Foundation Primitive Recognition Calculus Delta Real Lo Le Hi Cross

The cross-bound lemma

A real number is often described by an infinite chain of nested rational intervals, each one a little narrower than the last. The classical picture is the familiar one: to pin down π, take intervals like [3, 4], [3.1, 3.2], [3.14, 3.15], and so on, with the width shrinking toward zero. The framework's ledger, a discrete record of events, builds real numbers the same way. Its protocol is a sequence of closed rational intervals that are nested, each contained in the previous, and whose widths are bounded by 1/(n+1) at step n.

The declaration lo_le_hi_cross proves a simple but essential fact about such a protocol: the lower bound at any step is at most the upper bound at any other step. If the lower bound at step a ever exceeded the upper bound at step b, the intervals would be inconsistent, and no real number could lie inside all of them. The proof uses monotonicity of the lower bounds and antitone upper bounds, together with the fact that each interval has its lower bound below its upper bound. This cross-bound lemma is the bridge that lets the framework define the value of a protocol as the supremum of all its lower bounds, and then prove that this value lies inside every interval.

From this lemma, the framework derives the central uniqueness theorem: any real number that lies in every interval of a protocol must equal the protocol's value. The proof bounds the difference between the candidate and the value by 1/(n+1) for every n, forcing the difference to zero. This is what makes the ledger a faithful representation of the reals: each protocol has exactly one real number as its limit, and every real number arises from some protocol. The framework's library, a machine-checked collection of formal theorems, also proves that the value respects addition, negation, and subtraction, and that two protocols are observationally equivalent exactly when their values agree.

What the lemma does not do is give a constructive method for finding that value, nor does it say anything about the physical meaning of a protocol. It is a piece of pure mathematics about nested intervals, not a claim about measurement or the structure of space. The framework uses it as a foundation for its real-number calculus, but the lemma itself is just a careful statement about bounds.

/-- Any lower endpoint is below any upper endpoint. -/
theorem lo_le_hi_cross (x : Protocol) (a b : ℕ) : x.lo a ≤ x.hi b := by
  have h1 : x.lo a ≤ x.lo (max a b) := x.lo_mono (le_max_left a b)
  have h2 : x.lo (max a b) ≤ x.hi (max a b) := x.lo_le_hi _
  have h3 : x.hi (max a b) ≤ x.hi b := x.hi_anti (le_max_right a b)
  linarith

/-- Squeeze: any real in every interval equals the value. -/
theorem value_unique (x : Protocol) (y : ℝ)
    (hy : ∀ n, x.lo n ≤ y ∧ y ≤ x.hi n) : y = x.value := by
  have hbound : ∀ n : ℕ, |y - x.value| ≤ 1 / (n + 1) := by
    intro n
    obtain ⟨h1l, h1r⟩ := hy n
    obtain ⟨h2l, h2r⟩ := x.value_mem n
    have hw := x.width_real_bound n
    rw [abs_le]
    constructor <;> linarith
  have : |y - x.value| = 0 := tiny_le_zero (abs_nonneg _) hbound
  have := abs_eq_zero.mp this
  linarith

/-- The value lies in every interval. -/
theorem value_mem (x : Protocol) (n : ℕ) : x.lo n ≤ x.value ∧ x.value ≤ x.hi n :=
  ⟨x.lo_le_value n, x.value_le_hi n⟩

/-- `value` is surjective onto ℝ. -/
theorem value_surjective : Function.Surjective Protocol.value :=
  fun r => ⟨canonical r, value_canonical r⟩

What this page does not claim

The lemma does not provide a constructive algorithm for computing the value of a protocol. The lemma does not make any physical claim about measurement or the structure of space. The lemma does not imply that the framework's real numbers are the only possible formalization.

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

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