WORST_CASE(?,O(n^3)) * Step 1: InnermostRuleRemoval WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) activate(n__len(X)) -> len(activate(X)) activate(n__s(X)) -> s(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X add(s(X),Y) -> s(n__add(activate(X),Y)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) fst(X1,X2) -> n__fst(X1,X2) fst(0(),Z) -> nil() fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z))) len(X) -> n__len(X) len(cons(X,Z)) -> s(n__len(activate(Z))) len(nil()) -> 0() s(X) -> n__s(X) - Signature: {activate/1,add/2,from/1,fst/2,len/1,s/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len,s} and constructors {0,cons ,n__add,n__from,n__fst,n__len,n__s,nil} + Applied Processor: InnermostRuleRemoval + Details: Arguments of following rules are not normal-forms. add(s(X),Y) -> s(n__add(activate(X),Y)) fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z))) All above mentioned rules can be savely removed. * Step 2: WeightGap WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) activate(n__len(X)) -> len(activate(X)) activate(n__s(X)) -> s(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) fst(X1,X2) -> n__fst(X1,X2) fst(0(),Z) -> nil() len(X) -> n__len(X) len(cons(X,Z)) -> s(n__len(activate(Z))) len(nil()) -> 0() s(X) -> n__s(X) - Signature: {activate/1,add/2,from/1,fst/2,len/1,s/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len,s} and constructors {0,cons ,n__add,n__from,n__fst,n__len,n__s,nil} + 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(add) = {1,2}, uargs(from) = {1}, uargs(fst) = {1,2}, uargs(len) = {1}, uargs(n__len) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(activate) = [1] x1 + [2] p(add) = [1] x1 + [1] x2 + [0] p(cons) = [1] x2 + [3] p(from) = [1] x1 + [0] p(fst) = [1] x1 + [1] x2 + [0] p(len) = [1] x1 + [0] p(n__add) = [1] x1 + [1] x2 + [0] p(n__from) = [1] x1 + [7] p(n__fst) = [1] x1 + [1] x2 + [0] p(n__len) = [1] x1 + [0] p(n__s) = [1] x1 + [2] p(nil) = [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: activate(X) = [1] X + [2] > [1] X + [0] = X activate(n__from(X)) = [1] X + [9] > [1] X + [2] = from(activate(X)) activate(n__s(X)) = [1] X + [4] > [1] X + [0] = s(X) len(cons(X,Z)) = [1] Z + [3] > [1] Z + [2] = s(n__len(activate(Z))) Following rules are (at-least) weakly oriented: activate(n__add(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [4] = add(activate(X1),activate(X2)) activate(n__fst(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [4] = fst(activate(X1),activate(X2)) activate(n__len(X)) = [1] X + [2] >= [1] X + [2] = len(activate(X)) add(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__add(X1,X2) add(0(),X) = [1] X + [0] >= [1] X + [0] = X from(X) = [1] X + [0] >= [1] X + [12] = cons(X,n__from(n__s(X))) from(X) = [1] X + [0] >= [1] X + [7] = n__from(X) fst(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__fst(X1,X2) fst(0(),Z) = [1] Z + [0] >= [0] = nil() len(X) = [1] X + [0] >= [1] X + [0] = n__len(X) len(nil()) = [0] >= [0] = 0() s(X) = [1] X + [0] >= [1] X + [2] = n__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^3)) + Considered Problem: - Strict TRS: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) activate(n__len(X)) -> len(activate(X)) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) fst(X1,X2) -> n__fst(X1,X2) fst(0(),Z) -> nil() len(X) -> n__len(X) len(nil()) -> 