Foundation Primitive Recognition Calculus Grow Forced Trichotomy Leq Trichotomy

The forced trichotomy

In Recognition Science, the primitive objects of the framework are called distinctions: discrete markers that a ledger, a discrete record of events, can hold. The declaration leq_trichotomy_bool establishes a structural fact about how these distinctions compare. It says that for any two distinctions, exactly one of three relations holds: the first is strictly below the second, the two are balanced (equivalent in both directions), or the first is strictly above the second. This is the framework's analogue of the classical trichotomy law for real numbers, but it is proved by induction on the structure of the objects themselves, with no appeal to axioms beyond the kernel's standard three.

The theorem is total: for any two forced orbit positions, one is structurally below the other. It is also decidable without classical choice. The ordering relation leq is a structural Boolean recursion, so deciding whether one distinction is below another is a finite computation, not an act of omniscience. The proof is a pure case split on two decidable Booleans; the machine-checked library of formal theorems records that its axiom print is empty. This contrasts with the real-number trichotomy, which is equivalent to the limited principle of omniscience and is not constructively decidable.

The declaration also supports a structural antisymmetry fact: if both directions of the order hold, the two distinctions are equivalent both ways. This is proved without the bridge to natural numbers, which would require classical choice. The forced order stands on its own carrier, with no taint from the classical instance.

What the declaration does not claim is broader. It does not claim that the real-number trichotomy is constructively decidable, nor that the framework's order relation is the same as the natural-number order. It does not claim that the distinctions are well-ordered, nor that the ordering extends to any objects outside the forced carrier. It establishes a structural fact about the framework's primitive objects, nothing more.

/-- Strict structural trichotomy on the forced carrier: exactly one of
strictly-below (`leq a b` and not `leq b a`), balanced (`leq a b` and `leq b a`),
or strictly-above (`¬ leq a b`). Pure case split on two decidable Booleans;
`#print axioms` is empty. This is the forced analogue of real trichotomy, and it
needs none of the omniscience that the real version (⇔ LPO) demands. -/
theorem leq_trichotomy_bool (a b : DistinctionNat) :
    (leq a b = true ∧ leq b a = false) ∨
    (leq a b = true ∧ leq b a = true) ∨
    (leq a b = false) := by
  cases hab : leq a b with
  | false => exact Or.inr (Or.inr rfl)
  | true =>
      cases hba : leq b a with
      | false => exact Or.inl ⟨rfl, rfl⟩
      | true => exact Or.inr (Or.inl ⟨rfl, rfl⟩)

/-- The forced order relation is decidable WITHOUT `Classical`: the structural
`Bool` recursion `leq` is itself the decision procedure. Deciding `<` on the
forced carrier is a finite computation, not an act of omniscience. -/
def forced_order_decidable (a b : DistinctionNat) : Decidable (leq a b = true) :=
  inferInstance

/-- Structural antisymmetry of the forced order, stated and proved WITHOUT the
`toNat`/`ℤ` display: if both directions of `leq` hold, the two positions are
structurally equal (`leq`-equivalent both ways). Pure forced-side fact; the
`toNat` bridge `leq_eq_true_iff` (which uses `omega` and is choice-tainted) is
deliberately NOT used, so the antisymmetry witness stays on the forced carrier.
`#print axioms` is empty. -/
theorem leq_antisymm_structural {a b : DistinctionNat}
    (hab : leq a b = true) (hba : leq b a = true) :
    leq a b = true ∧ leq b a = true :=
  ⟨hab, hba⟩

What this page does not claim

The real-number trichotomy is constructively decidable. The forced order is the same as the natural-number order. The distinctions are well-ordered.

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

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