Foundation Primitive Recognition Calculus Prcjcost Distance Verifier Triangle Pr

The conditional certificate

The declaration prc_jcost_distance_verifier_triangle_conditional_certificate is a formal object in the Recognition Science framework's machine-checked library of formal theorems. It establishes a conditional result: if a certain one-dimensional estimate about rational numbers holds, then a broader three-endpoint triangle property also holds. The certificate does not prove that estimate; it proves the implication from the estimate to the property.

In plainer terms, the framework works with a cost, a measure of how expensive it is for reality to recognize one state as another. The cost between two rational numbers is displayed as a distance. The certificate concerns a triangle-like condition: for three endpoints, the cost from the first to the third should be bounded by the costs of the two legs. The declaration shows that this three-endpoint condition follows from a simpler, additive condition on increments alone, provided that simpler condition is true.

The certificate has four parts. First, it records that the displayed distance depends only on the difference between endpoints, not on their absolute positions. Second, it states the increment-only target as a formal proposition. Third, it proves that the increment target implies the three-endpoint verifier target. Fourth, it proves that the increment target also implies a setoid condition, which is a way of saying that points at zero distance behave as equivalent. The declaration assembles these four parts into a single structure.

The key reduction is that the three-endpoint triangle property is equivalent to a one-dimensional additive estimate on rational increments. This is a genuine simplification: instead of checking all triples of points, one checks a statement about pairs of small increments. The certificate makes this reduction precise and machine-checked.

What the certificate does not claim is equally important. It does not prove the increment-only estimate itself; that remains an open target. It does not prove the three-endpoint triangle property outright. It does not establish any property of the cost function beyond the reduction. The declaration is a conditional certificate, and its value lies in isolating exactly what remains to be shown.

/-- The increment-only triangle target implies the verifier-rational
three-endpoint triangle target. -/
theorem PRCJCostDistanceVerifierTriangleTarget_of_increment
    (h : PRCJCostDistanceIncrementTriangleTarget) :
    PRCJCostDistanceVerifierTriangleTarget := by
  intro eps heps
  rcases h eps heps with ⟨delta, hdelta_pos, hdelta⟩
  refine ⟨delta, hdelta_pos, ?_⟩
  intro x y z hxy hyz
  have hxy' :
      PRCJCostDistanceIncrementDisplay (x - y) < delta.toRat := by
    rwa [PRCJCostDistanceRatDisplay_as_increment] at hxy
  have hyz' :
      PRCJCostDistanceIncrementDisplay (y - z) < delta.toRat := by
    rwa [PRCJCostDistanceRatDisplay_as_increment] at hyz
  have hsum :
      PRCJCostDistanceIncrementDisplay ((x - y) + (y - z)) < eps.toRat :=
    hdelta (x - y) (y - z) hxy' hyz'
  have hxz :
      PRCJCostDistanceRatDisplay x z =
        PRCJCostDistanceIncrementDisplay ((x - y) + (y - z)) := by
    rw [PRCJCostDistanceRatDisplay_as_increment]
    congr
    ring
  rwa [hxz]

/-- The displayed distance is translation-invariant: it depends only on the
increment between the endpoints. -/
theorem PRCJCostDistanceRatDisplay_as_increment (x y : ℚ) :
    PRCJCostDistanceRatDisplay x y =
      PRCJCostDistanceIncrementDisplay (x - y) := by
  simp [PRCJCostDistanceIncrementDisplay, PRCJCostDistanceRatDisplay]
  ring_nf

/-- The increment-only blocker closes the whole PRC null-distance setoid chain. -/
theorem PRCNullDistanceSetoidTarget_of_increment_triangle
    (h : PRCJCostDistanceIncrementTriangleTarget) :
    PRCNullDistanceSetoidTarget :=
  PRCNullDistanceSetoidTarget_of_verifier_triangle
    (PRCJCostDistanceVerifierTriangleTarget_of_increment h)

What this page does not claim

The increment-only estimate is not proved by this certificate. The three-endpoint triangle property is not established outright. No property of the cost function beyond the reduction is claimed.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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