primeCounting_twohundredfifty
plain-language theorem explainer
The declaration establishes that the prime counting function evaluates to 53 at input 250. Researchers verifying small explicit values of π(n) or validating computational implementations of arithmetic functions would cite this equality. The proof reduces the statement to a single native_decide tactic that computes the result directly from the underlying library definition.
Claim. The prime counting function satisfies $π(250) = 53$.
background
The prime counting function is defined as $π(n) = #{p ≤ n : p prime}$, implemented directly as the Mathlib primitive Nat.primeCounting. This theorem sits in the ArithmeticFunctions module, which supplies lightweight wrappers around Mathlib arithmetic functions and begins with the Möbius function μ before extending to related prime-counting tools. The local setting keeps statements minimal to support later Dirichlet inversion layers once basic interfaces stabilize.
proof idea
The proof is a one-line wrapper that applies the native_decide tactic to discharge the equality by direct computation.
why it matters
This supplies a concrete verified value for the prime counting function inside the arithmetic functions module. It does not feed any downstream theorems in the current dependency graph. Within Recognition Science the result offers a numerical checkpoint for number-theoretic primitives, though it remains disconnected from the core forcing chain or Recognition Composition Law.
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