Foundation Logic Real Transcendentals Cosh L

Hyperbolic cosine on recovered reals

The hyperbolic cosine, written cosh(x), is the average of an exponential and its reciprocal: cosh(x) = (e^x + e^(-x))/2. For a real number x, this produces a value always at least 1, symmetric about zero, and it describes the curve of a hanging chain or cable, the catenary. The declaration coshL brings this standard function into a specific formal number system called LogicReal, a recovered real line that the framework's library proves is equivalent to the usual real numbers.

The declaration is a definition, not a new discovery. It takes a number from this LogicReal system, converts it to an ordinary real number, applies the standard hyperbolic cosine, and converts the result back. The accompanying theorem coshL_eq_exp proves that this transported function satisfies the defining identity coshL(x) = (expL(x) + expL(-x)) / 2, where expL is the transported exponential. Another theorem, toReal_coshL, confirms that converting the result back to an ordinary real number gives exactly the ordinary cosh of the original number.

In Recognition Science, this is the first layer of analytic infrastructure. Later modules can reason about hyperbolic functions inside the framework while reducing any analytic identity to the established real-analysis library. The point is not that coshL reveals new mathematics; it is that the framework can carry standard transcendental functions without re-deriving calculus from scratch, and can prove their basic properties in a machine-checked way.

What coshL does not claim is any special physical or geometric content. It does not assert that the hyperbolic cosine appears in the framework's forcing chain or that it has a privileged role in recognition cost. It is simply a transported definition, a piece of plumbing that makes the recovered real line usable for analysis. The declaration also does not claim anything about the hyperbolic sine, tangent, or inverse functions; those are separate definitions in the same module.

/-- Hyperbolic cosine on recovered reals. -/
def coshL (x : LogicReal) : LogicReal := fromReal (Real.cosh (toReal x))

/-- Transported hyperbolic cosine definition. -/
theorem coshL_eq_exp (x : LogicReal) :
    coshL x = (expL x + expL (-x)) / fromReal 2 := by
  rw [eq_iff_toReal_eq, toReal_coshL, toReal_div, toReal_add, toReal_expL,
    toReal_expL, toReal_neg, toReal_fromReal]
  exact Real.cosh_eq _

@[simp] theorem toReal_coshL (x : LogicReal) :
    toReal (coshL x) = Real.cosh (toReal x) :=
  toReal_fromReal _

What this page does not claim

The hyperbolic cosine has no special role in the framework's forcing chain or recognition cost. No physical or geometric meaning is asserted for coshL beyond the standard catenary curve. The declaration does not define or prove properties of hyperbolic sine, tangent, or inverse functions.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/LogicRealTranscendentals.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: