Foundation Primitive Recognition Calculus Prctype Theory Parse Tt System Expr Re

What reflexivity means here

In the Recognition Science framework, a ledger is a discrete record of events, and the framework's machine-checked library of formal theorems studies which ledgers can serve as foundations. One basic property a ledger can have is reflexivity: every expression in the ledger can be traced back to itself along the ledger's own ordering. The declaration ttSystem_exprReflexive proves, in the framework's library, that the two-element type of Martin-Löf type theory, here represented by the Boolean type with values false and true, has exactly this property.

Concretely, the theorem states that for every natural number n, the expression n extends itself in the system's ordering, which is the usual less-than-or-equal relation on natural numbers. The proof is immediate: n ≤ n by reflexivity of the natural order. The declaration is part of a larger package that parses type theory into the framework's FormalSystem interface, where expressions are natural numbers, tokens are the two Boolean values, and the discrimination relation is inequality. The framework also proves that this system is expressive, meaning the two tokens are distinct, and that it is not degenerate, meaning it realizes the framework's delta core.

What the declaration does not claim is broader. It does not claim that reflexivity holds for every formal system in the framework; it is a property of this particular system, proved for it. It does not claim that the two-element type is the only reflexive system, nor that reflexivity alone carries the framework's content. The theorem is a small structural fact: the natural-number expressions of this parsed type theory satisfy the reflexive condition. Its role is to certify that the type theory, as parsed into the framework, meets a basic requirement for being a well-formed ledger, alongside the canonicity and no-confusion theorems that establish the two-element type has exactly two inhabitants and that they are distinct.

theorem ttSystem_exprReflexive : DistinctionDichotomy.ExprReflexive ttSystem :=
  fun n => Nat.le_refl n

/-- **The faithful parse, packaged.** Type theory's two-element type satisfies
canonicity (exactly two closed terms) and no-confusion (they are distinct), and the
foundation realizes the δ core. -/
theorem type_theory_realizes_delta :
    (∀ b : Two, b = false ∨ b = true)
      ∧ ((false : Two) ≠ true)
      ∧ Nonempty (PRCEmbeddingInto ttSystem) :=
  ⟨canonicity, no_confusion, ttSystem_embeds_delta⟩

theorem ttSystem_not_degenerate : ¬ DistinctionDichotomy.Degenerate ttSystem :=
  DistinctionDichotomy.not_degenerate_of_realizesDelta ttSystem ttSystem_embeds_delta

What this page does not claim

Reflexivity holds for every formal system in the framework. The two-element type is the only reflexive system. Reflexivity alone carries the framework's substantive content.

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

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