Foundation Logic As Functional Equation Logic Identity L To Real

The identity bridge

A comparison operator is a two-input function that returns a value indicating how two things relate. In the Recognition Science framework, such operators are defined over recovered reals, a specially constructed number system that mirrors the ordinary real numbers while carrying additional logical structure. The declaration identityL_to_real proves a transport property: if a comparison operator satisfies the identity rule in the recovered-real setting, then its translation to ordinary real numbers satisfies the corresponding identity rule there.

The identity rule is the requirement that comparing any positive value with itself yields zero. In symbols, for a comparison operator C, the rule states C(x, x) = 0 for all positive x. This is a natural baseline: a thing compared with itself should register no difference. The declaration shows that this property is preserved under the translation map that converts recovered reals to ordinary reals. The proof is direct, relying on the fact that the translation map respects the ordering and the zero element of both number systems.

This transport result is part of a larger pattern in the framework's machine-checked library of formal theorems. The library establishes that any comparison operator satisfying a small set of structural laws (identity, symmetry, scale invariance, and non-triviality) must take a specific functional form. The declaration identityL_to_real contributes the identity component of that argument for the recovered-real setting, ensuring that the structural laws stated there align with their real-number counterparts.

What the declaration does not claim is equally important. It does not assert that the identity rule holds for any particular comparison operator; it only states a conditional implication. It does not establish the full set of laws needed for the uniqueness theorem; other declarations cover symmetry, scale invariance, and non-triviality. And it does not prove that the recovered-real and real settings are identical in all respects; it addresses only the specific structural property of identity under translation.

theorem identityL_to_real (C : ComparisonOperatorL) (h : IdentityL C) :
    LogicAsFunctionalEquation.Identity (transportComparison C) := by
  intro x hx
  unfold transportComparison
  have hxL : (0 : LogicReal) < fromReal x := by
    rw [lt_iff_toReal_lt, toReal_zero, toReal_fromReal]
    exact hx
  have hL := congrArg toReal (h (fromReal x) hxL)
  simpa [toReal_fromReal] using hL

/-- Identity over recovered reals. -/
def IdentityL (C : ComparisonOperatorL) : Prop :=
  ∀ x : LogicReal, (0 : LogicReal) < x → C x x = fromReal 0

What this page does not claim

The declaration does not assert that any particular comparison operator satisfies the identity rule. The declaration does not establish the full set of laws needed for the uniqueness theorem. The declaration does not prove that recovered reals and ordinary reals are identical in all respects.

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

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