0() s(X) -> n__s(X) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(X) len(cons(X,Z)) -> s(n__len(activate(Z))) - Signature: {activate/1,add/2,from/1,fst/2,len/1,s/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len,s} and constructors {0,cons ,n__add,n__from,n__fst,n__len,n__s,nil} + 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(add) = {1,2}, uargs(from) = {1}, uargs(fst) = {1,2}, uargs(len) = {1}, uargs(n__len) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [8] p(activate) = [1] x1 + [2] p(add) = [1] x1 + [1] x2 + [2] p(cons) = [1] x2 + [4] p(from) = [1] x1 + [0] p(fst) = [1] x1 + [1] x2 + [0] p(len) = [1] x1 + [0] p(n__add) = [1] x1 + [1] x2 + [0] p(n__from) = [1] x1 + [0] p(n__fst) = [1] x1 + [1] x2 + [0] p(n__len) = [1] x1 + [1] p(n__s) = [1] x1 + [9] p(nil) = [0] p(s) = [1] x1 + [1] Following rules are strictly oriented: activate(n__len(X)) = [1] X + [3] > [1] X + [2] = len(activate(X)) add(X1,X2) = [1] X1 + [1] X2 + [2] > [1] X1 + [1] X2 + [0] = n__add(X1,X2) add(0(),X) = [1] X + [10] > [1] X + [0] = X fst(0(),Z) = [1] Z + [8] > [0] = nil() Following rules are (at-least) weakly oriented: activate(X) = [1] X + [2] >= [1] X + [0] = X activate(n__add(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [6] = add(activate(X1),activate(X2)) activate(n__from(X)) = [1] X + [2] >= [1] X + [2] = from(activate(X)) activate(n__fst(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [4] = fst(activate(X1),activate(X2)) activate(n__s(X)) = [1] X + [11] >= [1] X + [1] = s(X) from(X) = [1] X + [0] >= [1] X + [13] = cons(X,n__from(n__s(X))) from(X) = [1] X + [0] >= [1] X + [0] = n__from(X) fst(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__fst(X1,X2) len(X) = [1] X + [0] >= [1] X + [1] = n__len(X) len(cons(X,Z)) = [1] Z + [4] >= [1] Z + [4] = s(n__len(activate(Z))) len(nil()) = [0] >= [8] = 0() s(X) = [1] X + [1] >= [1] X + [9] = n__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^3)) + Considered Problem: - Strict TRS: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) fst(X1,X2) -> n__fst(X1,X2) len(X) -> n__len(X) len(nil()) -> 0() s(X) -> n__s(X) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__len(X)) -> len(activate(X)) activate(n__s(X)) -> s(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X fst(0(),Z) -> nil() len(cons(X,Z)) -> s(n__len(activate(Z))) - Signature: {activate/1,add/2,from/1,fst/2,len/1,s/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len,s} and constructors {0,cons ,n__add,n__from,n__fst,n__len,n__s,nil} + 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(add) = {1,2}, uargs(from) = {1}, uargs(fst) = {1,2}, uargs(len) = {1}, uargs(n__len) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(activate) = [1] x1 + [6] p(add) = [1] x1 + [1] x2 + [0] p(cons) = [1] x2 + [7] p(from) = [1] x1 + [1] p(fst) = [1] x1 + [1] x2 + [1] p(len) = [1] x1 + [1] p(n__add) = [1] x1 + [1] x2 + [0] p(n__from) = [1] x1 + [4] p(n__fst) = [1] x1 + [1] x2 + [0] p(n__len) = [1] x1 + [2] p(n__s) = [1] x1 + [2] p(nil) = [1] p(s) = [1] x1 + [0] Following rules are strictly oriented: fst(X1,X2) = [1] X1 + [1] X2 + [1] > [1] X1 + [1] X2 + [0] = n__fst(X1,X2) len(nil()) = [2] > [1] = 0() Following rules are (at-least) weakly oriented: activate(X) = [1] X + [6] >= [1] X + [0] = X activate(n__add(X1,X2)) = [1] X1 + [1] X2 + [6] >= [1] X1 + [1] X2 + [12] = add(activate(X1),activate(X2)) activate(n__from(X)) = [1] X + [10] >= [1] X + [7] = from(activate(X)) activate(n__fst(X1,X2)) = [1] X1 + [1] X2 + [6] >= [1] X1 + [1] X2 + [13] = fst(activate(X1),activate(X2)) activate(n__len(X)) = [1] X + [8] >= [1] X + [7] = len(activate(X)) activate(n__s(X)) = [1] X + [8] >= [1] X + [0] = s(X) add(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__add(X1,X2) add(0(),X) = [1] X + [1] >= [1] X + [0] = X from(X) = [1] X + [1] >= [1] X + [13] = cons(X,n__from(n__s(X))) from(X) = [1] X + [1] >= [1] X + [4] = n__from(X) fst(0(),Z) = [1] Z + [2] >= [1] = nil() len(X) = [1] X + [1] >= [1] X + [2] = n__len(X) len(cons(X,Z)) = [1] Z + [8] >= [1] Z + [8] = s(n__len(activate(Z))) s(X) = [1] X + [0] >= [1] X + [2] = n__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^3)) + Considered Problem: - Strict TRS: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) len(X) -> n__len(X) s(X) -> n__s(X) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__len(X)) -> len(activate(X)) activate(n__s(X)) -> s(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X fst(X1,X2) -> n__fst(X1,X2) fst(0(),Z) -> nil() len(cons(X,Z)) -> s(n__len(activate(Z))) len(nil()) -> 0() - Signature: {activate/1,add/2,from/1,fst/2,len/1,s/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len,s} and constructors {0,cons ,n__add,n__from,n__fst,n__len,n__s,nil} + 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(add) = {1,2}, uargs(from) = {1}, uargs(fst) = {1,2}, uargs(len) = {1}, uargs(n__len) = {1}, uargs(s) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(activate) = [1] x1 + [2] p(add) = [1] x1 + [1] x2 + [0] p(cons) = [1] x2 + [5] p(from) = [1] x1 + [0] p(fst) = [1] x1 + [1] x2 + [0] p(len) = [1] x1 + [0] p(n__add) = [1] x1 + [1] x2 + [0] p(n__from) = [1] x1 + [0] p(n__fst) = [1] x1 + [1] x2 + [0] p(n__len) = [1] x1 + [2] p(n__s) = [1] x1 + [0] p(nil) = [2] p(s) = [1] x1 + [1] Following rules are strictly oriented: s(X) = [1] X + [1] > [1] X + [0] = n__s(X) Following rules are (at-least) weakly oriented: activate(X) = [1] X + [2] >= [1] X + [0] = X activate(n__add(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [4] = add(activate(X1),activate(X2)) activate(n__from(X)) = [1] X + [2] >= [1] X + [2] = from(activate(X)) activate(n__fst(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [4] = fst(activate(X1),activate(X2)) activate(n__len(X)) = [1] X + [4] >= [1] X + [2] = len(activate(X)) activate(n__s(X)) = [1] X + [2] >= [1] X + [1] = s(X) add(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__add(X1,X2) add(0(),X) = [1] X + [2] >= [1] X + [0] = X from(X) = [1] X + [0] >= [1] X + [5] = cons(X,n__from(n__s(X))) from(X) = [1] X + [0] >= [1] X + [0] = n__from(X) fst(X1,X2) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = n__fst(X1,X2) fst(0(),Z) = [1] Z + [2] >= [2] = nil() len(X) = [1] X + [0] >= [1] X + [2] = n__len(X) len(cons(X,Z)) = [1] Z + [5] >= [1] Z + [5] = s(n__len(activate(Z))) len(nil()) = [2] >= [2] = 0() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) len(X) -> n__len(X) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__len(X)) -> len(activate(X)) activate(n__s(X)) -> s(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X fst(X1,X2) -> n__fst(X1,X2) fst(0(),Z) -> nil() len(cons(X,Z)) -> s(n__len(activate(Z))) len(nil()) -> 0() s(X) -> n__s(X) - Signature: {activate/1,add/2,from/1,fst/2,len/1,s/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len,s} and constructors {0,cons ,n__add,n__from,n__fst,n__len,n__s,nil} + 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(add) = {1,2}, uargs(from) = {1}, uargs(fst) = {1,2}, uargs(len) = {1}, uargs(n__len) = {1}, uargs(s) = {1} Following symbols are considered usable: {activate,add,from,fst,len,s} TcT has computed the following interpretation: p(0) = [0] [0] p(activate) = [1 4] x1 + [0] [0 1] [0] p(add) = [1 4] x1 + [1 0] x2 + [0] [0 1] [0 1] [0] p(cons) = [1 4] x2 + [0] [0 0] [0] p(from) = [1 0] x1 + [0] [0 1] [0] p(fst) = [1 0] x1 + [1 0] x2 + [2] [0 1] [0 1] [2] p(len) = [1 0] x1 + [0] [0 1] [0] p(n__add) = [1 4] x1 + [1 0] x2 + [0] [0 1] [0 1] [0] p(n__from) = [1 0] x1 + [0] [0 1] [0] p(n__fst) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 1] [2] p(n__len) = [1 0] x1 + [0] [0 1] [0] p(n__s) = [1 0] x1 + [0] [0 0] [0] p(nil) = [2] [0] p(s) = [1 0] x1 + [0] [0 0] [0] Following rules are strictly oriented: activate(n__fst(X1,X2)) = [1 4] X1 + [1 4] X2 + [8] [0 1] [0 1] [2] > [1 4] X1 + [1 4] X2 + [2] [0 1] [0 1] [2] = fst(activate(X1),activate(X2)) Following rules are (at-least) weakly oriented: activate(X) = [1 4] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X activate(n__add(X1,X2)) = [1 8] X1 + [1 4] X2 + [0] [0 1] [0 1] [0] >= [1 8] X1 + [1 4] X2 + [0] [0 1] [0 1] [0] = add(activate(X1),activate(X2)) activate(n__from(X)) = [1 4] X + [0] [0 1] [0] >= [1 4] X + [0] [0 1] [0] = from(activate(X)) activate(n__len(X)) = [1 4] X + [0] [0 1] [0] >= [1 4] X + [0] [0 1] [0] = len(activate(X)) activate(n__s(X)) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = s(X) add(X1,X2) = [1 4] X1 + [1 0] X2 + [0] [0 1] [0 1] [0] >= [1 4] X1 + [1 0] X2 + [0] [0 1] [0 1] [0] = n__add(X1,X2) add(0(),X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X from(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 0] [0] = cons(X,n__from(n__s(X))) from(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = n__from(X) fst(X1,X2) = [1 0] X1 + [1 0] X2 + [2] [0 1] [0 1] [2] >= [1 0] X1 + [1 0] X2 + [0] [0 1] [0 1] [2] = n__fst(X1,X2) fst(0(),Z) = [1 0] Z + [2] [0 1] [2] >= [2] [0] = nil() len(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = n__len(X) len(cons(X,Z)) = [1 4] Z + [0] [0 0] [0] >= [1 4] Z + [0] [0 0] [0] = s(n__len(activate(Z))) len(nil()) = [2] [0] >= [0] [0] = 0() s(X) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = n__s(X) * Step 7: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) len(X) -> n__len(X) - Weak TRS: activate(X) -> X activate(n__from(X)) -> from(activate(X)) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) activate(n__len(X)) -> len(activate(X)) activate(n__s(X)) -> s(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X fst(X1,X2) -> n__fst(X1,X2) fst(0(),Z) -> nil() len(cons(X,Z)) -> s(n__len(activate(Z))) len(nil()) -> 