WORST_CASE(?,O(n^2)) * Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(X) -> len(X) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(a__add) = [1] x1 + [1] x2 + [0] p(a__from) = [1] x1 + [0] p(a__fst) = [1] x1 + [1] x2 + [0] p(a__len) = [1] x1 + [0] p(add) = [1] x2 + [0] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [0] p(fst) = [1] x1 + [0] p(len) = [1] x1 + [0] p(mark) = [0] p(nil) = [0] p(s) = [0] Following rules are strictly oriented: a__add(0(),X) = [1] X + [1] > [0] = mark(X) a__fst(0(),Z) = [1] Z + [1] > [0] = nil() Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X2 + [0] = add(X1,X2) a__add(s(X),Y) = [1] Y + [0] >= [0] = s(add(X,Y)) a__from(X) = [1] X + [0] >= [0] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [0] >= [1] X + [0] = from(X) a__fst(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [0] = fst(X1,X2) a__fst(s(X),cons(Y,Z)) = [1] Y + [0] >= [0] = cons(mark(Y),fst(X,Z)) a__len(X) = [1] X + [0] >= [1] X + [0] = len(X) a__len(cons(X,Z)) = [1] X + [0] >= [0] = s(len(Z)) a__len(nil()) = [0] >= [1] = 0() mark(0()) = [0] >= [1] = 0() mark(add(X1,X2)) = [0] >= [0] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [0] >= [0] = cons(mark(X1),X2) mark(from(X)) = [0] >= [0] = a__from(mark(X)) mark(fst(X1,X2)) = [0] >= [0] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [0] >= [0] = a__len(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [0] = s(X) 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: a__add(X1,X2) -> add(X1,X2) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(X) -> len(X) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Weak TRS: a__add(0(),X) -> mark(X) a__fst(0(),Z) -> nil() - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [4] p(a__add) = [1] x1 + [1] x2 + [1] p(a__from) = [1] x1 + [0] p(a__fst) = [1] x1 + [1] x2 + [0] p(a__len) = [1] x1 + [0] p(add) = [1] x1 + [1] x2 + [0] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [0] p(fst) = [1] x1 + [1] x2 + [0] p(len) = [1] x1 + [0] p(mark) = [1] x1 + [0] p(nil) = [0] p(s) = [0] Following rules are strictly oriented: a__add(X1,X2) = [1] X1 + [1] X2 + [1] > [1] X1 + [1] X2 + [0] = add(X1,X2) a__add(s(X),Y) = [1] Y + [1] > [0] = s(add(X,Y)) Following rules are (at-least) weakly oriented: a__add(0(),X) = [1] X + [5] >= [1] X + [0] = mark(X) a__from(X) = [1] X + [0] >= [1] X + [0] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [0] >= [1] X + [0] = from(X) a__fst(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = fst(X1,X2) a__fst(0(),Z) = [1] Z + [4] >= [0] = nil() a__fst(s(X),cons(Y,Z)) = [1] Y + [0] >= [1] Y + [0] = cons(mark(Y),fst(X,Z)) a__len(X) = [1] X + [0] >= [1] X + [0] = len(X) a__len(cons(X,Z)) = [1] X + [0] >= [0] = s(len(Z)) a__len(nil()) = [0] >= [4] = 0() mark(0()) = [4] >= [4] = 0() mark(add(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [1] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = cons(mark(X1),X2) mark(from(X)) = [1] X + [0] >= [1] X + [0] = a__from(mark(X)) mark(fst(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [1] X + [0] >= [1] X + [0] = a__len(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [0] = s(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: a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(X) -> len(X) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__fst(0(),Z) -> nil() - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__add) = [1] x1 + [1] x2 + [0] p(a__from) = [1] x1 + [4] p(a__fst) = [1] x1 + [1] x2 + [0] p(a__len) = [1] x1 + [0] p(add) = [1] x1 + [1] x2 + [0] p(cons) = [1] x1 + [0] p(from) = [1] x1 + [0] p(fst) = [1] x1 + [1] x2 + [0] p(len) = [1] x1 + [0] p(mark) = [1] x1 + [0] p(nil) = [0] p(s) = [0] Following rules are strictly oriented: a__from(X) = [1] X + [4] > [1] X + [0] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [4] > [1] X + [0] = from(X) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = add(X1,X2) a__add(0(),X) = [1] X + [0] >= [1] X + [0] = mark(X) a__add(s(X),Y) = [1] Y + [0] >= [0] = s(add(X,Y)) a__fst(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = fst(X1,X2) a__fst(0(),Z) = [1] Z + [0] >= [0] = nil() a__fst(s(X),cons(Y,Z)) = [1] Y + [0] >= [1] Y + [0] = cons(mark(Y),fst(X,Z)) a__len(X) = [1] X + [0] >= [1] X + [0] = len(X) a__len(cons(X,Z)) = [1] X + [0] >= [0] = s(len(Z)) a__len(nil()) = [0] >= [0] = 0() mark(0()) = [0] >= [0] = 0() mark(add(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = cons(mark(X1),X2) mark(from(X)) = [1] X + [0] >= [1] X + [4] = a__from(mark(X)) mark(fst(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [1] X + [0] >= [1] X + [0] = a__len(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [0] = s(X) 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: a__fst(X1,X2) -> fst(X1,X2) a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(X) -> len(X) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(0(),Z) -> nil() - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__add) = [1] x1 + [1] x2 + [1] p(a__from) = [1] x1 + [3] p(a__fst) = [1] x1 + [1] x2 + [1] p(a__len) = [1] x1 + [0] p(add) = [1] x1 + [1] x2 + [1] p(cons) = [1] x1 + [2] p(from) = [1] x1 + [0] p(fst) = [1] x1 + [1] x2 + [0] p(len) = [1] x1 + [0] p(mark) = [1] x1 + [0] p(nil) = [0] p(s) = [0] Following rules are strictly oriented: a__fst(X1,X2) = [1] X1 + [1] X2 + [1] > [1] X1 + [1] X2 + [0] = fst(X1,X2) a__fst(s(X),cons(Y,Z)) = [1] Y + [3] > [1] Y + [2] = cons(mark(Y),fst(X,Z)) a__len(cons(X,Z)) = [1] X + [2] > [0] = s(len(Z)) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = add(X1,X2) a__add(0(),X) = [1] X + [1] >= [1] X + [0] = mark(X) a__add(s(X),Y) = [1] Y + [1] >= [0] = s(add(X,Y)) a__from(X) = [1] X + [3] >= [1] X + [2] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [3] >= [1] X + [0] = from(X) a__fst(0(),Z) = [1] Z + [1] >= [0] = nil() a__len(X) = [1] X + [0] >= [1] X + [0] = len(X) a__len(nil()) = [0] >= [0] = 0() mark(0()) = [0] >= [0] = 0() mark(add(X1,X2)) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1] X1 + [2] >= [1] X1 + [2] = cons(mark(X1),X2) mark(from(X)) = [1] X + [0] >= [1] X + [3] = a__from(mark(X)) mark(fst(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [1] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [1] X + [0] >= [1] X + [0] = a__len(mark(X)) mark(nil()) = [0] >= [0] = nil() mark(s(X)) = [0] >= [0] = s(X) 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: a__len(X) -> len(X) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(cons(X,Z)) -> s(len(Z)) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [4] p(a__add) = [1] x1 + [1] x2 + [7] p(a__from) = [1] x1 + [2] p(a__fst) = [1] x1 + [1] x2 + [2] p(a__len) = [1] x1 + [0] p(add) = [1] x1 + [1] x2 + [7] p(cons) = [1] x1 + [0] p(from) = [0] p(fst) = [0] p(len) = [0] p(mark) = [2] p(nil) = [6] p(s) = [0] Following rules are strictly oriented: a__len(nil()) = [6] > [4] = 0() mark(s(X)) = [2] > [0] = s(X) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [1] X2 + [7] >= [1] X1 + [1] X2 + [7] = add(X1,X2) a__add(0(),X) = [1] X + [11] >= [2] = mark(X) a__add(s(X),Y) = [1] Y + [7] >= [0] = s(add(X,Y)) a__from(X) = [1] X + [2] >= [2] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [2] >= [0] = from(X) a__fst(X1,X2) = [1] X1 + [1] X2 + [2] >= [0] = fst(X1,X2) a__fst(0(),Z) = [1] Z + [6] >= [6] = nil() a__fst(s(X),cons(Y,Z)) = [1] Y + [2] >= [2] = cons(mark(Y),fst(X,Z)) a__len(X) = [1] X + [0] >= [0] = len(X) a__len(cons(X,Z)) = [1] X + [0] >= [0] = s(len(Z)) mark(0()) = [2] >= [4] = 0() mark(add(X1,X2)) = [2] >= [11] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [2] >= [2] = cons(mark(X1),X2) mark(from(X)) = [2] >= [4] = a__from(mark(X)) mark(fst(X1,X2)) = [2] >= [6] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [2] >= [2] = a__len(mark(X)) mark(nil()) = [2] >= [6] = nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__len(X) -> len(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(s(X)) -> s(X) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(a__add) = [1] x1 + [1] x2 + [2] p(a__from) = [1] x1 + [4] p(a__fst) = [1] x1 + [1] x2 + [4] p(a__len) = [1] x1 + [0] p(add) = [0] p(cons) = [1] x1 + [0] p(from) = [0] p(fst) = [0] p(len) = [0] p(mark) = [4] p(nil) = [2] p(s) = [0] Following rules are strictly oriented: mark(0()) = [4] > [2] = 0() mark(nil()) = [4] > [2] = nil() Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [1] X2 + [2] >= [0] = add(X1,X2) a__add(0(),X) = [1] X + [4] >= [4] = mark(X) a__add(s(X),Y) = [1] Y + [2] >= [0] = s(add(X,Y)) a__from(X) = [1] X + [4] >= [4] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [4] >= [0] = from(X) a__fst(X1,X2) = [1] X1 + [1] X2 + [4] >= [0] = fst(X1,X2) a__fst(0(),Z) = [1] Z + [6] >= [2] = nil() a__fst(s(X),cons(Y,Z)) = [1] Y + [4] >= [4] = cons(mark(Y),fst(X,Z)) a__len(X) = [1] X + [0] >= [0] = len(X) a__len(cons(X,Z)) = [1] X + [0] >= [0] = s(len(Z)) a__len(nil()) = [2] >= [2] = 0() mark(add(X1,X2)) = [4] >= [10] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [4] >= [4] = cons(mark(X1),X2) mark(from(X)) = [4] >= [8] = a__from(mark(X)) mark(fst(X1,X2)) = [4] >= [12] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [4] >= [4] = a__len(mark(X)) mark(s(X)) = [4] >= [0] = s(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 7: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__len(X) -> len(X) mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + 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__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(a__add) = [1] x1 + [1] x2 + [0] p(a__from) = [1] x1 + [5] p(a__fst) = [1] x1 + [1] x2 + [1] p(a__len) = [1] x1 + [1] p(add) = [0] p(cons) = [1] x1 + [4] p(from) = [0] p(fst) = [1] x2 + [1] p(len) = [0] p(mark) = [1] p(nil) = [0] p(s) = [0] Following rules are strictly oriented: a__len(X) = [1] X + [1] > [0] = len(X) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1] X1 + [1] X2 + [0] >= [0] = add(X1,X2) a__add(0(),X) = [1] X + [1] >= [1] = mark(X) a__add(s(X),Y) = [1] Y + [0] >= [0] = s(add(X,Y)) a__from(X) = [1] X + [5] >= [5] = cons(mark(X),from(s(X))) a__from(X) = [1] X + [5] >= [0] = from(X) a__fst(X1,X2) = [1] X1 + [1] X2 + [1] >= [1] X2 + [1] = fst(X1,X2) a__fst(0(),Z) = [1] Z + [2] >= [0] = nil() a__fst(s(X),cons(Y,Z)) = [1] Y + [5] >= [5] = cons(mark(Y),fst(X,Z)) a__len(cons(X,Z)) = [1] X + [5] >= [0] = s(len(Z)) a__len(nil()) = [1] >= [1] = 0() mark(0()) = [1] >= [1] = 0() mark(add(X1,X2)) = [1] >= [2] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1] >= [5] = cons(mark(X1),X2) mark(from(X)) = [1] >= [6] = a__from(mark(X)) mark(fst(X1,X2)) = [1] >= [3] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [1] >= [2] = a__len(mark(X)) mark(nil()) = [1] >= [0] = nil() mark(s(X)) = [1] >= [0] = s(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 8: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(X) -> len(X) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + Applied Processor: NaturalMI {miDimension = 3, 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 (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: {a__add,a__from,a__fst,a__len,mark} TcT has computed the following interpretation: p(0) = [0] [0] [1] p(a__add) = [1 0 0] [1 0 1] [1] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__from) = [1 1 1] [0] [0 0 0] x1 + [0] [0 1 1] [0] p(a__fst) = [1 1 0] [1 0 1] [0] [0 0 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__len) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(add) = [1 0 0] [1 0 1] [1] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(cons) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(from) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(fst) = [1 1 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(len) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(mark) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [1] p(s) = [0] [0] [0] Following rules are strictly oriented: mark(fst(X1,X2)) = [1 1 1] [1 0 2] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] > [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = a__fst(mark(X1),mark(X2)) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 