WORST_CASE(?,O(n^1)) * Step 1: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(X1,X2) activate(n__from(X)) -> from(X) activate(n__fst(X1,X2)) -> fst(X1,X2) activate(n__len(X)) -> len(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(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() - Signature: {activate/1,add/2,from/1,fst/2,len/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons ,n__add,n__from,n__fst,n__len,nil,s} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(cons) = {2}, uargs(n__add) = {1}, uargs(n__fst) = {1,2}, uargs(n__len) = {1}, uargs(s) = {1} Following symbols are considered usable: {activate,add,from,fst,len} TcT has computed the following interpretation: p(0) = 0 p(activate) = 7 + 4*x1 p(add) = 2 + 4*x1 + 4*x2 p(cons) = 1 + x2 p(from) = 5 + 4*x1 p(fst) = 3 + 4*x1 + 4*x2 p(len) = 7 + 4*x1 p(n__add) = x1 + x2 p(n__from) = x1 p(n__fst) = x1 + x2 p(n__len) = 1 + x1 p(nil) = 0 p(s) = 2 + x1 Following rules are strictly oriented: activate(X) = 7 + 4*X > X = X activate(n__add(X1,X2)) = 7 + 4*X1 + 4*X2 > 2 + 4*X1 + 4*X2 = add(X1,X2) activate(n__from(X)) = 7 + 4*X > 5 + 4*X = from(X) activate(n__fst(X1,X2)) = 7 + 4*X1 + 4*X2 > 3 + 4*X1 + 4*X2 = fst(X1,X2) activate(n__len(X)) = 11 + 4*X > 7 + 4*X = len(X) add(X1,X2) = 2 + 4*X1 + 4*X2 > X1 + X2 = n__add(X1,X2) add(0(),X) = 2 + 4*X > X = X add(s(X),Y) = 10 + 4*X + 4*Y > 9 + 4*X + Y = s(n__add(activate(X),Y)) from(X) = 5 + 4*X > 3 + X = cons(X,n__from(s(X))) from(X) = 5 + 4*X > X = n__from(X) fst(X1,X2) = 3 + 4*X1 + 4*X2 > X1 + X2 = n__fst(X1,X2) fst(0(),Z) = 3 + 4*Z > 0 = nil() len(X) = 7 + 4*X > 1 + X = n__len(X) len(cons(X,Z)) = 11 + 4*Z > 10 + 4*Z = s(n__len(activate(Z))) len(nil()) = 7 > 0 = 0() Following rules are (at-least) weakly oriented: fst(s(X),cons(Y,Z)) = 15 + 4*X + 4*Z >= 15 + 4*X + 4*Z = cons(Y,n__fst(activate(X),activate(Z))) * Step 2: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: fst(s(X),cons(Y,Z)) -> cons(Y,n__fst(activate(X),activate(Z))) - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(X1,X2) activate(n__from(X)) -> from(X) activate(n__fst(X1,X2)) -> fst(X1,X2) activate(n__len(X)) -> len(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(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() - Signature: {activate/1,add/2,from/1,fst/2,len/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons ,n__add,n__from,n__fst,n__len,nil,s} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(cons) = {2}, uargs(n__add) = {1}, uargs(n__fst) = {1,2}, uargs(n__len) = {1}, uargs(s) = {1} Following symbols are considered usable: {activate,add,from,fst,len} TcT has computed the following interpretation: p(0) = 0 p(activate) = 4*x1 p(add) = 4*x1 + 2*x2 p(cons) = x2 p(from) = 0 p(fst) = 1 + 4*x1 + 4*x2 p(len) = 5 + 4*x1 p(n__add) = x1 + x2 p(n__from) = 0 p(n__fst) = 1 + x1 + x2 p(n__len) = 2 + x1 p(nil) = 0 p(s) = 2 + x1 Following rules are strictly oriented: fst(s(X),cons(Y,Z)) = 9 + 4*X + 4*Z > 1 + 4*X + 4*Z = cons(Y,n__fst(activate(X),activate(Z))) Following rules are (at-least) weakly oriented: activate(X) = 4*X >= X = X activate(n__add(X1,X2)) = 4*X1 + 4*X2 >= 4*X1 + 2*X2 = add(X1,X2) activate(n__from(X)) = 0 >= 0 = from(X) activate(n__fst(X1,X2)) = 4 + 4*X1 + 4*X2 >= 1 + 4*X1 + 4*X2 = fst(X1,X2) activate(n__len(X)) = 8 + 4*X >= 5 + 4*X = len(X) add(X1,X2) = 4*X1 + 2*X2 >= X1 + X2 = n__add(X1,X2) add(0(),X) = 2*X >= X = X add(s(X),Y) = 8 + 4*X + 2*Y >= 2 + 4*X + Y = s(n__add(activate(X),Y)) from(X) = 0 >= 0 = cons(X,n__from(s(X))) from(X) = 0 >= 0 = n__from(X) fst(X1,X2) = 1 + 4*X1 + 4*X2 >= 1 + X1 + X2 = n__fst(X1,X2) fst(0(),Z) = 1 + 4*Z >= 0 = nil() len(X) = 5 + 4*X >= 2 + X = n__len(X) len(cons(X,Z)) = 5 + 4*Z >= 4 + 4*Z = s(n__len(activate(Z))) len(nil()) = 5 >= 0 = 0() * Step 3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__add(X1,X2)) -> add(X1,X2) activate(n__from(X)) -> from(X) activate(n__fst(X1,X2)) -> fst(X1,X2) activate(n__len(X)) -> len(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(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() - Signature: {activate/1,add/2,from/1,fst/2,len/1} / {0/0,cons/2,n__add/2,n__from/1,n__fst/2,n__len/1,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {activate,add,from,fst,len} and constructors {0,cons ,n__add,n__from,n__fst,n__len,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))