0() s(X) -> n__s(X) - Signature: {activate/1,add/2,from/1,fst/2,len/1,s/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len,s} and constructors {0,cons ,n__add,n__from,n__fst,n__len,n__s,nil} + 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(add) = {1,2}, uargs(from) = {1}, uargs(fst) = {1,2}, uargs(len) = {1}, uargs(n__len) = {1}, uargs(s) = {1} Following symbols are considered usable: {activate,add,from,fst,len,s} TcT has computed the following interpretation: p(0) = [3] [2] p(activate) = [1 1] x1 + [0] [0 4] [0] p(add) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 1] [7] p(cons) = [0 0] x1 + [1 1] x2 + [0] [0 1] [0 1] [0] p(from) = [1 0] x1 + [2] [0 1] [0] p(fst) = [1 0] x1 + [1 0] x2 + [3] [0 1] [0 1] [4] p(len) = [1 0] x1 + [2] [0 1] [0] p(n__add) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 1] [2] p(n__from) = [1 0] x1 + [2] [0 1] [0] p(n__fst) = [1 0] x1 + [1 0] x2 + [3] [0 1] [0 1] [1] p(n__len) = [1 0] x1 + [2] [0 1] [0] p(n__s) = [1 0] x1 + [0] [0 0] [0] p(nil) = [1] [5] p(s) = [1 0] x1 + [0] [0 0] [0] Following rules are strictly oriented: activate(n__add(X1,X2)) = [1 1] X1 + [1 1] X2 + [2] [0 4] [0 4] [8] > [1 1] X1 + [1 1] X2 + [0] [0 4] [0 4] [7] = add(activate(X1),activate(X2)) Following rules are (at-least) weakly oriented: activate(X) = [1 1] X + [0] [0 4] [0] >= [1 0] X + [0] [0 1] [0] = X activate(n__from(X)) = [1 1] X + [2] [0 4] [0] >= [1 1] X + [2] [0 4] [0] = from(activate(X)) activate(n__fst(X1,X2)) = [1 1] X1 + [1 1] X2 + [4] [0 4] [0 4] [4] >= [1 1] X1 + [1 1] X2 + [3] [0 4] [0 4] [4] = fst(activate(X1),activate(X2)) activate(n__len(X)) = [1 1] X + [2] [0 4] [0] >= [1 1] X + [2] [0 4] [0] = len(activate(X)) activate(n__s(X)) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = s(X) add(X1,X2) = [1 0] X1 + [1 0] X2 + [0] [0 1] [0 1] [7] >= [1 0] X1 + [1 0] X2 + [0] [0 1] [0 1] [2] = n__add(X1,X2) add(0(),X) = [1 0] X + [3] [0 1] [9] >= [1 0] X + [0] [0 1] [0] = X from(X) = [1 0] X + [2] [0 1] [0] >= [1 0] X + [2] [0 1] [0] = cons(X,n__from(n__s(X))) from(X) = [1 0] X + [2] [0 1] [0] >= [1 0] X + [2] [0 1] [0] = n__from(X) fst(X1,X2) = [1 0] X1 + [1 0] X2 + [3] [0 1] [0 1] [4] >= [1 0] X1 + [1 0] X2 + [3] [0 1] [0 1] [1] = n__fst(X1,X2) fst(0(),Z) = [1 0] Z + [6] [0 1] [6] >= [1] [5] = nil() len(X) = [1 0] X + [2] [0 1] [0] >= [1 0] X + [2] [0 1] [0] = n__len(X) len(cons(X,Z)) = [0 0] X + [1 1] Z + [2] [0 1] [0 1] [0] >= [1 1] Z + [2] [0 0] [0] = s(n__len(activate(Z))) len(nil()) = [3] [5] >= [3] [2] = 0() s(X) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = n__s(X) * Step 8: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) len(X) -> n__len(X) - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) activate(n__len(X)) -> len(activate(X)) activate(n__s(X)) -> s(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X fst(X1,X2) -> n__fst(X1,X2) fst(0(),Z) -> nil() len(cons(X,Z)) -> s(n__len(activate(Z))) len(nil()) -> 0() s(X) -> n__s(X) - Signature: {activate/1,add/2,from/1,fst/2,len/1,s/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len,s} and constructors {0,cons ,n__add,n__from,n__fst,n__len,n__s,nil} + 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(add) = {1,2}, uargs(from) = {1}, uargs(fst) = {1,2}, uargs(len) = {1}, uargs(n__len) = {1}, uargs(s) = {1} Following