0 0] [1 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = add(X1,X2) a__add(0(),X) = [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(X) a__add(s(X),Y) = [1 0 1] [1] [0 0 0] Y + [0] [0 0 1] [0] >= [0] [0] [0] = s(add(X,Y)) a__from(X) = [1 1 1] [0] [0 0 0] X + [0] [0 1 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 1 1] [0] [0 0 0] X + [0] [0 1 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = from(X) a__fst(X1,X2) = [1 1 0] [1 0 1] [0] [0 0 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 1 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = fst(X1,X2) a__fst(0(),Z) = [1 0 1] [0] [0 1 0] Z + [0] [0 0 1] [2] >= [0] [0] [1] = nil() a__fst(s(X),cons(Y,Z)) = [1 0 1] [0] [0 0 0] Y + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] Y + [0] [0 0 1] [0] = cons(mark(Y),fst(X,Z)) a__len(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = len(X) a__len(cons(X,Z)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0] [0] [0] = s(len(Z)) a__len(nil()) = [0] [0] [1] >= [0] [0] [1] = 0() mark(0()) = [1] [0] [1] >= [0] [0] [1] = 0() mark(add(X1,X2)) = [1 0 1] [1 0 2] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 2] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(from(X)) = [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [0] = a__from(mark(X)) mark(len(X)) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__len(mark(X)) mark(nil()) = [1] [0] [1] >= [0] [0] [1] = nil() mark(s(X)) = [0] [0] [0] >= [0] [0] [0] = s(X) * Step 9: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(len(X)) -> a__len(mark(X)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(X) -> len(X) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + Applied Processor: NaturalMI {miDimension = 3, 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 (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: {a__add,a__from,a__fst,a__len,mark} TcT has computed the following interpretation: p(0) = [0] [0] [1] p(a__add) = [1 0 1] [1 0 1] [1] [0 1 0] x1 + [0 0 1] x2 + [1] [0 0 1] [0 0 1] [1] p(a__from) = [1 0 1] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(a__fst) = [1 0 0] [1 0 1] [1] [0 0 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__len) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(add) = [1 0 1] [1 0 1] [1] [0 0 0] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [1] p(cons) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(from) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(fst) = [1 0 0] [1 0 1] [1] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(len) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(mark) = [1 0 1] [0] [0 0 1] x1 + [1] [0 0 1] [0] p(nil) = [0] [0] [1] p(s) = [0 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: mark(add(X1,X2)) = [1 0 2] [1 0 2] [2] [0 0 1] X1 + [0 0 1] X2 + [2] [0 0 1] [0 0 1] [1] > [1 0 2] [1 0 2] [1] [0 0 1] X1 + [0 0 1] X2 + [2] [0 0 1] [0 0 1] [1] = a__add(mark(X1),mark(X2)) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 0 1] [1 0 1] [1] [0 1 0] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [1] >= [1 0 1] [1 0 1] [1] [0 0 0] X1 + [0 0 1] X2 + [0] [0 0 1] [0 0 1] [1] = add(X1,X2) a__add(0(),X) = [1 0 1] [2] [0 0 1] X + [1] [0 0 1] [2] >= [1 0 1] [0] [0 0 1] X + [1] [0 0 1] [0] = mark(X) a__add(s(X),Y) = [0 1 0] [1 0 1] [1] [0 0 0] X + [0 0 1] Y + [1] [0 0 0] [0 0 1] [1] >= [0 0 1] [0] [0 0 0] Y + [0] [0 0 0] [0] = s(add(X,Y)) a__from(X) = [1 0 1] [0] [0 0 0] X + [1] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 0 1] [0] [0 0 0] X + [1] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = from(X) a__fst(X1,X2) = [1 0 0] [1 0 1] [1] [0 0 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = fst(X1,X2) a__fst(0(),Z) = [1 0 1] [1] [0 1 0] Z + [0] [0 0 1] [1] >= [0] [0] [1] = nil() a__fst(s(X),cons(Y,Z)) = [0 1 0] [1 0 1] [1] [0 0 0] X + [0 0 0] Y + [0] [0 0 0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] Y + [0] [0 0 