Foundation Logic Real Transcendentals Cosh L Eq Exp

The recovered hyperbolic cosine

The hyperbolic cosine, cosh, is a standard function from classical mathematics. For any real number x, it is defined as (e^x + e^(-x)) / 2, the average of the exponential function and its reciprocal. This identity is a fundamental property taught in calculus and analysis courses, and it connects the hyperbolic functions to the ordinary exponential function.

In the Recognition Science framework, the recovered real numbers are built to be equivalent to the standard real numbers. The framework's library of formal theorems, a machine-checked collection, defines its own versions of the transcendental functions by transporting them from the standard real line. This means that the framework's exponential function, expL, and its hyperbolic cosine, coshL, are defined so that they behave exactly like their classical counterparts.

The declaration coshL_eq_exp is a theorem in this library. It proves that for any recovered real number x, the framework's hyperbolic cosine satisfies the same formula as the classical one: coshL x = (expL x + expL (-x)) / 2. The proof is direct: it shows that the framework's definition agrees with the standard real definition, and then applies the classical identity. This is a transport lemma, a bridge that lets later work use the familiar properties of hyperbolic functions.

What this theorem does not claim is any new physical or mathematical content. It does not derive the hyperbolic cosine from the framework's forcing chain, nor does it assign any special meaning to cosh within the framework's recognition ledger. It is purely a technical consistency result: the framework's recovered functions match the standard ones. The theorem is a foundation for future work, not a discovery in itself.

/-- 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 theorem derives the hyperbolic cosine from the framework's forcing chain. The theorem assigns any special recognition-ledger meaning to the hyperbolic cosine. The theorem establishes any new mathematical content beyond transport of the classical identity.

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)

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