symbols are considered usable: {activate,add,from,fst,len,s} TcT has computed the following interpretation: p(0) = [4] [1] p(activate) = [1 1] x1 + [0] [0 4] [0] p(add) = [1 0] x1 + [1 0] x2 + [2] [0 1] [0 1] [6] p(cons) = [0 0] x1 + [1 2] x2 + [0] [0 1] [0 0] [0] p(from) = [1 0] x1 + [0] [0 1] [0] p(fst) = [1 0] x1 + [1 0] x2 + [7] [0 1] [0 1] [2] p(len) = [1 0] x1 + [1] [0 1] [2] p(n__add) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 1] [3] p(n__from) = [1 0] x1 + [0] [0 1] [0] p(n__fst) = [1 0] x1 + [1 0] x2 + [7] [0 1] [0 1] [1] p(n__len) = [1 0] x1 + [0] [0 1] [2] p(n__s) = [1 0] x1 + [0] [0 0] [0] p(nil) = [5] [1] p(s) = [1 0] x1 + [0] [0 0] [0] Following rules are strictly oriented: len(X) = [1 0] X + [1] [0 1] [2] > [1 0] X + [0] [0 1] [2] = n__len(X) Following rules are (at-least) weakly oriented: activate(X) = [1 1] X + [0] [0 4] [0] >= [1 0] X + [0] [0 1] [0] = X activate(n__add(X1,X2)) = [1 1] X1 + [1 1] X2 + [3] [0 4] [0 4] [12] >= [1 1] X1 + [1 1] X2 + [2] [0 4] [0 4] [6] = add(activate(X1),activate(X2)) activate(n__from(X)) = [1 1] X + [0] [0 4] [0] >= [1 1] X + [0] [0 4] [0] = from(activate(X)) activate(n__fst(X1,X2)) = [1 1] X1 + [1 1] X2 + [8] [0 4] [0 4] [4] >= [1 1] X1 + [1 1] X2 + [7] [0 4] [0 4] [2] = fst(activate(X1),activate(X2)) activate(n__len(X)) = [1 1] X + [2] [0 4] [8] >= [1 1] X + [1] [0 4] [2] = len(activate(X)) activate(n__s(X)) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = s(X) add(X1,X2) = [1 0] X1 + [1 0] X2 + [2] [0 1] [0 1] [6] >= [1 0] X1 + [1 0] X2 + [0] [0 1] [0 1] [3] = n__add(X1,X2) add(0(),X) = [1 0] X + [6] [0 1] [7] >= [1 0] X + [0] [0 1] [0] = X from(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = cons(X,n__from(n__s(X))) from(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = n__from(X) fst(X1,X2) = [1 0] X1 + [1 0] X2 + [7] [0 1] [0 1] [2] >= [1 0] X1 + [1 0] X2 + [7] [0 1] [0 1] [1] = n__fst(X1,X2) fst(0(),Z) = [1 0] Z + [11] [0 1] [3] >= [5] [1] = nil() len(cons(X,Z)) = [0 0] X + [1 2] Z + [1] [0 1] [0 0] [2] >= [1 1] Z + [0] [0 0] [0] = s(n__len(activate(Z))) len(nil()) = [6] [3] >= [4] [1] = 0() s(X) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = n__s(X) * Step 9: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) activate(n__len(X)) -> len(activate(X)) activate(n__s(X)) -> s(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X fst(X1,X2) -> n__fst(X1,X2) fst(0(),Z) -> nil() len(X) -> n__len(X) len(cons(X,Z)) -> s(n__len(activate(Z))) len(nil()) -> 0() s(X) -> n__s(X) - Signature: {activate/1,add/2,from/1,fst/2,len/1,s/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len,s} and constructors {0,cons ,n__add,n__from,n__fst,n__len,n__s,nil} + 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(add) = {1,2}, uargs(from) = {1}, uargs(fst) = {1,2}, uargs(len) = {1}, uargs(n__len) = {1}, uargs(s) = {1} Following symbols are considered usable: {activate,add,from,fst,len,s} TcT has computed the following interpretation: p(0) = [4] [0] p(activate) = [1 4] x1 + [4] [0 4] [0] p(add) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 1] [2] p(cons) = [0 0] x1 + [1 4] x2 + [0] [0 1] [0 0] [1] p(from) = [1 0] x1 + [4] [0 1] [1] p(fst) = [1 0] x1 + [1 0] x2 + [7] [0 