WORST_CASE(Omega(n^1),O(n^1)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^1)) + 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__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if ,mark} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + 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__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if ,mark} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: mark(x){x -> add(x,y)} = mark(add(x,y)) ->^+ a__add(mark(x),y) = C[mark(x) = mark(x){}] ** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1)) + 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__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if ,mark} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + 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(a__add) = {1}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1} Following symbols are considered usable: {a__add,a__and,a__first,a__from,a__if,mark} TcT has computed the following interpretation: p(0) = 1 p(a__add) = x1 + 4*x2 p(a__and) = x1 + 4*x2 p(a__first) = 2 + x1 + x2 p(a__from) = x1 p(a__if) = 7 + x1 + 4*x2 + 4*x3 p(add) = x1 + x2 p(and) = x1 + x2 p(cons) = 0 p(false) = 0 p(first) = 2 + x1 + x2 p(from) = x1 p(if) = 2 + x1 + x2 + x3 p(mark) = 4*x1 p(nil) = 0 p(s) = 0 p(true) = 1 Following rules are strictly oriented: a__add(0(),X) = 1 + 4*X > 4*X = mark(X) a__and(true(),X) = 1 + 4*X > 4*X = mark(X) a__first(0(),X) = 3 + X > 0 = nil() a__first(s(X),cons(Y,Z)) = 2 > 0 = cons(Y,first(X,Z)) a__if(X1,X2,X3) = 7 + X1 + 4*X2 + 4*X3 > 2 + X1 + X2 + X3 = if(X1,X2,X3) a__if(false(),X,Y) = 7 + 4*X + 4*Y > 4*Y = mark(Y) a__if(true(),X,Y) = 8 + 4*X + 4*Y > 4*X = mark(X) mark(0()) = 4 > 1 = 0() mark(first(X1,X2)) = 8 + 4*X1 + 4*X2 > 2 + 4*X1 + 4*X2 = a__first(mark(X1),mark(X2)) mark(if(X1,X2,X3)) = 8 + 4*X1 + 4*X2 + 4*X3 > 7 + 4*X1 + 4*X2 + 4*X3 = a__if(mark(X1),X2,X3) mark(true()) = 4 > 1 = true() Following rules are (at-least) weakly oriented: a__add(X1,X2) = X1 + 4*X2 >= X1 + X2 = add(X1,X2) a__add(s(X),Y) = 4*Y >= 0 = s(add(X,Y)) a__and(X1,X2) = X1 + 4*X2 >= X1 + X2 = and(X1,X2) a__and(false(),Y) = 4*Y >= 0 = false() a__first(X1,X2) = 2 + X1 + X2 >= 2 + X1 + X2 = first(X1,X2) a__from(X) = X >= 0 = cons(X,from(s(X))) a__from(X) = X >= X = from(X) mark(add(X1,X2)) = 4*X1 + 4*X2 >= 4*X1 + 4*X2 = a__add(mark(X1),X2) mark(and(X1,X2)) = 4*X1 + 4*X2 >= 4*X1 + 4*X2 = a__and(mark(X1),X2) mark(cons(X1,X2)) = 0 >= 0 = cons(X1,X2) mark(false()) = 0 >= 0 = false() mark(from(X)) = 4*X >= X = a__from(X) mark(nil()) = 0 >= 0 = nil() mark(s(X)) = 0 >= 0 = s(X) ** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__first(X1,X2) -> first(X1,X2) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(from(X)) -> a__from(X) mark(nil()) -> nil() mark(s(X)) -> s(X) - Weak TRS: a__add(0(),X) -> mark(X) a__and(true(),X) -> mark(X) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if ,mark} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + 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(a__add) = {1}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1} Following