3 + [8] p(b) = [2] p(f) = [1] x_1 + [1] x_2 + [1] x_3 + [4] p(mark) = [4] x_1 + [4] Following rules are strictly oriented: a__b() = [6] > [4] = a() Following rules are (at-least) weakly oriented: a__b() = [6] >= [2] = b() a__f(X1,X2,X3) = [4] X1 + [1] X2 + [4] X3 + [8] >= [1] X1 + [1] X2 + [1] X3 + [4] = f(X1,X2,X3) a__f(a(),X,X) = [5] X + [24] >= [4] X + [22] = a__f(X,a__b(),b()) mark(a()) = [20] >= [4] = a() mark(b()) = [12] >= [6] = a__b() mark(f(X1,X2,X3)) = [4] X1 + [4] X2 + [4] X3 + [20] >= [4] X1 + [4] X2 + [4] X3 + [12] = a__f(X1,mark(X2),X3) * Step 5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__b() -> a() a__b() -> b() a__f(X1,X2,X3) -> f(X1,X2,X3) a__f(a(),X,X) -> a__f(X,a__b(),b()) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Signature: {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))