Foundation Primitive Recognition Calculus Basic Extends Refl

The reflexivity of extension

A trace is a finite sequence of primitive distinction acts, the most basic unit of recognition in the Recognition Science framework. The declaration extends_refl proves a simple property of the relation Extends on traces: every trace extends itself. Formally, for any trace T, we have Extends T T, meaning there exists a suffix trace V such that appending V to T yields T. The proof is immediate: take V to be the empty trace, and the append operation returns T unchanged.

This is a theorem in the framework's machine-checked library of formal theorems, proved by reflexivity of equality. It establishes that the extension relation is reflexive, a standard property of partial orders and preorders. The result is foundational in the sense that it is one of the first facts proved about traces, but it is deliberately modest: it says nothing about whether a trace can be extended by a non-empty suffix, nor about the direction of time or the order of recognition events. It is a purely syntactic statement about the append operation and the empty trace.

In the broader context of the framework, this reflexivity result is a building block. Together with the transitivity of extension, which is also proved in the same file, it shows that the extension relation behaves like a preorder on traces. This is a necessary step for later developments, but it does not by itself force any physical content. The theorem does not claim that every trace is extendable, that extension is antisymmetric, or that the relation is total. Those are separate questions, and some of them are not addressed in this file.

What the result changes is the formal setting: it gives the framework a clean, verified starting point for reasoning about how traces relate to one another. A reader can now rely on the fact that the extension relation is reflexive without rechecking the definition. That is the practical payoff of a machine-checked proof, even for a statement that seems obvious from the definition.

/-- R4. Trace extension is reflexive. -/
theorem extends_refl (T : Trace) :
    Extends T T := by
  exact ⟨Trace.empty, rfl⟩

/-- R4. Trace extension is reflexive. -/
theorem extends_refl (T : Trace) :
    Extends T T := by
  exact ⟨Trace.empty, rfl⟩

What this page does not claim

The theorem does not claim that every trace can be extended by a non-empty suffix. It does not claim that extension is antisymmetric or that it forms a partial order. It does not claim anything about the physical direction of time or the order of recognition events.

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

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