Foundation Primitive Recognition Calculus Integer Rational Signed Orbit Equiv If

Signed orbits and their integers

A signed orbit is a way to keep a discrete record of a position that can also carry a direction. Classically, an integer is exactly such a record: the whole number tells you how far from zero you are, and the sign tells you which way. The framework builds its own integer-like objects, called signed orbits, from two counting numbers, one for the positive part and one for the negative part. The pair (3, 1) and the pair (2, 0) both describe the same position, namely 2, because the negative part cancels part of the positive part. The framework's declaration signedOrbitEquiv_iff_toInt_eq proves that two signed orbits are equivalent exactly when they convert to the same integer. In plain language, the machine-checked theorem says that the only thing that distinguishes two signed orbits is the integer they represent.

This is a structural fact about the framework's own construction, not a claim about ordinary arithmetic. The framework defines a type called PRCInt as a quotient of signed orbits under an equivalence relation. The theorem in question, signedOrbitEquiv_iff_toInt_eq, states that this equivalence relation coincides with having equal integer values. That means the quotient construction is faithful: no information is lost or added when passing from a signed orbit to its integer. The declaration appears in the machine-checked library of formal theorems, specifically in the IntegerRational module of the PrimitiveRecognitionCalculus file. The proof is a theorem in the library, meaning it is a proved statement, not a definitional choice.

What the declaration does not claim is just as important. It does not claim that signed orbits are the same thing as integers in every respect. A signed orbit carries extra structure, namely the two counting numbers that make up the pair, and the equivalence relation deliberately identifies many pairs with the same integer. The theorem does not say that the map from signed orbits to integers is injective on the raw pairs; it says the map is injective on equivalence classes. Nor does the declaration claim anything about the rational version, PRCRat, which is built from ratio orbits in a parallel but separate construction. The rational case has its own definitions and its own theorems, and the integer theorem does not transfer automatically.

Within the framework, this theorem matters because it gives a clean bridge between the primitive combinatorial objects and the familiar arithmetic they are meant to model. When the framework's later results speak of integers, they can rely on this equivalence to know that the signed-orbit representation is not introducing hidden distinctions. A reader who wants to know what the framework means by an integer can look at this theorem and see the answer: an integer is exactly an equivalence class of signed orbits that agree on their integer value. The declaration is a small but load-bearing piece of the foundation, because it pins down the sense in which the primitive objects really do behave like the numbers they resemble.

/-- K4.8. PRC integers as signed-orbit quotient classes. The quotient is
taken by the internal balanced-length relation; the verifier display into
`ℤ` is a downstream theorem. -/
def PRCInt : Type :=
  Quot signedOrbitSetoid

/-- K4.8. PRC integers as signed-orbit quotient classes. The quotient is
taken by the internal balanced-length relation; the verifier display into
`ℤ` is a downstream theorem. -/
def PRCInt : Type :=
  Quot signedOrbitSetoid

What this page does not claim

Signed orbits are identical to integers in every structural respect. The theorem applies to the rational type PRCRat. The equivalence relation makes the raw pair map injective.

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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