Foundation Primitive Recognition Calculus Basic Extends Trans

The extension relation

A trace is a finite record of distinction acts, where a distinction act is the primitive event of marking a difference. The record starts empty, and each act appends one mark. The relation extends holds between two traces when one is the other followed by some further suffix of acts. This is a syntactic relation on the records themselves, defined by the existence of that suffix.

The declaration extends_trans proves that this relation is transitive: if trace T extends trace U, and U extends trace V, then T extends V. The proof is direct: the suffix that takes T to U and the suffix that takes U to V can be appended, and associativity of trace append guarantees the combined suffix takes T to V. The theorem is machine-checked, meaning it holds by the rules of the formal system with no hidden assumptions.

Transitivity is the property that makes the extension relation an ordering. It lets the framework speak of a chain of traces, each building on the previous one, without ambiguity about whether the relation composes. This is the foundation for later claims about recognition cycles and scaling, where the order of records matters.

The theorem does not claim that every pair of traces is comparable. Two traces may be unrelated: neither extends the other. It does not claim that extension is antisymmetric, so two distinct traces could extend each other only if they were equal. It does not claim anything about the content of the acts, only their order. The relation is about the shape of the record, not what the marks mean.

/-- R4. Trace extension is transitive. -/
theorem extends_trans {T U V : Trace}
    (hTU : Extends T U) (hUV : Extends U V) :
    Extends T V := by
  rcases hTU with ⟨A, hA⟩
  rcases hUV with ⟨B, hB⟩
  refine ⟨append A B, ?_⟩
  rw [← append_assoc, hA, hB]

What this page does not claim

The theorem does not claim that any two traces are comparable by extension. It does not claim that extension is antisymmetric. It does not assign any meaning to the distinction acts themselves.

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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