Foundation Primitive Recognition Calculus Certified Analytic Protocols Value Neg

Negation in the registry

In mathematics, a registry is a way of organizing a collection of expressions so that they can be evaluated and compared. The Recognition Science framework builds a countable registry of certified analytic protocol ingredients, meaning a list of constants and unary transformers that can be combined into finite expression trees. Each such expression evaluates to a protocol, which in turn displays a real value. The declaration value_neg is a small but load-bearing theorem about this registry: it states that the negation of an expression evaluates to the negation of the expression's value. In symbols, value R (.neg a) = - value R a. This is not a new operation; it is the same negation already defined on protocols, and the theorem confirms that the display function respects it.

The theorem is proved by a direct simplification using the definitions of evaluation and display. It is one of several similar results: value_rat, value_add, and value_sub state the same kind of respect for rational literals, addition, and subtraction. Together they form the algebraic backbone of the registry, ensuring that the display of a composite expression is exactly what ordinary arithmetic would produce from the displays of its parts. This matters because the registry is meant to be a faithful mirror of analytic content, not a source of surprises. The theorem also contributes to a larger closure result, transcendental_protocol_closure, which bundles these facts with countability and witness properties.

What the theorem does not claim is just as important as what it proves. It does not say that every real number appears as a value, nor that the registry can represent all analytic functions. It only concerns the specific operation of negation on expressions already in the registry. It does not introduce a new kind of negation, nor does it claim anything about the values of unary transformers or constants beyond what their definitions already provide. The theorem is a consistency check, not an expansion of expressive power.

/-- Native negation remains negation under the value display. -/
theorem value_neg (R : Registry) (a : Expr) :
    value R (.neg a) = - value R a := by
  simp [value, eval, Protocol.value_neg]

/-- Native negation remains negation under the value display. -/
theorem value_neg (R : Registry) (a : Expr) :
    value R (.neg a) = - value R a := by
  simp [value, eval, Protocol.value_neg]

What this page does not claim

The theorem does not assert that every real number is a value of some expression in the registry. The theorem does not claim anything about the values of unary transformers or constants beyond their definitions. The theorem does not introduce a new kind of negation distinct from the protocol-level negation.

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

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