pi_pow5_eq
plain-language theorem explainer
The algebraic identity π⁵ = π² ⋅ π² ⋅ π holds by direct rearrangement of exponents. Researchers deriving interval bounds on powers of π in numerical analysis cite this step when simplifying expressions ahead of applying Mathlib's π inequalities. The proof is a one-line application of the ring tactic that matches monomials without external lemmas.
Claim. $π^5 = π^2 ⋅ π^2 ⋅ π$
background
The module supplies interval bounds on π drawn from Mathlib's library of proven inequalities, including 3.14 < π < 3.15 and tighter variants up to 20 decimal places. These bounds support construction of intervals for π and its powers to enable rigorous numerical claims. The present declaration supplies a basic algebraic reduction used inside such interval constructions.
proof idea
The proof is a one-line wrapper that invokes the ring tactic to expand both sides and confirm monomial equality.
why it matters
The identity closes a trivial algebraic step inside the PiBounds module, allowing subsequent interval theorems on π^5 to avoid manual expansion. It supports the module's goal of providing Mathlib-backed bounds that can feed into Recognition Science numerics for constants such as α^{-1}. No downstream uses are recorded yet.
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