* Step 1: MI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: implies(x,or(y,z)) -> or(y,implies(x,z)) implies(not(x),y) -> or(x,y) implies(not(x),or(y,z)) -> implies(y,or(x,z)) - Signature: {implies/2} / {not/1,or/2} - Obligation: innermost runtime complexity wrt. defined symbols {implies} and constructors {not,or} + 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(or) = {2} Following symbols are considered usable: {implies} TcT has computed the following interpretation: p(implies) = [2] x_2 + [10] p(not) = [0] p(or) = [1] x_2 + [10] Following rules are strictly oriented: implies(x,or(y,z)) = [2] z + [30] > [2] z + [20] = or(y,implies(x,z)) Following rules are (at-least) weakly oriented: implies(not(x),y) = [2] y + [10] >= [1] y + [10] = or(x,y) implies(not(x),or(y,z)) = [2] z + [30] >= [2] z + [30] = implies(y,or(x,z)) * Step 2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: implies(not(x),y) -> or(x,y) implies(not(x),or(y,z)) -> implies(y,or(x,z)) - Weak TRS: implies(x,or(y,z)) -> or(y,implies(x,z)) - Signature: {implies/2} / {not/1,or/2} - Obligation: innermost runtime complexity wrt. defined symbols {implies} and constructors {not,or} + 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(or) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(implies) = [2] x2 + [1] p(not) = [1] x1 + [0] p(or) = [1] x2 + [0] Following rules are strictly oriented: implies(not(x),y) = [2] y + [1] > [1] y + [0] = or(x,y) Following rules are (at-least) weakly oriented: implies(x,or(y,z)) = [2] z + [1] >= [2] z + [1] = or(y,implies(x,z)) implies(not(x),or(y,z)) = [2] z + [1] >= [2] z + [1] = implies(y,or(x,z)) 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: implies(not(x),or(y,z)) -> implies(y,or(x,z)) - Weak TRS: implies(x,or(y,z)) -> or(y,implies(x,z)) implies(not(x),y) -> or(x,y) - Signature: {implies/2} / {not/1,or/2} - Obligation: innermost runtime complexity wrt. defined symbols {implies} and constructors {not,or} + 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(or) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(implies) = [1] x1 + [1] x2 + [0] p(not) = [1] x1 + [2] p(or) = [1] x1 + [1] x2 + [1] Following rules are strictly oriented: implies(not(x),or(y,z)) = [1] x + [1] y + [1] z + [3] > [1] x + [1] y + [1] z + [1] = implies(y,or(x,z)) Following rules are (at-least) weakly oriented: implies(x,or(y,z)) = [1] x + [1] y + [1] z + [1] >= [1] x + [1] y + [1] z + [1] = or(y,implies(x,z)) implies(not(x),y) = [1] x + [1] y + [2] >= [1] x + [1] y + [1] = or(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: implies(x,or(y,z)) -> or(y,implies(x,z)) implies(not(x),y) -> or(x,y) implies(not(x),or(y,z)) -> implies(y,or(x,z)) - Signature: {implies/2} / {not/1,or/2} - Obligation: innermost runtime complexity wrt. defined symbols {implies} and constructors {not,or} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))