How to define a large number of assumptions?

Question

Please consider the following simple and larger example:

varSimple = ToExpression@CharacterRange["a", "b"];
fSimple = Total@Map[Sqrt[#^2] &, varSimple];
assumpSimple = And[Element[varSimple, Integers] , a >= 0 , b >= 0];
simpfSimple = Simplify[fSimple, assumpSimple]
(*a+b*)

When one defines a larger function such as fLarge, it is very inconvenient to type character1>=0 && ... && character#>=0 for the whole CharacterRange to define the assumptions.

varLarge = ToExpression@CharacterRange["a", "z"];
fLarge = Total@Map[Sqrt[#^2] &, varLarge];
assumpLarge = And[Element[varLarge, Integers] , a >= 0 , ... , z=>0];
simpfSimple = Simplify[fSimple, assumpLarge]
(*a+...+z*)

Question 1

Since And (as well as Alternatives) cannot cope with List as an argument, how can I combine a larger number of expressions with And (or Alternatives)?

Question 2

With ToExpression@CharacterRange["a", "z"] the number of variables is limited to 24. Is there another (easy) way to define a larger number of variables?

Answer 1

For question 1: To And together a list you can use Apply (it is usually written as @@, see the docs for more information):

And@@Thread[{a, b, c} >= 0]
(* a >= 0 && b >= 0 && c >= 0 *)

For Alternatives you can:

Alternatives@@{a, b, c}
(* a | b | c *)

For question 2: You can call your variables x[i]

And@@Thread[(x[#] & /@ Range[50]) >= 0]
(* x[1] >= 0 && ... && x[50] >= 0  *)

Full example:

varLarge = x[#] & /@ Range[50];
fLarge = Total@Map[Sqrt[#^2] &, varLarge];
assumpLarge = And[Element[varLarge, Integers], And @@ Thread@(varLarge >= 0)];
simpfSimple = Simplify[fLarge, assumpLarge]
(* x[1] + ... + x[50] *)

When calling variables x[i] you can also use x[_] >= 0 as your assumption, but for some reason that runs really slow for me.

If you have the same assumptions for everything involved you can put something like _ \[Element] Integers && _ >= 0 as your assumption since _ is a pattern that matches everything.