1] [0 1] [4] p(len) = [1 0] x1 + [4] [0 1] [4] p(n__add) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 1] [1] p(n__from) = [1 0] x1 + [0] [0 1] [1] p(n__fst) = [1 0] x1 + [1 0] x2 + [3] [0 1] [0 1] [2] p(n__len) = [1 0] x1 + [0] [0 1] [2] p(n__s) = [1 0] x1 + [0] [0 0] [0] p(nil) = [2] [2] p(s) = [1 0] x1 + [0] [0 0] [0] Following rules are strictly oriented: from(X) = [1 0] X + [4] [0 1] [1] > [1 0] X + [0] [0 1] [1] = n__from(X) Following rules are (at-least) weakly oriented: activate(X) = [1 4] X + [4] [0 4] [0] >= [1 0] X + [0] [0 1] [0] = X activate(n__add(X1,X2)) = [1 4] X1 + [1 4] X2 + [8] [0 4] [0 4] [4] >= [1 4] X1 + [1 4] X2 + [8] [0 4] [0 4] [2] = add(activate(X1),activate(X2)) activate(n__from(X)) = [1 4] X + [8] [0 4] [4] >= [1 4] X + [8] [0 4] [1] = from(activate(X)) activate(n__fst(X1,X2)) = [1 4] X1 + [1 4] X2 + [15] [0 4] [0 4] [8] >= [1 4] X1 + [1 4] X2 + [15] [0 4] [0 4] [4] = fst(activate(X1),activate(X2)) activate(n__len(X)) = [1 4] X + [12] [0 4] [8] >= [1 4] X + [8] [0 4] [4] = len(activate(X)) activate(n__s(X)) = [1 0] X + [4] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = s(X) add(X1,X2) = [1 0] X1 + [1 0] X2 + [0] [0 1] [0 1] [2] >= [1 0] X1 + [1 0] X2 + [0] [0 1] [0 1] [1] = n__add(X1,X2) add(0(),X) = [1 0] X + [4] [0 1] [2] >= [1 0] X + [0] [0 1] [0] = X from(X) = [1 0] X + [4] [0 1] [1] >= [1 0] X + [4] [0 1] [1] = cons(X,n__from(n__s(X))) fst(X1,X2) = [1 0] X1 + [1 0] X2 + [7] [0 1] [0 1] [4] >= [1 0] X1 + [1 0] X2 + [3] [0 1] [0 1] [2] = n__fst(X1,X2) fst(0(),Z) = [1 0] Z + [11] [0 1] [4] >= [2] [2] = nil() len(X) = [1 0] X + [4] [0 1] [4] >= [1 0] X + [0] [0 1] [2] = n__len(X) len(cons(X,Z)) = [0 0] X + [1 4] Z + [4] [0 1] [0 0] [5] >= [1 4] Z + [4] [0 0] [0] = s(n__len(activate(Z))) len(nil()) = [6] [6] >= [4] [0] = 0() s(X) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = n__s(X) * Step 10: NaturalMI WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: from(X) -> cons(X,n__from(n__s(X))) - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) activate(n__len(X)) -> len(activate(X)) activate(n__s(X)) -> s(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X from(X) -> n__from(X) fst(X1,X2) -> n__fst(X1,X2) fst(0(),Z) -> nil() len(X) -> n__len(X) len(cons(X,Z)) -> s(n__len(activate(Z))) len(nil()) -> 0() s(X) -> n__s(X) - Signature: {activate/1,add/2,from/1,fst/2,len/1,s/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len,s} and constructors {0,cons ,n__add,n__from,n__fst,n__len,n__s,nil} + Applied Processor: NaturalMI {miDimension = 3, 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: The following argument positions are considered usable: uargs(add) = {1,2}, uargs(from) = {1}, uargs(fst) = {1,2}, uargs(len) = {1}, uargs(n__len) = {1}, uargs(s) = {1} Following symbols are considered usable: {activate,add,from,fst,len,s} TcT has computed the following interpretation: p(0) = [1] [1] [0] p(activate) = [1 0 1] [0] [0 1 0] x1 + [2] [0 0 2] [0] p(add) = [1 0 0] [1 0 0] [2] [0 0 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [0] p(cons) = [1 2 0] [0] [0 0 1] x2 + [1] [0 0 0] [1] p(from) = [1 0 0] [1] [0 0 0] x1 + [2] [0 0 1] [1] p(fst) = [1 0 0] [1 0 0] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(len) = [1 1 0] [1] [0 