1] [0] = cons(mark(Y),fst(X,Z)) a__len(X) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] = len(X) a__len(cons(X,Z)) = [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [0] >= [0] [0] [0] = s(len(Z)) a__len(nil()) = [0] [0] [1] >= [0] [0] [1] = 0() mark(0()) = [1] [2] [1] >= [0] [0] [1] = 0() mark(cons(X1,X2)) = [1 0 1] [0] [0 0 1] X1 + [1] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X1 + [0] [0 0 1] [0] = cons(mark(X1),X2) mark(from(X)) = [1 0 2] [0] [0 0 1] X + [1] [0 0 1] [0] >= [1 0 2] [0] [0 0 0] X + [1] [0 0 1] [0] = a__from(mark(X)) mark(fst(X1,X2)) = [1 0 1] [1 0 2] [1] [0 0 1] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 2] [1] [0 0 0] X1 + [0 0 1] X2 + [1] [0 0 1] [0 0 1] [0] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [1 0 1] [0] [0 0 1] X + [1] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = a__len(mark(X)) mark(nil()) = [1] [2] [1] >= [0] [0] [1] = nil() mark(s(X)) = [0 1 0] [0] [0 0 0] X + [1] [0 0 0] [0] >= [0 1 0] [0] [0 0 0] X + [0] [0 0 0] [0] = s(X) * Step 10: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(len(X)) -> a__len(mark(X)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(X) -> len(X) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + Applied Processor: NaturalMI {miDimension = 3, 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 (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: {a__add,a__from,a__fst,a__len,mark} TcT has computed the following interpretation: p(0) = [1] [0] [1] p(a__add) = [1 0 0] [1 0 1] [1] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__from) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [1] p(a__fst) = [1 0 0] [1 0 1] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__len) = [1 0 0] [1] [0 1 0] x1 + [0] [0 0 1] [0] p(add) = [1 0 0] [1 0 1] [1] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(cons) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(from) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(fst) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(len) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 1] [0] p(mark) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [0] p(nil) = [1] [0] [1] p(s) = [0] [0] [0] Following rules are strictly oriented: mark(cons(X1,X2)) = [1 0 1] [2] [0 0 0] X1 + [0] [0 0 1] [1] > [1 0 1] [1] [0 0 0] X1 + [0] [0 0 1] [1] = cons(mark(X1),X2) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 0 0] [1 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 0] [1 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = add(X1,X2) a__add(0(),X) = [1 0 1] [2] [0 0 0] X + [0] [0 0 1] [2] >= [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [0] = mark(X) a__add(s(X),Y) = [1 0 1] [1] [0 0 0] Y + [0] [0 0 1] [1] >= [0] [0] [0] = s(add(X,Y)) a__from(X) = [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [1] = cons(mark(X),from(s(X))) a__from(X) = [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = from(X) a__fst(X1,X2) = [1 0 0] [1 0 1] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = fst(X1,X2) a__fst(0(),Z) = [1 0 1] [1] [0 1 0] Z + [0] [0 0 1] [2] >= [1] [0] [1] = nil() a__fst(s(X),cons(Y,Z)) = [1 0 1] [1] [0 0 0] Y + [0] [0 0 1] [2] >= [1 0 1] [1] [0 0 0] Y + [0] [0 0 1] [1] = cons(mark(Y),fst(X,Z)) a__len(X) = [1 0 0] [1] [0 1 0] X + [0] [0 0 1] [0] >= [1 0 0] [1] [0 0 0] X + [0] [0 0 1] [0] = len(X) a__len(cons(X,Z)) = [1 0 0] [1] [0 0 0] X + [0] [0 0 1] [1] >= [0] [0] [0] = s(len(Z)) a__len(nil()) = [2] [0] [1] >= [1] [0] [1] = 0() mark(0()) = [3] [0] [1] >= [1] [0] [1] = 0() mark(add(X1,X2)) = [1 0 1] [1 0 2] [3] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 1] [1 0 2] [3] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = a__add(mark(X1),mark(X2)) mark(from(X)) = [1 0 2] [2] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 2] [2] [0 0 0] X + [0] [0 0 1] [1] = a__from(mark(X)) mark(fst(X1,X2)) = [1 0 1] [1 0 2] [2] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 1] [1 0 2] [2] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [1 0 