primeCounting_seven
plain-language theorem explainer
The declaration establishes that the prime counting function evaluates to exactly 4 at input 7. Number theorists building explicit arithmetic libraries or verifying small prime counts would cite this base case. The proof is a one-line native decision that confirms the equality by direct computation.
Claim. The prime counting function satisfies $π(7) = 4$.
background
The module supplies lightweight wrappers around Mathlib arithmetic functions, beginning with the Möbius function μ and related objects for later Dirichlet inversion. The upstream definition states that the prime counting function π(n) equals the cardinality of primes p ≤ n, implemented as Nat.primeCounting n. This theorem records the explicit value at n = 7.
proof idea
The proof is a one-line wrapper that applies the native_decide tactic to verify the numerical equality by direct computation.
why it matters
This supplies a concrete base case for the prime counting function inside the arithmetic functions module. It supports further prime-related constructions such as Möbius applications, though no immediate dependents appear. In the Recognition Science setting such explicit counts provide number-theoretic footholds that can feed into larger algebraic developments.
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