Foundation Primitive Recognition Calculus Integer Rational Abs Ne Zero Of To Int

The absolute value lemma

The declaration abs_ne_zero_of_toInt_ne_zero is a small theorem inside the Recognition Science framework's machine-checked library of formal theorems. It states a fact about the framework's own integer type, called PRCInt, which is built from signed orbits, a discrete record of positions along a recognition cycle. The theorem says: if the integer value of a PRCInt element is not zero, then its absolute value is also not zero. In plainer terms, if a number is not zero, then its distance from zero is not zero either. This is a basic property of absolute values that holds in ordinary arithmetic, and the framework proves it holds for its own constructed integers as well.

The proof is a direct consequence of how the framework defines absolute value and zero. The declaration PRCInt is a quotient type, meaning it identifies different representations of the same integer. The theorem zero_toInt establishes that the zero element maps to the integer 0, and toInt_zero' shows the reverse direction. The lemma abs_ne_zero_of_toInt_ne_zero then connects these facts: it shows that the absolute value function preserves the distinction between zero and non-zero elements. This is a foundational consistency check, ensuring that the framework's notion of absolute value behaves as expected on its own integer type.

What the declaration does not claim is important. It does not say anything about the absolute value of a rational number, which is a different type in the framework called PRCRat. It does not establish any property about the sign of a number, only about whether its absolute value is zero or not. It also does not claim that the framework's integers are the same as the usual integers, only that this particular property holds for them. The theorem is a building block, not a standalone result about the nature of numbers.

In the context of the framework, this lemma serves as internal support for comparing positions along a signed orbit, which is a sequence of positions that can be traversed in either direction. It is part of the scaffolding that lets the framework reason about order and distance in its own number system. The practical consequence is that the framework can reliably use absolute values in its proofs without worrying about zero slipping through the cracks. It is a small but necessary piece of the larger structure.

@[simp] theorem zero_toInt :
    zero.toInt = 0 := by
  rfl

@[simp] theorem zero_toInt :
    zero.toInt = 0 := by
  rfl

What this page does not claim

The declaration does not claim any property about the absolute value of rational numbers. It does not establish that the framework's integers are identical to the usual integers. It does not say anything about the sign of a number, only about whether its absolute value is zero.

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

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