WORST_CASE(?,O(n^1)) * Step 1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__c() -> b() a__c() -> c() a__f(X1,X2,X3) -> f(X1,X2,X3) a__f(b(),X,c()) -> a__f(X,a__c(),X) mark(b()) -> b() mark(c()) -> a__c() mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Signature: {a__c/0,a__f/3,mark/1} / {b/0,c/0,f/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {b,c,f} + 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(a__f) = {2} Following symbols are considered usable: {a__c,a__f,mark} TcT has computed the following interpretation: p(a__c) = [5] p(a__f) = [1] x2 + [1] x3 + [4] p(b) = [0] p(c) = [5] p(f) = [1] x2 + [1] x3 + [4] p(mark) = [1] x1 + [6] Following rules are strictly oriented: a__c() = [5] > [0] = b() mark(b()) = [6] > [0] = b() mark(c()) = [11] > [5] = a__c() Following rules are (at-least) weakly oriented: a__c() = [5] >= [5] = c() a__f(X1,X2,X3) = [1] X2 + [1] X3 + [4] >= [1] X2 + [1] X3 + [4] = f(X1,X2,X3) a__f(b(),X,c()) = [1] X + [9] >= [1] X + [9] = a__f(X,a__c(),X) mark(f(X1,X2,X3)) = [1] X2 + [1] X3 + [10] >= [1] X2 + [1] X3 + [10] = a__f(X1,mark(X2),X3) * Step 2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__c() -> c() a__f(X1,X2,X3) -> f(X1,X2,X3) a__f(b(),X,c()) -> a__f(X,a__c(),X) mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Weak TRS: a__c() -> b() mark(b()) -> b() mark(c()) -> a__c() - Signature: {a__c/0,a__f/3,mark/1} / {b/0,c/0,f/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {b,c,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__c) = [0] p(a__f) = [1] x1 + [1] x2 + [1] p(b) = [0] p(c) = [1] p(f) = [1] x1 + [1] x2 + [2] p(mark) = [8] x1 + [9] Following rules are strictly oriented: mark(f(X1,X2,X3)) = [8] X1 + [8] X2 + [25] > [1] X1 + [8] X2 + [10] = a__f(X1,mark(X2),X3) Following rules are (at-least) weakly oriented: a__c() = [0] >= [0] = b() a__c() = [0] >= [1] = c() a__f(X1,X2,X3) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [2] = f(X1,X2,X3) a__f(b(),X,c()) = [1] X + [1] >= [1] X + [1] = a__f(X,a__c(),X) mark(b()) = [9] >= [0] = b() mark(c()) = [17] >= [0] = a__c() 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__c() -> c() a__f(X1,X2,X3) -> f(X1,X2,X3) a__f(b(),X,c()) -> a__f(X,a__c(),X) - Weak TRS: a__c() -> b() mark(b()) -> b() mark(c()) -> a__c() mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Signature: {a__c/0,a__f/3,mark/1} / {b/0,c/0,f/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {b,c,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__c) = [0] p(a__f) = [1] x1 + [1] x2 + [8] p(b) = [0] p(c) = [0] p(f) = [1] x1 + [1] x2 + [2] p(mark) = [4] x1 + [4] Following rules are strictly oriented: a__f(X1,X2,X3) = [1] X1 + [1] X2 + [8] > [1] X1 + [1] X2 + [2] = f(X1,X2,X3) Following rules are (at-least) weakly oriented: a__c() = [0] >= [0] = b() a__c() = [0] >= [0] = c() a__f(b(),X,c()) = [1] X + [8] >= [1] X + [8] = a__f(X,a__c(),X) mark(b()) = [4] >= [0] = b() mark(c()) = [4] >= [0] = a__c() mark(f(X1,X2,X3)) = [4] X1 + [4] X2 + [12] >= [1] X1 + [4] X2 + [12] = a__f(X1,mark(X2),X3) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__c() -> c() a__f(b(),X,c()) -> a__f(X,a__c(),X) - Weak TRS: a__c() -> b() a__f(X1,X2,X3) -> f(X1,X2,X3) mark(b()) -> b() mark(c()) -> a__c() mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Signature: {a__c/0,a__f/3,mark/1} / {b/0,c/0,f/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {b,c,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__c) = [0] p(a__f) = [1] x2 + [1] x3 + [0] p(b) = [0] p(c) = [5] p(f) = [1] x2 + [1] x3 + [0] p(mark) = [2] x1 + [4] Following rules are strictly oriented: a__f(b(),X,c()) = [1] X + [5] > [1] X + [0] = a__f(X,a__c(),X) Following rules are (at-least) weakly oriented: a__c() = [0] >= [0] = b() a__c() = [0] >= [5] = c() a__f(X1,X2,X3) = [1] X2 + [1] X3 + [0] >= [1] X2 + [1] X3 + [0] = f(X1,X2,X3) mark(b()) = [4] >= [0] = b() mark(c()) = [14] >= [0] = a__c() mark(f(X1,X2,X3)) = [2] X2 + [2] X3 + [4] >= [2] 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 5: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__c() -> c() - Weak TRS: a__c() -> b() a__f(X1,X2,X3) -> f(X1,X2,X3) a__f(b(),X,c()) -> a__f(X,a__c(),X) mark(b()) -> b() mark(c()) -> a__c() mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Signature: {a__c/0,a__f/3,mark/1} / {b/0,c/0,f/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {b,c,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__c) = [1] p(a__f) = [1] x1 + [1] x2 + [0] p(b) = [1] p(c) = [0] p(f) = [1] x1 + [1] x2 + [0] p(mark) = [1] x1 + [11] Following rules are strictly oriented: a__c() = [1] > [0] = c() Following rules are (at-least) weakly oriented: a__c() = [1] >= [1] = b() a__f(X1,X2,X3) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = f(X1,X2,X3) a__f(b(),X,c()) = [1] X + [1] >= [1] X + [1] = a__f(X,a__c(),X) mark(b()) = [12] >= [1] = b() mark(c()) = [11] >= [1] = a__c() mark(f(X1,X2,X3)) = [1] X1 + [1] X2 + [11] >= [1] X1 + [1] X2 + [11] = a__f(X1,mark(X2),X3) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__c() -> b() a__c() -> c() a__f(X1,X2,X3) -> f(X1,X2,X3) a__f(b(),X,c()) -> a__f(X,a__c(),X) mark(b()) -> b() mark(c()) -> a__c() mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Signature: {a__c/0,a__f/3,mark/1} / {b/0,c/0,f/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__c,a__f,mark} and constructors {b,c,f} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))