Prime set

[Happypig375]

x in PP // is x a prime?
min(PP) // 2
min(x, x in PP and x > 14) // 17
PP subset ZZ* // true

The prime-counting function can be reformulated with the prime set.

Can we use the prime set to express the following:

• Integer factorisation
• The i-th prime number
  ?

[Rafael-SOWNet]

Taking this as a goal (untyped, from you). The shape, and what each line needs:

line	what it is	needs
x in PP	membership in a new special set PP (\mathbb{P}, SymPy's S.Primes), decided for a number by a primality test, left as written for a symbol	PP as a special set beside ZZ+; Integer.IsPrime exists (by counting divisors — a Miller–Rabin for large arguments comes with it)
PP subset ZZ*	the chain PP ⊂ ZZ+ ⊂ ZZ* ⊂ ZZ ⊂ QQ ⊂ RR ⊂ CC	one more link in the subset table
min(PP)	the least member of a set — min takes a set as its one argument	a min(S)/max(S) reading beside min(expr, x in S); 2 for PP, 1 for ZZ+, 0 for ZZ*, none for ZZ
min(x, x in PP and x > 14)	the least member of { x in PP : x > 14 } — a next prime	min(expr, x in S) accepting a conjunction as the range, and a search upward from the bound over a set of whole numbers that lists its members in order; 17
card(PP /\ [1; n])	the prime-counting function π(n)	the intersection of PP with a bounded interval is finite and listed — card then counts; π(100) = 25
PP /\ [1; 30]	the primes up to 30, listed	the same listing

On the two questions:

• Integer factorisation is a statement about PP but not an operation on it: 360 = 2^3 3^2 5 is product(p^v_p(360), p in PP) with v_p the multiplicity, which needs the indexed product over a set (product(f, p in S), the multiplicative twin of An integer range is listed, and a family of sets has a union and an intersection #1435's union(A_k, k in I)) and a valuation v_p(n) — the second is a small function (valuation(n, p), Sullivan and Mackey's Ex 5.5.1 wants the 2-adic one), the first is a binder that exists for sets already. factorize(360) printing 2^3 * 3^2 * 5 is the honest surface for the answer and it exists; the prime-set spelling is the definition of it.
• The i-th prime is an enumeration of PP in its order, which a set does not carry; prime(i) (SymPy's spelling) is a function of its own, and min(x, x in PP and x > prime(i)) is prime(i + 1) — that is the relation the set gives. I would add prime(n) beside phi in the number-theory functions rather than pretend a set is a sequence.

PR to follow with the first four rows and the listing; prime(n), valuation(n, p) and product(f, p in S) after, as their own.