symbols are considered usable: {a__add,a__and,a__first,a__from,a__if,mark} TcT has computed the following interpretation: p(0) = 0 p(a__add) = x1 + 4*x2 p(a__and) = x1 + 4*x2 p(a__first) = x1 + x2 p(a__from) = 0 p(a__if) = x1 + 4*x2 + 4*x3 p(add) = x1 + x2 p(and) = x1 + x2 p(cons) = 0 p(false) = 2 p(first) = x1 + x2 p(from) = 0 p(if) = x1 + x2 + x3 p(mark) = 4*x1 p(nil) = 0 p(s) = 0 p(true) = 3 Following rules are strictly oriented: mark(false()) = 8 > 2 = false() Following rules are (at-least) weakly oriented: a__add(X1,X2) = X1 + 4*X2 >= X1 + X2 = add(X1,X2) a__add(0(),X) = 4*X >= 4*X = mark(X) a__add(s(X),Y) = 4*Y >= 0 = s(add(X,Y)) a__and(X1,X2) = X1 + 4*X2 >= X1 + X2 = and(X1,X2) a__and(false(),Y) = 2 + 4*Y >= 2 = false() a__and(true(),X) = 3 + 4*X >= 4*X = mark(X) a__first(X1,X2) = X1 + X2 >= X1 + X2 = first(X1,X2) a__first(0(),X) = X >= 0 = nil() a__first(s(X),cons(Y,Z)) = 0 >= 0 = cons(Y,first(X,Z)) a__from(X) = 0 >= 0 = cons(X,from(s(X))) a__from(X) = 0 >= 0 = from(X) a__if(X1,X2,X3) = X1 + 4*X2 + 4*X3 >= X1 + X2 + X3 = if(X1,X2,X3) a__if(false(),X,Y) = 2 + 4*X + 4*Y >= 4*Y = mark(Y) a__if(true(),X,Y) = 3 + 4*X + 4*Y >= 4*X = mark(X) mark(0()) = 0 >= 0 = 0() mark(add(X1,X2)) = 4*X1 + 4*X2 >= 4*X1 + 4*X2 = a__add(mark(X1),X2) mark(and(X1,X2)) = 4*X1 + 4*X2 >= 4*X1 + 4*X2 = a__and(mark(X1),X2) mark(cons(X1,X2)) = 0 >= 0 = cons(X1,X2) mark(first(X1,X2)) = 4*X1 + 4*X2 >= 4*X1 + 4*X2 = a__first(mark(X1),mark(X2)) mark(from(X)) = 0 >= 0 = a__from(X) mark(if(X1,X2,X3)) = 4*X1 + 4*X2 + 4*X3 >= 4*X1 + 4*X2 + 4*X3 = a__if(mark(X1),X2,X3) mark(nil()) = 0 >= 0 = nil() mark(s(X)) = 0 >= 0 = s(X) mark(true()) = 12 >= 3 = true() ** Step 1.b:3: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__first(X1,X2) -> first(X1,X2) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(from(X)) -> a__from(X) mark(nil()) -> nil() mark(s(X)) -> s(X) - Weak TRS: a__add(0(),X) -> mark(X) a__and(true(),X) -> mark(X) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if ,mark} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + 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(a__add) = {1}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1} Following symbols are considered usable: {a__add,a__and,a__first,a__from,a__if,mark} TcT has computed the following interpretation: p(0) = 1 p(a__add) = x1 + 2*x2 p(a__and) = 7 + x1 + 2*x2 p(a__first) = 1 + x1 + x2 p(a__from) = 2 + 2*x1 p(a__if) = x1 + 2*x2 + 2*x3 p(add) = x1 + x2 p(and) = 5 + x1 + x2 p(cons) = 2 + x1 p(false) = 4 p(first) = 1 + x1 + x2 p(from) = 2 + x1 p(if) = x1 + x2 + x3 p(mark) = 1 + 2*x1 p(nil) = 0 p(s) = x1 p(true) = 1 Following rules are strictly oriented: a__and(X1,X2) = 7 + X1 + 2*X2 > 5 + X1 + X2 = and(X1,X2) a__and(false(),Y) = 11 + 2*Y > 4 = false() mark(and(X1,X2)) = 11 + 2*X1 + 2*X2 > 8 + 2*X1 + 2*X2 = a__and(mark(X1),X2) mark(cons(X1,X2)) = 5 + 2*X1 > 2 + X1 = cons(X1,X2) mark(from(X)) = 5 + 2*X > 2 + 2*X = a__from(X) mark(nil()) = 1 > 0 = nil() mark(s(X)) = 1 + 2*X > X = s(X) Following rules are (at-least) weakly oriented: a__add(X1,X2) = X1 + 2*X2 >= X1 + X2 = add(X1,X2) a__add(0(),X) = 1 + 2*X >= 1 + 2*X = mark(X) a__add(s(X),Y) = X + 2*Y >= X + Y = s(add(X,Y)) a__and(true(),X) = 8 + 2*X >= 1 + 2*X = mark(X) a__first(X1,X2) = 1 + X1 + X2 >= 1 + X1 + X2 = first(X1,X2) a__first(0(),X) = 2 + X >= 0 = nil() a__first(s(X),cons(Y,Z)) = 3 + X + Y >= 2 + Y = cons(Y,first(X,Z)) a__from(X) = 2 + 2*X >= 2 + X = cons(X,from(s(X))) a__from(X) = 2 + 2*X >= 2 + X = from(X) a__if(X1,X2,X3) = X1 + 2*X2 + 2*X3 >= X1 + X2 + X3 = if(X1,X2,X3) a__if(false(),X,Y) = 4 + 2*X + 2*Y >= 1 + 2*Y = mark(Y) a__if(true(),X,Y) = 1 + 2*X + 2*Y >= 1 + 2*X = mark(X) mark(0()) = 3 >= 1 = 0() mark(add(X1,X2)) = 1 + 2*X1 + 2*X2 >= 1 + 2*X1 + 2*X2 = a__add(mark(X1),X2) mark(false()) = 9 >= 4 = false() mark(first(X1,X2)) = 3 + 2*X1 + 2*X2 >= 3 + 2*X1 + 2*X2 = a__first(mark(X1),mark(X2)) mark(if(X1,X2,X3)) = 1 + 2*X1 + 2*X2 + 2*X3 >= 1 + 2*X1 + 2*X2 + 2*X3 = a__if(mark(X1),X2,X3) mark(true()) = 3 >= 1 = true() ** Step 1.b:4: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) a__add(s(X),Y) -> s(add(X,Y)) a__first(X1,X2) -> first(X1,X2) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) mark(add(X1,X2)) -> a__add(mark(X1),X2) - Weak TRS: a__add(0(),X) -> mark(X) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if ,mark} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + 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(a__add) = {1}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1} Following symbols are considered usable: {a__add,a__and,a__first,a__from,a__if,mark} TcT has computed the following interpretation: p(0) = 1 p(a__add) = 2 + x1 + 4*x2 p(a__and) = 2 + x1 + 4*x2 p(a__first) = 4 + x1 + x2 p(a__from) = 4 p(a__if) = x1 + 4*x2 + 4*x3 p(add) = 2 + x1 + x2 p(and) = 2 + x1 + x2 p(cons) = 0 p(false) = 0 p(first) = 1 + x1 + x2 p(from) = 3 p(if) = x1 + x2 + x3 p(mark) = 4*x1 p(nil) = 0 p(s) = 0 p(true) = 0 Following rules are strictly oriented: a__add(s(X),Y) = 2 + 4*Y > 0 = s(add(X,Y)) a__first(X1,X2) = 4 + X1 + X2 > 1 + X1 + X2 = first(X1,X2) a__from(X) = 4 > 0 = cons(X,from(s(X))) a__from(X) = 4 > 3 = from(X) mark(add(X1,X2)) = 8 + 4*X1 + 4*X2 > 2 + 4*X1 + 4*X2 = a__add(mark(X1),X2) Following rules are (at-least) weakly oriented: a__add(X1,X2) = 2 + X1 + 4*X2 >= 2 + X1 + X2 = add(X1,X2) a__add(0(),X) = 3 + 4*X >= 4*X = mark(X) a__and(X1,X2) = 2 + X1 + 4*X2 >= 2 + X1 + X2 = and(X1,X2) a__and(false(),Y) = 2 + 4*Y >= 0 = false() a__and(true(),X) = 2 + 4*X >= 4*X = mark(X) a__first(0(),X) = 5 + X >= 0 = nil() a__first(s(X),cons(Y,Z)) = 4 >= 0 = cons(Y,first(X,Z)) a__if(X1,X2,X3) = X1 + 4*X2 + 4*X3 >= X1 + X2 + X3 = if(X1,X2,X3) a__if(false(),X,Y) = 4*X + 4*Y >= 4*Y = mark(Y) a__if(true(),X,Y) = 4*X + 4*Y >= 4*X = mark(X) mark(0()) = 4 >= 1 = 0() mark(and(X1,X2)) = 8 + 4*X1 + 4*X2 >= 2 + 4*X1 + 4*X2 = a__and(mark(X1),X2) mark(cons(X1,X2)) = 0 >= 0 = cons(X1,X2) mark(false()) = 0 >= 0 = false() mark(first(X1,X2)) = 4 + 4*X1 + 4*X2 >= 4 + 4*X1 + 4*X2 = a__first(mark(X1),mark(X2)) mark(from(X)) = 12 >= 4 = a__from(X) mark(if(X1,X2,X3)) = 4*X1 + 4*X2 + 4*X3 >= 4*X1 + 4*X2 + 