1] [2] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [2] [0 0 0] X + [0] [0 0 1] [0] = a__len(mark(X)) mark(nil()) = [3] [0] [1] >= [1] [0] [1] = nil() mark(s(X)) = [1] [0] [0] >= [0] [0] [0] = s(X) * Step 11: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(from(X)) -> a__from(mark(X)) mark(len(X)) -> a__len(mark(X)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(X) -> len(X) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + Applied Processor: NaturalMI {miDimension = 3, 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 (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: {a__add,a__from,a__fst,a__len,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__add) = [1 1 0] [1 1 0] [0] [0 1 1] x1 + [0 1 0] x2 + [0] [0 0 0] [0 0 0] [0] p(a__from) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(a__fst) = [1 0 0] [1 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 0] [0 0 1] [0] p(a__len) = [1 0 0] [0] [0 1 1] x1 + [1] [0 0 0] [0] p(add) = [1 1 0] [1 1 0] [0] [0 1 1] x1 + [0 1 0] x2 + [0] [0 0 0] [0 0 0] [0] p(cons) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(from) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(fst) = [1 0 0] [1 1 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0] [0 0 0] [0 0 0] [0] p(len) = [1 0 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(mark) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(nil) = [0] [0] [0] p(s) = [0] [1] [0] Following rules are strictly oriented: mark(len(X)) = [1 1 0] [1] [0 1 0] X + [1] [0 0 0] [0] > [1 1 0] [0] [0 1 0] X + [1] [0 0 0] [0] = a__len(mark(X)) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 1 0] [1 1 0] [0] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 1 0] [1 1 0] [0] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] = add(X1,X2) a__add(0(),X) = [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = mark(X) a__add(s(X),Y) = [1 1 0] [1] [0 1 0] Y + [1] [0 0 0] [0] >= [0] [1] [0] = s(add(X,Y)) a__from(X) = [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = cons(mark(X),from(s(X))) a__from(X) = [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X + [0] [0 0 0] [0] = from(X) a__fst(X1,X2) = [1 0 0] [1 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 1] [0] >= [1 0 0] [1 1 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] = fst(X1,X2) a__fst(0(),Z) = [1 1 0] [0] [0 1 0] Z + [0] [0 0 1] [0] >= [0] [0] [0] = nil() a__fst(s(X),cons(Y,Z)) = [1 1 0] [0] [0 1 0] Y + [1] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] Y + [0] [0 0 0] [0] = cons(mark(Y),fst(X,Z)) a__len(X) = [1 0 0] [0] [0 1 1] X + [1] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [1] [0 0 0] [0] = len(X) a__len(cons(X,Z)) = [1 0 0] [0] [0 1 0] X + [1] [0 0 0] [0] >= [0] [1] [0] = s(len(Z)) a__len(nil()) = [0] [1] [0] >= [0] [0] [0] = 0() mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(add(X1,X2)) = [1 2 1] [1 2 0] [0] [0 1 1] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 2 0] [1 2 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 1 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] >= [1 1 0] [0] [0 1 0] X1 + [0] [0 0 0] [0] = cons(mark(X1),X2) mark(from(X)) = [1 2 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 2 0] [0] [0 1 0] X + [0] [0 0 0] [0] = a__from(mark(X)) mark(fst(X1,X2)) = [1 1 0] [1 2 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] >= [1 1 0] [1 2 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0] [0 0 0] [0 0 0] [0] = a__fst(mark(X1),mark(X2)) mark(nil()) = [0] [0] [0] >= [0] [0] [0] = nil() mark(s(X)) = [1] [1] [0] >= [0] [1] [0] = s(X) * Step 12: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(from(X)) -> a__from(mark(X)) - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(X) -> len(X) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + Applied Processor: NaturalMI {miDimension = 3, 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 (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(a__add) = {1,2}, uargs(a__from) = {1}, uargs(a__fst) = {1,2}, uargs(a__len) = {1}, uargs(cons) = {1} Following