primeCounting_zero
plain-language theorem explainer
The prime counting function evaluates to zero at input zero. Number theorists using base cases for arithmetic functions in Recognition Science contexts would cite this result. The proof is a direct one-line simplification from the definition of primeCounting.
Claim. The prime counting function satisfies $π(0) = 0$.
background
This theorem appears in the module supplying lightweight wrappers around Mathlib arithmetic functions, beginning with the Möbius function μ. The prime counting function is defined by π(n) = #{p ≤ n : p prime} and implemented as a direct call to Nat.primeCounting. The module setting keeps statements minimal while interfaces stabilize before adding Dirichlet inversion layers.
proof idea
The proof is a one-line term-mode wrapper that applies simp to the definition of primeCounting.
why it matters
This base case anchors the prime counting function inside the NumberTheory.Primes.ArithmeticFunctions module. It supplies the zero evaluation required for summations or inductive steps over primes. No downstream uses are recorded, yet the result completes the initial arithmetic-function interface before deeper Recognition Science number theory is layered on.
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