WORST_CASE(?,O(n^2)) * Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: append(@l1,@l2) -> append#1(@l1,@l2) append#1(::(@x,@xs),@l2) -> ::(@x,append(@xs,@l2)) append#1(nil(),@l2) -> @l2 subtrees(@t) -> subtrees#1(@t) subtrees#1(leaf()) -> nil() subtrees#1(node(@x,@t1,@t2)) -> subtrees#2(subtrees(@t1),@t1,@t2,@x) subtrees#2(@l1,@t1,@t2,@x) -> subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x) subtrees#3(@l2,@l1,@t1,@t2,@x) -> ::(node(@x,@t1,@t2),append(@l1,@l2)) - Signature: {append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {::/2,leaf/0,nil/0,node/3} - Obligation: innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2 ,subtrees#3} and constructors {::,leaf,nil,node} + 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(::) = {2}, uargs(subtrees#2) = {1}, uargs(subtrees#3) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(::) = [1] x2 + [10] p(append) = [1] x2 + [1] p(append#1) = [1] x2 + [0] p(leaf) = [0] p(nil) = [0] p(node) = [1] x2 + [1] x3 + [0] p(subtrees) = [0] p(subtrees#1) = [0] p(subtrees#2) = [1] x1 + [0] p(subtrees#3) = [1] x1 + [1] x2 + [0] Following rules are strictly oriented: append(@l1,@l2) = [1] @l2 + [1] > [1] @l2 + [0] = append#1(@l1,@l2) Following rules are (at-least) weakly oriented: append#1(::(@x,@xs),@l2) = [1] @l2 + [0] >= [1] @l2 + [11] = ::(@x,append(@xs,@l2)) append#1(nil(),@l2) = [1] @l2 + [0] >= [1] @l2 + [0] = @l2 subtrees(@t) = [0] >= [0] = subtrees#1(@t) subtrees#1(leaf()) = [0] >= [0] = nil() subtrees#1(node(@x,@t1,@t2)) = [0] >= [0] = subtrees#2(subtrees(@t1),@t1,@t2,@x) subtrees#2(@l1,@t1,@t2,@x) = [1] @l1 + [0] >= [1] @l1 + [0] = subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x) subtrees#3(@l2,@l1,@t1,@t2,@x) = [1] @l1 + [1] @l2 + [0] >= [1] @l2 + [11] = ::(node(@x,@t1,@t2),append(@l1,@l2)) 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: append#1(::(@x,@xs),@l2) -> ::(@x,append(@xs,@l2)) append#1(nil(),@l2) -> @l2 subtrees(@t) -> subtrees#1(@t) subtrees#1(leaf()) -> nil() subtrees#1(node(@x,@t1,@t2)) -> subtrees#2(subtrees(@t1),@t1,@t2,@x) subtrees#2(@l1,@t1,@t2,@x) -> subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x) subtrees#3(@l2,@l1,@t1,@t2,@x) -> ::(node(@x,@t1,@t2),append(@l1,@l2)) - Weak TRS: append(@l1,@l2) -> append#1(@l1,@l2) - Signature: {append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {::/2,leaf/0,nil/0,node/3} - Obligation: innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2 ,subtrees#3} and constructors {::,leaf,nil,node} + 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(::) = {2}, uargs(subtrees#2) = {1}, uargs(subtrees#3) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(::) = [1] x2 + [0] p(append) = [1] x1 + [1] x2 + [0] p(append#1) = [1] x1 + [1] x2 + [0] p(leaf) = [0] p(nil) = [0] p(node) = [1] x1 + [1] x2 + [1] x3 + [0] p(subtrees) = [0] p(subtrees#1) = [0] p(subtrees#2) = [1] x1 + [0] p(subtrees#3) = [1] x1 + [1] x2 + [1] Following rules are strictly oriented: subtrees#3(@l2,@l1,@t1,@t2,@x) = [1] @l1 + [1] @l2 + [1] > [1] @l1 + [1] @l2 + [0] = ::(node(@x,@t1,@t2),append(@l1,@l2)) Following rules are (at-least) weakly oriented: append(@l1,@l2) = [1] @l1 + [1] @l2 + [0] >= [1] @l1 + [1] @l2 + [0] = append#1(@l1,@l2) append#1(::(@x,@xs),@l2) = [1] @l2 + [1] @xs + [0] >= [1] @l2 + [1] @xs + [0] = ::(@x,append(@xs,@l2)) append#1(nil(),@l2) = [1] @l2 + [0] >= [1] @l2 + [0] = @l2 subtrees(@t) = [0] >= [0] = subtrees#1(@t) subtrees#1(leaf()) = [0] >= [0] = nil() subtrees#1(node(@x,@t1,@t2)) = [0] >= [0] = subtrees#2(subtrees(@t1),@t1,@t2,@x) subtrees#2(@l1,@t1,@t2,@x) = [1] @l1 + [0] >= [1] @l1 + [1] = subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: append#1(::(@x,@xs),@l2) -> ::(@x,append(@xs,@l2)) append#1(nil(),@l2) -> @l2 subtrees(@t) -> subtrees#1(@t) subtrees#1(leaf()) -> nil() subtrees#1(node(@x,@t1,@t2)) -> subtrees#2(subtrees(@t1),@t1,@t2,@x) subtrees#2(@l1,@t1,@t2,@x) -> subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x) - Weak TRS: append(@l1,@l2) -> append#1(@l1,@l2) subtrees#3(@l2,@l1,@t1,@t2,@x) -> ::(node(@x,@t1,@t2),append(@l1,@l2)) - Signature: {append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {::/2,leaf/0,nil/0,node/3} - Obligation: innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2 ,subtrees#3} and constructors {::,leaf,nil,node} + 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(::) = {2}, uargs(subtrees#2) = {1}, uargs(subtrees#3) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(::) = [1] x2 + [0] p(append) = [1] x2 + [0] p(append#1) = [1] x2 + [0] p(leaf) = [0] p(nil) = [0] p(node) = [1] x3 + [1] p(subtrees) = [8] p(subtrees#1) = [2] p(subtrees#2) = [1] x1 + [8] p(subtrees#3) = [1] x1 + [1] x2 + [0] Following rules are strictly oriented: subtrees(@t) = [8] > [2] = subtrees#1(@t) subtrees#1(leaf()) = [2] > [0] = nil() Following rules are (at-least) weakly oriented: append(@l1,@l2) = [1] @l2 + [0] >= [1] @l2 + [0] = append#1(@l1,@l2) append#1(::(@x,@xs),@l2) = [1] @l2 + [0] >= [1] @l2 + [0] = ::(@x,append(@xs,@l2)) append#1(nil(),@l2) = [1] @l2 + [0] >= [1] @l2 + [0] = @l2 subtrees#1(node(@x,@t1,@t2)) = [2] >= [16] = subtrees#2(subtrees(@t1),@t1,@t2,@x) subtrees#2(@l1,@t1,@t2,@x) = [1] @l1 + [8] >= [1] @l1 + [8] = subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x) subtrees#3(@l2,@l1,@t1,@t2,@x) = [1] @l1 + [1] @l2 + [0] >= [1] @l2 + [0] = ::(node(@x,@t1,@t2),append(@l1,@l2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: append#1(::(@x,@xs),@l2) -> ::(@x,append(@xs,@l2)) append#1(nil(),@l2) -> @l2 subtrees#1(node(@x,@t1,@t2)) -> subtrees#2(subtrees(@t1),@t1,@t2,@x) subtrees#2(@l1,@t1,@t2,@x) -> subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x) - Weak TRS: append(@l1,@l2) -> append#1(@l1,@l2) subtrees(@t) -> subtrees#1(@t) subtrees#1(leaf()) -> nil() subtrees#3(@l2,@l1,@t1,@t2,@x) -> ::(node(@x,@t1,@t2),append(@l1,@l2)) - Signature: {append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {::/2,leaf/0,nil/0,node/3} - Obligation: innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2 ,subtrees#3} and constructors {::,leaf,nil,node} + 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(::) = {2}, uargs(subtrees#2) = {1}, uargs(subtrees#3) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(::) = [1] x2 + [0] p(append) = [1] x1 + [1] x2 + [0] p(append#1) = [1] x1 + [1] x2 + [0] p(leaf) = [0] p(nil) = [0] p(node) = [1] x2 + [1] x3 + [5] p(subtrees) = [1] x1 + [0] p(subtrees#1) = [1] x1 + [0] p(subtrees#2) = [1] x1 + [1] x3 + [1] p(subtrees#3) = [1] x1 + [1] x2 + [0] Following rules are strictly oriented: subtrees#1(node(@x,@t1,@t2)) = [1] @t1 + [1] @t2 + [5] > [1] @t1 + [1] @t2 + [1] = subtrees#2(subtrees(@t1),@t1,@t2,@x) subtrees#2(@l1,@t1,@t2,@x) = [1] @l1 + [1] @t2 + [1] > [1] @l1 + [1] @t2 + [0] = subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x) Following rules are (at-least) weakly oriented: append(@l1,@l2) = [1] @l1 + [1] @l2 + [0] >= [1] @l1 + [1] @l2 + [0] = append#1(@l1,@l2) append#1(::(@x,@xs),@l2) = [1] @l2 + [1] @xs + [0] >= [1] @l2 + [1] @xs + [0] = ::(@x,append(@xs,@l2)) append#1(nil(),@l2) = [1] @l2 + [0] >= [1] @l2 + [0] = @l2 subtrees(@t) = [1] @t + [0] >= [1] @t + [0] = subtrees#1(@t) subtrees#1(leaf()) = [0] >= [0] = nil() subtrees#3(@l2,@l1,@t1,@t2,@x) = [1] @l1 + [1] @l2 + [0] >= [1] @l1 + [1] @l2 + [0] = ::(node(@x,@t1,@t2),append(@l1,@l2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: append#1(::(@x,@xs),@l2) -> ::(@x,append(@xs,@l2)) append#1(nil(),@l2) -> @l2 - Weak TRS: append(@l1,@l2) -> append#1(@l1,@l2) subtrees(@t) -> subtrees#1(@t) subtrees#1(leaf()) -> nil() subtrees#1(node(@x,@t1,@t2)) -> subtrees#2(subtrees(@t1),@t1,@t2,@x) subtrees#2(@l1,@t1,@t2,@x) -> subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x) subtrees#3(@l2,@l1,@t1,@t2,@x) -> ::(node(@x,@t1,@t2),append(@l1,@l2)) - Signature: {append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {::/2,leaf/0,nil/0,node/3} - Obligation: innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2 ,subtrees#3} and constructors {::,leaf,nil,node} + 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(::) = {2}, uargs(subtrees#2) = {1}, uargs(subtrees#3) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(::) = [1] x2 + [6] p(append) = [1] x1 + [1] x2 + [0] p(append#1) = [1] x1 + [1] x2 + [0] p(leaf) = [1] p(nil) = [1] p(node) = [1] x2 + [1] x3 + [6] p(subtrees) = [1] x1 + [0] p(subtrees#1) = [1] x1 + [0] p(subtrees#2) = [1] x1 + [1] x3 + [6] p(subtrees#3) = [1] x1 + [1] x2 + [6] Following rules are strictly oriented: append#1(nil(),@l2) = [1] @l2 + [1] > [1] @l2 + [0] = @l2 Following rules are (at-least) weakly oriented: append(@l1,@l2) = [1] @l1 + [1] @l2 + [0] >= [1] @l1 + [1] @l2 + [0] = append#1(@l1,@l2) append#1(::(@x,@xs),@l2) = [1] @l2 + [1] @xs + [6] >= [1] @l2 + [1] @xs + [6] = ::(@x,append(@xs,@l2)) subtrees(@t) = [1] @t + [0] >= [1] @t + [0] = subtrees#1(@t) subtrees#1(leaf()) = [1] >= [1] = nil() subtrees#1(node(@x,@t1,@t2)) = [1] @t1 + [1] @t2 + [6] >= [1] @t1 + [1] @t2 + [6] = subtrees#2(subtrees(@t1),@t1,@t2,@x) subtrees#2(@l1,@t1,@t2,@x) = [1] @l1 + [1] @t2 + [6] >= [1] @l1 + [1] @t2 + [6] = subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x) subtrees#3(@l2,@l1,@t1,@t2,@x) = [1] @l1 + [1] @l2 + [6] >= [1] @l1 + [1] @l2 + [6] = ::(node(@x,@t1,@t2),append(@l1,@l2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: append#1(::(@x,@xs),@l2) -> ::(@x,append(@xs,@l2)) - Weak