Foundation Modular Logic Realization Fin Cost Symm

The symmetry of simple costs

In mathematics, a cost function assigns a number to a pair of objects, often measuring how different they are. The simplest possible example is a two-valued cost on a finite set: it charges 0 when the two objects are identical, and 1 otherwise. This is the equality cost, and it is the subject of the declaration finCost_symm.

The declaration proves a symmetry property: for any two elements x and y in a finite set, the cost of going from x to y equals the cost of going from y to x. In symbols, finCost x y = finCost y x. The proof is a short case analysis: if x equals y, both sides are 0; if they differ, both sides are 1. This is a theorem in the machine-checked library of formal theorems, meaning it has been verified step by step by a computer.

In Recognition Science, this lemma appears inside a broader construction. The framework models a ledger, a discrete record of events, and defines a cost of recognition, a number measuring how hard it is to match one event to another. The equality cost is one possible such cost. The lemma finCost_symm establishes that this particular cost treats both directions the same way: matching A to B costs the same as matching B to A.

The lemma does not claim that all recognition costs are symmetric. It applies only to the specific equality cost defined in the same file. It also does not derive the unique cost function J(x) = (x + 1/x)/2 - 1 that appears elsewhere in the framework; that is a separate theorem with different hypotheses. The lemma is a small, precise building block, not a general statement about the nature of recognition.

theorem finCost_symm {m : ℕ} (x y : Fin m) : finCost x y = finCost y x := by
  by_cases h : x = y
  · subst h
    simp [finCost]
  · have h' : y ≠ x := by intro hyx; exact h hyx.symm
    simp [finCost, h, h']

theorem finCost_symm {m : ℕ} (x y : Fin m) : finCost x y = finCost y x := by
  by_cases h : x = y
  · subst h
    simp [finCost]
  · have h' : y ≠ x := by intro hyx; exact h hyx.symm
    simp [finCost, h, h']

What this page does not claim

This lemma does not prove that the unique cost function J(x) is symmetric. This lemma does not apply to any cost function other than the equality cost defined in the same file.

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

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