WORST_CASE(?,O(n^2)) * Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) rest(cons(x,y)) -> sent(y) rest(nil()) -> sent(nil()) top(sent(x)) -> top(check(rest(x))) - Signature: {check/1,rest/1,top/1} / {cons/2,nil/0,sent/1} - Obligation: innermost runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent} + 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: Following symbols are considered usable: all TcT has computed the following interpretation: p(check) = [1] x1 + [6] p(cons) = [1] x1 + [1] x2 + [0] p(nil) = [0] p(rest) = [1] x1 + [0] p(sent) = [1] x1 + [15] p(top) = [1] x1 + [0] Following rules are strictly oriented: check(cons(x,y)) = [1] x + [1] y + [6] > [1] x + [1] y + [0] = cons(x,y) top(sent(x)) = [1] x + [15] > [1] x + [6] = top(check(rest(x))) Following rules are (at-least) weakly oriented: check(cons(x,y)) = [1] x + [1] y + [6] >= [1] x + [1] y + [6] = cons(x,check(y)) check(cons(x,y)) = [1] x + [1] y + [6] >= [1] x + [1] y + [6] = cons(check(x),y) check(rest(x)) = [1] x + [6] >= [1] x + [6] = rest(check(x)) check(sent(x)) = [1] x + [21] >= [1] x + [21] = sent(check(x)) rest(cons(x,y)) = [1] x + [1] y + [0] >= [1] y + [15] = sent(y) rest(nil()) = [0] >= [15] = sent(nil()) 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: check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) rest(cons(x,y)) -> sent(y) rest(nil()) -> sent(nil()) - Weak TRS: check(cons(x,y)) -> cons(x,y) top(sent(x)) -> top(check(rest(x))) - Signature: {check/1,rest/1,top/1} / {cons/2,nil/0,sent/1} - Obligation: innermost runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent} + 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: Following symbols are considered usable: all TcT has computed the following interpretation: p(check) = [1] x1 + [1] p(cons) = [1] x1 + [1] x2 + [8] p(nil) = [0] p(rest) = [1] x1 + [0] p(sent) = [1] x1 + [1] p(top) = [1] x1 + [0] Following rules are strictly oriented: rest(cons(x,y)) = [1] x + [1] y + [8] > [1] y + [1] = sent(y) Following rules are (at-least) weakly oriented: check(cons(x,y)) = [1] x + [1] y + [9] >= [1] x + [1] y + [8] = cons(x,y) check(cons(x,y)) = [1] x + [1] y + [9] >= [1] x + [1] y + [9] = cons(x,check(y)) check(cons(x,y)) = [1] x + [1] y + [9] >= [1] x + [1] y + [9] = cons(check(x),y) check(rest(x)) = [1] x + [1] >= [1] x + [1] = rest(check(x)) check(sent(x)) = [1] x + [2] >= [1] x + [2] = sent(check(x)) rest(nil()) = [0] >= [1] = sent(nil()) top(sent(x)) = [1] x + [1] >= [1] x + [1] = top(check(rest(x))) 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: check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) rest(nil()) -> sent(nil()) - Weak TRS: check(cons(x,y)) -> cons(x,y) rest(cons(x,y)) -> sent(y) top(sent(x)) -> top(check(rest(x))) - Signature: {check/1,rest/1,top/1} / {cons/2,nil/0,sent/1} - Obligation: innermost runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent} + 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: Following symbols are considered usable: {check,rest,top} TcT has computed the following interpretation: p(check) = [1 3] x1 + [0] [0 1] [0] p(cons) = [1 0] x1 + [1 3] x2 + [2] [0 1] [0 1] [3] p(nil) = [0] [0] p(rest) = [1 