Foundation Primitive Recognition Calculus Integer Rational Abs Eq Zero Iff To In

Zero and absolute value

The declaration abs_eq_zero_iff_toInt_eq_zero belongs to the Primitive Recognition Calculus, the framework's internal arithmetic for signed and ratio orbits, which are discrete records of events. In plain terms, the declaration proves a familiar fact about this arithmetic: a signed orbit, the framework's representation of an integer, has absolute value zero if and only if the integer it maps to is zero. The proof is not a new mathematical discovery; it is a machine-checked confirmation that the framework's definition of absolute value agrees with its definition of zero.

The declaration appears in the framework's machine-checked library of formal theorems, a collection of statements verified by a proof checker. It is a theorem, meaning it has been proved from the framework's axioms with no gaps. The statement is narrow: it concerns only the framework's own signed orbits, not the ordinary integers of everyday arithmetic. The framework defines a signed orbit as a pair of natural numbers, one marking nonnegative and one marking negative, and the declaration shows that the absolute value of such a pair is zero exactly when the pair represents zero.

In Recognition Science, the framework models reality as a ledger, a discrete record of recognition events, and its arithmetic is built from these orbits. The declaration is a small but necessary step in that construction: it ensures that the absolute value function behaves as expected on the framework's integers. Without it, the framework could not rely on absolute value to compare sizes or to reason about distances between orbits.

The declaration does not claim that the framework's integers are the same as the ordinary integers, nor that absolute value has any meaning outside the framework's own definitions. It does not assert that every integer has a unique absolute value, nor that the absolute value function is defined for all possible inputs. It is a local fact, proved for the framework's specific construction, and it does not generalize to other number systems without additional proof.

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

What this page does not claim

The declaration does not claim that the framework's integers are identical to ordinary integers. It does not claim that absolute value is defined for all possible inputs in the framework. It does not claim that the equivalence holds for ratio orbits or any other number system.

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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