4*X3 = a__if(mark(X1),X2,X3) mark(nil()) = 0 >= 0 = nil() mark(s(X)) = 0 >= 0 = s(X) mark(true()) = 0 >= 0 = true() ** Step 1.b:5: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__add(X1,X2) -> add(X1,X2) - Weak TRS: a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if ,mark} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + 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(a__add) = {1}, uargs(a__and) = {1}, uargs(a__first) = {1,2}, uargs(a__if) = {1} Following symbols are considered usable: {a__add,a__and,a__first,a__from,a__if,mark} TcT has computed the following interpretation: p(0) = 0 p(a__add) = 2 + x1 + 4*x2 p(a__and) = x1 + 4*x2 p(a__first) = 2 + x1 + x2 p(a__from) = 0 p(a__if) = x1 + 4*x2 + 4*x3 p(add) = 1 + x1 + x2 p(and) = x1 + x2 p(cons) = 0 p(false) = 1 p(first) = 2 + x1 + x2 p(from) = 0 p(if) = x1 + x2 + x3 p(mark) = 4*x1 p(nil) = 1 p(s) = 0 p(true) = 0 Following rules are strictly oriented: a__add(X1,X2) = 2 + X1 + 4*X2 > 1 + X1 + X2 = add(X1,X2) Following rules are (at-least) weakly oriented: a__add(0(),X) = 2 + 4*X >= 4*X = mark(X) a__add(s(X),Y) = 2 + 4*Y >= 0 = s(add(X,Y)) a__and(X1,X2) = X1 + 4*X2 >= X1 + X2 = and(X1,X2) a__and(false(),Y) = 1 + 4*Y >= 1 = false() a__and(true(),X) = 4*X >= 4*X = mark(X) a__first(X1,X2) = 2 + X1 + X2 >= 2 + X1 + X2 = first(X1,X2) a__first(0(),X) = 2 + X >= 1 = nil() a__first(s(X),cons(Y,Z)) = 2 >= 0 = cons(Y,first(X,Z)) a__from(X) = 0 >= 0 = cons(X,from(s(X))) a__from(X) = 0 >= 0 = from(X) a__if(X1,X2,X3) = X1 + 4*X2 + 4*X3 >= X1 + X2 + X3 = if(X1,X2,X3) a__if(false(),X,Y) = 1 + 4*X + 4*Y >= 4*Y = mark(Y) a__if(true(),X,Y) = 4*X + 4*Y >= 4*X = mark(X) mark(0()) = 0 >= 0 = 0() mark(add(X1,X2)) = 4 + 4*X1 + 4*X2 >= 2 + 4*X1 + 4*X2 = a__add(mark(X1),X2) mark(and(X1,X2)) = 4*X1 + 4*X2 >= 4*X1 + 4*X2 = a__and(mark(X1),X2) mark(cons(X1,X2)) = 0 >= 0 = cons(X1,X2) mark(false()) = 4 >= 1 = false() mark(first(X1,X2)) = 8 + 4*X1 + 4*X2 >= 2 + 4*X1 + 4*X2 = a__first(mark(X1),mark(X2)) mark(from(X)) = 0 >= 0 = a__from(X) mark(if(X1,X2,X3)) = 4*X1 + 4*X2 + 4*X3 >= 4*X1 + 4*X2 + 4*X3 = a__if(mark(X1),X2,X3) mark(nil()) = 4 >= 1 = nil() mark(s(X)) = 0 >= 0 = s(X) mark(true()) = 0 >= 0 = true() ** Step 1.b:6: 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__and(X1,X2) -> and(X1,X2) a__and(false(),Y) -> false() a__and(true(),X) -> mark(X) a__first(X1,X2) -> first(X1,X2) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) a__from(X) -> from(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) mark(0()) -> 0() mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(cons(X1,X2)) -> cons(X1,X2) mark(false()) -> false() mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(nil()) -> nil() mark(s(X)) -> s(X) mark(true()) -> true() - Signature: {a__add/2,a__and/2,a__first/2,a__from/1,a__if/3,mark/1} / {0/0,add/2,and/2,cons/2,false/0,first/2,from/1 ,if/3,nil/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__add,a__and,a__first,a__from,a__if ,mark} and constructors {0,add,and,cons,false,first,from,if,nil,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^1))