TRS: append(@l1,@l2) -> append#1(@l1,@l2) append#1(nil(),@l2) -> @l2 subtrees(@t) -> subtrees#1(@t) subtrees#1(leaf()) -> nil() subtrees#1(node(@x,@t1,@t2)) -> subtrees#2(subtrees(@t1),@t1,@t2,@x) subtrees#2(@l1,@t1,@t2,@x) -> subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x) subtrees#3(@l2,@l1,@t1,@t2,@x) -> ::(node(@x,@t1,@t2),append(@l1,@l2)) - Signature: {append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {::/2,leaf/0,nil/0,node/3} - Obligation: innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2 ,subtrees#3} and constructors {::,leaf,nil,node} + 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(::) = {2}, uargs(subtrees#2) = {1}, uargs(subtrees#3) = {1} Following symbols are considered usable: {append,append#1,subtrees,subtrees#1,subtrees#2,subtrees#3} TcT has computed the following interpretation: p(::) = [1 0] x2 + [0] [0 1] [4] p(append) = [0 1] x1 + [1 0] x2 + [0] [0 1] [0 1] [0] p(append#1) = [0 1] x1 + [1 0] x2 + [0] [0 1] [0 1] [0] p(leaf) = [5] [0] p(nil) = [1] [0] p(node) = [1 6] x2 + [1 0] x3 + [0] [0 1] [0 1] [4] p(subtrees) = [1 2] x1 + [4] [0 2] [2] p(subtrees#1) = [1 2] x1 + [4] [0 2] [0] p(subtrees#2) = [1 2] x1 + [1 2] x3 + [4] [0 1] [0 2] [6] p(subtrees#3) = [1 0] x1 + [1 2] x2 + [0] [0 1] [0 1] [4] Following rules are strictly oriented: append#1(::(@x,@xs),@l2) = [1 0] @l2 + [0 1] @xs + [4] [0 1] [0 1] [4] > [1 0] @l2 + [0 1] @xs + [0] [0 1] [0 1] [4] = ::(@x,append(@xs,@l2)) Following rules are (at-least) weakly oriented: append(@l1,@l2) = [0 1] @l1 + [1 0] @l2 + [0] [0 1] [0 1] [0] >= [0 1] @l1 + [1 0] @l2 + [0] [0 1] [0 1] [0] = append#1(@l1,@l2) append#1(nil(),@l2) = [1 0] @l2 + [0] [0 1] [0] >= [1 0] @l2 + [0] [0 1] [0] = @l2 subtrees(@t) = [1 2] @t + [4] [0 2] [2] >= [1 2] @t + [4] [0 2] [0] = subtrees#1(@t) subtrees#1(leaf()) = [9] [0] >= [1] [0] = nil() subtrees#1(node(@x,@t1,@t2)) = [1 8] @t1 + [1 2] @t2 + [12] [0 2] [0 2] [8] >= [1 6] @t1 + [1 2] @t2 + [12] [0 2] [0 2] [8] = subtrees#2(subtrees(@t1),@t1,@t2,@x) subtrees#2(@l1,@t1,@t2,@x) = [1 2] @l1 + [1 2] @t2 + [4] [0 1] [0 2] [6] >= [1 2] @l1 + [1 2] @t2 + [4] [0 1] [0 2] [6] = subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x) subtrees#3(@l2,@l1,@t1,@t2,@x) = [1 2] @l1 + [1 0] @l2 + [0] [0 1] [0 1] [4] >= [0 1] @l1 + [1 0] @l2 + [0] [0 1] [0 1] [4] = ::(node(@x,@t1,@t2),append(@l1,@l2)) * Step 7: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: append(@l1,@l2) -> append#1(@l1,@l2) append#1(::(@x,@xs),@l2) -> ::(@x,append(@xs,@l2)) append#1(nil(),@l2) -> @l2 subtrees(@t) -> subtrees#1(@t) subtrees#1(leaf()) -> nil() subtrees#1(node(@x,@t1,@t2)) -> subtrees#2(subtrees(@t1),@t1,@t2,@x) subtrees#2(@l1,@t1,@t2,@x) -> subtrees#3(subtrees(@t2),@l1,@t1,@t2,@x) subtrees#3(@l2,@l1,@t1,@t2,@x) -> ::(node(@x,@t1,@t2),append(@l1,@l2)) - Signature: {append/2,append#1/2,subtrees/1,subtrees#1/1,subtrees#2/4,subtrees#3/5} / {::/2,leaf/0,nil/0,node/3} - Obligation: innermost runtime complexity wrt. defined symbols {append,append#1,subtrees,subtrees#1,subtrees#2 ,subtrees#3} and constructors {::,leaf,nil,node} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))