0 0] x1 + [2] [0 0 1] [3] p(n__add) = [1 0 0] [1 0 0] [2] [0 0 0] x1 + [0 1 0] x2 + [0] [0 0 1] [0 0 1] [0] p(n__from) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(n__fst) = [1 0 0] [1 0 0] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [0] p(n__len) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 1] [3] p(n__s) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(nil) = [1] [0] [0] p(s) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: from(X) = [1 0 0] [1] [0 0 0] X + [2] [0 0 1] [1] > [1 0 0] [0] [0 0 0] X + [2] [0 0 0] [1] = cons(X,n__from(n__s(X))) Following rules are (at-least) weakly oriented: activate(X) = [1 0 1] [0] [0 1 0] X + [2] [0 0 2] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = X activate(n__add(X1,X2)) = [1 0 1] [1 0 1] [2] [0 0 0] X1 + [0 1 0] X2 + [2] [0 0 2] [0 0 2] [0] >= [1 0 1] [1 0 1] [2] [0 0 0] X1 + [0 1 0] X2 + [2] [0 0 2] [0 0 2] [0] = add(activate(X1),activate(X2)) activate(n__from(X)) = [1 0 1] [1] [0 0 0] X + [2] [0 0 2] [2] >= [1 0 1] [1] [0 0 0] X + [2] [0 0 2] [1] = from(activate(X)) activate(n__fst(X1,X2)) = [1 0 1] [1 0 1] [0] [0 1 0] X1 + [0 0 0] X2 + [2] [0 0 2] [0 0 2] [0] >= [1 0 1] [1 0 1] [0] [0 1 0] X1 + [0 0 0] X2 + [2] [0 0 2] [0 0 2] [0] = fst(activate(X1),activate(X2)) activate(n__len(X)) = [1 1 1] [3] [0 0 0] X + [2] [0 0 2] [6] >= [1 1 1] [3] [0 0 0] X + [2] [0 0 2] [3] = len(activate(X)) activate(n__s(X)) = [1 0 0] [0] [0 1 0] X + [2] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = s(X) add(X1,X2) = [1 0 0] [1 0 0] [2] [0 0 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 0] [2] [0 0 0] X1 + [0 1 0] X2 + [0] [0 0 1] [0 0 1] [0] = n__add(X1,X2) add(0(),X) = [1 0 0] [3] [0 1 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = X from(X) = [1 0 0] [1] [0 0 0] X + [2] [0 0 1] [1] >= [1 0 0] [0] [0 0 0] X + [0] [0 0 1] [1] = n__from(X) fst(X1,X2) = [1 0 0] [1 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [1 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [0] [0 0 1] [0 0 1] [0] = n__fst(X1,X2) fst(0(),Z) = [1 0 0] [1] [0 0 0] Z + [1] [0 0 1] [0] >= [1] [0] [0] = nil() len(X) = [1 1 0] [1] [0 0 0] X + [2] [0 0 1] [3] >= [1 1 0] [0] [0 0 0] X + [0] [0 0 1] [3] = n__len(X) len(cons(X,Z)) = [1 2 1] [2] [0 0 0] Z + [2] [0 0 0] [4] >= [1 1 1] [2] [0 0 0] Z + [0] [0 0 0] [0] = s(n__len(activate(Z))) len(nil()) = [2] [2] [3] >= [1] [1] [0] = 0() s(X) = [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 0] [0] = n__s(X) * Step 11: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(activate(X1),activate(X2)) activate(n__from(X)) -> from(activate(X)) activate(n__fst(X1,X2)) -> fst(activate(X1),activate(X2)) activate(n__len(X)) -> len(activate(X)) activate(n__s(X)) -> s(X) add(X1,X2) -> n__add(X1,X2) add(0(),X) -> X from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) fst(X1,X2) -> n__fst(X1,X2) fst(0(),Z) -> nil() len(X) -> n__len(X) len(cons(X,Z)) -> s(n__len(activate(Z))) len(nil()) -> 0() s(X) -> n__s(X) - Signature: {activate/1,add/2,from/1,fst/2,len/1,s/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,n__s/1,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len,s} and constructors {0,cons ,n__add,n__from,n__fst,n__len,n__s,nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))