0] x1 + [0] [0 1] [0] p(sent) = [1 3] x1 + [0] [0 1] [0] p(top) = [1 0] x1 + [0] [0 1] [0] Following rules are strictly oriented: check(cons(x,y)) = [1 3] x + [1 6] y + [11] [0 1] [0 1] [3] > [1 0] x + [1 6] y + [2] [0 1] [0 1] [3] = cons(x,check(y)) check(cons(x,y)) = [1 3] x + [1 6] y + [11] [0 1] [0 1] [3] > [1 3] x + [1 3] y + [2] [0 1] [0 1] [3] = cons(check(x),y) Following rules are (at-least) weakly oriented: check(cons(x,y)) = [1 3] x + [1 6] y + [11] [0 1] [0 1] [3] >= [1 0] x + [1 3] y + [2] [0 1] [0 1] [3] = cons(x,y) check(rest(x)) = [1 3] x + [0] [0 1] [0] >= [1 3] x + [0] [0 1] [0] = rest(check(x)) check(sent(x)) = [1 6] x + [0] [0 1] [0] >= [1 6] x + [0] [0 1] [0] = sent(check(x)) rest(cons(x,y)) = [1 0] x + [1 3] y + [2] [0 1] [0 1] [3] >= [1 3] y + [0] [0 1] [0] = sent(y) rest(nil()) = [0] [0] >= [0] [0] = sent(nil()) top(sent(x)) = [1 3] x + [0] [0 1] [0] >= [1 3] x + [0] [0 1] [0] = top(check(rest(x))) * Step 4: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) rest(nil()) -> sent(nil()) - Weak TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) rest(cons(x,y)) -> sent(y) top(sent(x)) -> top(check(rest(x))) - Signature: {check/1,rest/1,top/1} / {cons/2,nil/0,sent/1} - Obligation: innermost runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 3 non-zero interpretation-entries in the diagonal of the component-wise maxima): Following symbols are considered usable: {check,rest,top} TcT has computed the following interpretation: p(check) = [1 0 0 0] [0] [0 0 1 0] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(cons) = [1 0 0 1] [1 0 0 0] [1] [0 0 0 0] x1 + [0 0 1 1] x2 + [1] [0 0 0 0] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] p(nil) = [0] [1] [0] [1] p(rest) = [1 0 0 1] [0] [0 1 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(sent) = [1 0 0 0] [0] [0 0 1 1] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(top) = [1 1 0 0] [0] [1 1 1 0] x1 + [0] [1 1 0 0] [0] [0 0 0 0] [1] Following rules are strictly oriented: rest(nil()) = [1] [1] [0] [0] > [0] [1] [0] [0] = sent(nil()) Following rules are (at-least) weakly oriented: check(cons(x,y)) = [1 0 0 1] [1 0 0 0] [1] [0 0 0 0] x + [0 0 1 1] y + [1] [0 0 0 0] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 1] [1 0 0 0] [1] [0 0 0 0] x + [0 0 1 1] y + [1] [0 0 0 0] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] = cons(x,y) check(cons(x,y)) = [1 0 0 1] [1 0 0 0] [1] [0 0 0 0] x + [0 0 1 1] y + [1] [0 0 0 0] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 1] [1 0 0 0] [1] [0 0 0 0] x + [0 0 1 0] y + [1] [0 0 0 0] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0] = cons(x,check(y)) check(cons(x,y)) = [1 0 0 1] [1 0 0 0] [1] [0 0 0 0] x + [0 0 1 1] y + [1] [0 0 0 0] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 0] [1 0 0 0] [1] [0 0 0 0] x + [0 0 1 1] y + [1] [0 0 0 0] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] = cons(check(x),y) check(rest(x)) = [1 0 0 1] [0] [0 0 1 0] x + [0] [0 0 1 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [0] [0 0 1 0] x + [0] [0 0 1 0] [0] [0 0 0 0] [0] = rest(check(x)) check(sent(x)) = [1 0 0 0] [0] [0 0 1 0] x + [0] [0 0 1 0] [0] [0 0 0 0] [0] >= [1 