symbols are considered usable: {a__add,a__from,a__fst,a__len,mark} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__add) = [1 0 0] [1 0 1] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(a__from) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [1] p(a__fst) = [1 1 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(a__len) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(add) = [1 0 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(cons) = [1 0 0] [0 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 0] [1] p(from) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [1] p(fst) = [1 1 0] [1 0 1] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(len) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(mark) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(nil) = [0] [0] [1] p(s) = [0 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: mark(from(X)) = [1 0 2] [2] [0 0 0] X + [0] [0 0 1] [1] > [1 0 2] [1] [0 0 0] X + [0] [0 0 1] [1] = a__from(mark(X)) Following rules are (at-least) weakly oriented: a__add(X1,X2) = [1 0 0] [1 0 1] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = add(X1,X2) a__add(0(),X) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [0] = mark(X) a__add(s(X),Y) = [0 0 1] [1 0 1] [0] [0 0 0] X + [0 0 0] Y + [0] [0 0 0] [0 0 1] [0] >= [0 0 1] [0 0 1] [0] [0 0 0] X + [0 0 0] Y + [0] [0 0 0] [0 0 0] [0] = s(add(X,Y)) a__from(X) = [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [1] = cons(mark(X),from(s(X))) a__from(X) = [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [1] [0 0 0] X + [0] [0 0 1] [1] = from(X) a__fst(X1,X2) = [1 1 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 1 0] [1 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = fst(X1,X2) a__fst(0(),Z) = [1 0 1] [0] [0 0 0] Z + [0] [0 0 1] [1] >= [0] [0] [1] = nil() a__fst(s(X),cons(Y,Z)) = [0 0 1] [1 0 1] [0 0 1] [1] [0 0 0] X + [0 0 0] Y + [0 0 0] Z + [0] [0 0 0] [0 0 1] [0 0 0] [2] >= [0 0 1] [1 0 1] [0 0 1] [1] [0 0 0] X + [0 0 0] Y + [0 0 0] Z + [0] [0 0 0] [0 0 1] [0 0 0] [1] = cons(mark(Y),fst(X,Z)) a__len(X) = [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = len(X) a__len(cons(X,Z)) = [1 0 1] [0 0 1] [1] [0 0 0] X + [0 0 0] Z + [0] [0 0 1] [0 0 0] [2] >= [0 0 1] [1] [0 0 0] Z + [0] [0 0 0] [0] = s(len(Z)) a__len(nil()) = [1] [0] [2] >= [0] [0] [0] = 0() mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(add(X1,X2)) = [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) = [1 0 1] [0 0 1] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 0] [1] >= [1 0 1] [0 0 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 0] [1] = cons(mark(X1),X2) mark(fst(X1,X2)) = [1 1 1] [1 0 2] [1] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] >= [1 0 1] [1 0 2] [0] [0 0 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [1] = a__fst(mark(X1),mark(X2)) mark(len(X)) = [1 0 2] [1] [0 0 0] X + [0] [0 0 1] [1] >= [1 0 2] [0] [0 0 0] X + [0] [0 0 1] [1] = a__len(mark(X)) mark(nil()) = [1] [0] [1] >= [0] [0] [1] = nil() mark(s(X)) = [0 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] >= [0 0 1] [0] [0 0 0] X + [0] [0 0 0] [0] = s(X) * Step 13: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__add(X1,X2) -> add(X1,X2) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__from(X) -> cons(mark(X),from(s(X))) a__from(X) -> from(X) a__fst(X1,X2) -> fst(X1,X2) a__fst(0(),Z) -> nil() a__fst(s(X),cons(Y,Z)) -> cons(mark(Y),fst(X,Z)) a__len(X) -> len(X) a__len(cons(X,Z)) -> s(len(Z)) a__len(nil()) -> 0() mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(from(X)) -> a__from(mark(X)) mark(fst(X1,X2)) -> a__fst(mark(X1),mark(X2)) mark(len(X)) -> a__len(mark(X)) mark(nil()) -> nil() mark(s(X)) -> s(X) - Signature: {a__add/2,a__from/1,a__fst/2,a__len/1,mark/1} / {0/0,add/2,cons/2,from/1,fst/2,len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__from,a__fst,a__len,mark} and constructors {0 ,add,cons,from,fst,len,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))