0 0 0] [0] [0 0 1 0] x + [0] [0 0 1 0] [0] [0 0 0 0] [0] = sent(check(x)) rest(cons(x,y)) = [1 0 0 1] [1 0 0 0] [1] [0 0 0 0] x + [0 0 1 1] y + [1] [0 0 0 0] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 0] [0] [0 0 1 1] y + [0] [0 0 1 0] [0] [0 0 0 0] [0] = sent(y) top(sent(x)) = [1 0 1 1] [0] [1 0 2 1] x + [0] [1 0 1 1] [0] [0 0 0 0] [1] >= [1 0 1 1] [0] [1 0 2 1] x + [0] [1 0 1 1] [0] [0 0 0 0] [1] = top(check(rest(x))) * Step 5: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) - Weak TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) rest(cons(x,y)) -> sent(y) rest(nil()) -> sent(nil()) top(sent(x)) -> top(check(rest(x))) - Signature: {check/1,rest/1,top/1} / {cons/2,nil/0,sent/1} - Obligation: innermost runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 3 non-zero interpretation-entries in the diagonal of the component-wise maxima): Following symbols are considered usable: {check,rest,top} TcT has computed the following interpretation: p(check) = [1 0 1 0] [0] [0 0 1 0] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(cons) = [1 0 0 0] [1 0 1 1] [0] [0 0 1 0] x1 + [0 0 1 1] x2 + [1] [0 0 1 1] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] p(nil) = [0] [1] [0] [1] p(rest) = [1 0 0 0] [1] [0 1 0 1] x1 + [0] [0 0 1 1] [0] [0 0 0 0] [0] p(sent) = [1 0 1 1] [0] [0 0 1 1] x1 + [1] [0 0 1 0] [1] [0 0 0 0] [0] p(top) = [1 1 0 0] [0] [0 0 0 0] x1 + [0] [0 1 0 0] [1] [1 1 0 0] [0] Following rules are strictly oriented: check(sent(x)) = [1 0 2 1] [1] [0 0 1 0] x + [1] [0 0 1 0] [1] [0 0 0 0] [0] > [1 0 2 0] [0] [0 0 1 0] x + [1] [0 0 1 0] [1] [0 0 0 0] [0] = sent(check(x)) Following rules are (at-least) weakly oriented: check(cons(x,y)) = [1 0 1 1] [1 0 2 2] [1] [0 0 1 1] x + [0 0 1 1] y + [1] [0 0 1 1] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 0] [1 0 1 1] [0] [0 0 1 0] x + [0 0 1 1] y + [1] [0 0 1 1] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] = cons(x,y) check(cons(x,y)) = [1 0 1 1] [1 0 2 2] [1] [0 0 1 1] x + [0 0 1 1] y + [1] [0 0 1 1] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 0 0] [1 0 2 0] [0] [0 0 1 0] x + [0 0 1 0] y + [1] [0 0 1 1] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0] = cons(x,check(y)) check(cons(x,y)) = [1 0 1 1] [1 0 2 2] [1] [0 0 1 1] x + [0 0 1 1] y + [1] [0 0 1 1] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 1 0] [1 0 1 1] [0] [0 0 1 0] x + [0 0 1 1] y + [1] [0 0 1 0] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] = cons(check(x),y) check(rest(x)) = [1 0 1 1] [1] [0 0 1 1] x + [0] [0 0 1 1] [0] [0 0 0 0] [0] >= [1 0 1 0] [1] [0 0 1 0] x + [0] [0 0 1 0] [0] [0 0 0 0] [0] = rest(check(x)) rest(cons(x,y)) = [1 0 0 0] [1 0 1 1] [1] [0 0 1 0] x + [0 0 1 1] y + [1] [0 0 1 1] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 1 1] [0] [0 0 1 1] y + [1] [0 0 1 0] [1] [0 0 0 0] [0] = sent(y) rest(nil()) = [1] [2] [1] [0] >= [1] [2] [1] [0] = sent(nil()) top(sent(x)) = [1 0 2 2] [1] [0 0 0 0] x + [0] [0 0 1 1] [2] [1 0 2 2] [1] >= [1 0 2 2] [1] [0 0 0 0] x + [0] [0 0 1 1] [1] [1 0 2 2] [1] = top(check(rest(x))) * Step 6: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: check(rest(x)) -> rest(check(x)) - Weak TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(sent(x)) -> sent(check(x)) rest(cons(x,y)) -> sent(y) rest(nil()) -> sent(nil()) top(sent(x)) -> top(check(rest(x))) - Signature: {check/1,rest/1,top/1} / {cons/2,nil/0,sent/1} - Obligation: innermost runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 3, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 3 non-zero interpretation-entries in the diagonal of the component-wise maxima): Following symbols are considered usable: {check,rest,top} TcT has computed the following interpretation: p(check) = [1 0 1 0] [0] [0 0 1 0] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] p(cons) = [1 0 1 0] [1 0 1 0] [1] [0 0 1 1] x1 + [0 0 1 1] x2 + [1] [0 0 1 1] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] p(nil) = [1] [1] [0] [1] p(rest) = [1 0 0 1] [0] [0 1 0 1] x1 + [0] [0 0 1 0] [1] [0 0 0 0] [0] p(sent) = [1 0 1 0] [1] [0 0 1 1] x1 + [1] [0 0 1 0] [1] [0 0 0 0] [0] p(top) = [1 1 0 0] [1] [0 0 0 0] x1 + [0] [1 1 0 0] [1] [0 0 0 0] [0] Following rules are strictly oriented: check(rest(x)) = [1 0 1 1] [1] [0 0 1 0] x + [1] [0 0 1 0] [1] [0 0 0 0] [0] > [1 0 1 0] [0] [0 0 1 0] x + [0] [0 0 1 0] [1] [0 0 0 0] [0] = rest(check(x)) Following rules are (at-least) weakly oriented: check(cons(x,y)) = [1 0 2 1] [1 0 2 1] [2] [0 0 1 1] x + [0 0 1 1] y + [1] [0 0 1 1] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 1 0] [1 0 1 0] [1] [0 0 1 1] x + [0 0 1 1] y + [1] [0 0 1 1] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] = cons(x,y) check(cons(x,y)) = [1 0 2 1] [1 0 2 1] [2] [0 0 1 1] x + [0 0 1 1] y + [1] [0 0 1 1] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 1 0] [1 0 2 0] [1] [0 0 1 1] x + [0 0 1 0] y + [1] [0 0 1 1] [0 0 1 0] [1] [0 0 0 0] [0 0 0 0] [0] = cons(x,check(y)) check(cons(x,y)) = [1 0 2 1] [1 0 2 1] [2] [0 0 1 1] x + [0 0 1 1] y + [1] [0 0 1 1] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] >= [1 0 2 0] [1 0 1 0] [1] [0 0 1 0] x + [0 0 1 1] y + [1] [0 0 1 0] [0 0 1 1] [1] [0 0 0 0] [0 0 0 0] [0] = cons(check(x),y) check(sent(x)) = [1 0 2 0] [2] [0 0 1 0] x + [1] [0 0 1 0] [1] [0 0 0 0] [0] >= [1 0 2 0] [1] [0 0 1 0] x + [1] [0 0 1 0] [1] [0 0 0 0] [0] = sent(check(x)) rest(cons(x,y)) = [1 0 1 0] [1 0 1 0] [1] [0 0 1 1] x + [0 0 1 1] y + [1] [0 0 1 1] [0 0 1 1] [2] [0 0 0 0] [0 0 0 0] [0] >= [1 0 1 0] [1] [0 0 1 1] y + [1] [0 0 1 0] [1] [0 0 0 0] [0] = sent(y) rest(nil()) = [2] [2] [1] [0] >= [2] [2] [1] [0] = sent(nil()) top(sent(x)) = [1 0 2 1] [3] [0 0 0 0] x + [0] [1 0 2 1] [3] [0 0 0 0] [0] >= [1 0 2 1] [3] [0 0 0 0] x + [0] [1 0 2 1] [3] [0 0 0 0] [0] = top(check(rest(x))) * Step 7: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: check(cons(x,y)) -> cons(x,y) check(cons(x,y)) -> cons(x,check(y)) check(cons(x,y)) -> cons(check(x),y) check(rest(x)) -> rest(check(x)) check(sent(x)) -> sent(check(x)) rest(cons(x,y)) -> sent(y) rest(nil()) -> sent(nil()) top(sent(x)) -> top(check(rest(x))) - Signature: {check/1,rest/1,top/1} / {cons/2,nil/0,sent/1} - Obligation: innermost runtime complexity wrt. defined symbols {check,rest,top} and constructors {cons,nil,sent} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))