Foundation Discrete Logic Realization Bool Cost Symm

The Boolean comparison cost

A Boolean value is a truth value: true or false, 1 or 0, yes or no. A cost function on Booleans assigns a number to each pair of such values. The most basic choice, written boolCost, charges 0 when the two values are the same and 1 when they differ. That is the entire definition: equality costs nothing, distinction costs one unit. This is the discrete, propositional carrier mentioned in the framework's second Law-of-Logic realization, the first non-continuous test case for its universal forcing construction.

The theorem boolCost_symm states that this cost is symmetric: boolCost p q = boolCost q p for any two Booleans p and q. The proof is a case split. If p equals q, both sides are 0 by the definition. If p differs from q, then q differs from p, and both sides are 1. That is the whole argument. The theorem is machine-checked in the framework's library of formal theorems, with no hidden assumptions beyond the standard logical axioms.

What the theorem does not claim is just as important. It does not claim that this Boolean cost is the unique cost on two values, nor that it satisfies any composition law. It does not claim that the discrete realization is the only one, nor that it is preferred over the continuous realizations. The theorem only establishes symmetry for this particular definition. It is a building block, not a complete theory.

The symmetry property matters because it is one of the five plain conditions that, in the continuous case, force the unique cost function J(x) = (x + 1/x)/2 - 1. Here, in the discrete case, symmetry holds trivially. The discrete realization also carries a forced arithmetic structure, equivalent to Peano arithmetic, and has a non-trivial identity-step shadow. These are separate theorems in the same file, each with its own proof and scope.

theorem boolCost_symm (p q : Bool) : boolCost p q = boolCost q p := by
  by_cases h : p = q
  · subst h
    simp [boolCost]
  · have h' : q ≠ p := by intro hqp; exact h hqp.symm
    simp [boolCost, h, h']

theorem boolCost_symm (p q : Bool) : boolCost p q = boolCost q p := by
  by_cases h : p = q
  · subst h
    simp [boolCost]
  · have h' : q ≠ p := by intro hqp; exact h hqp.symm
    simp [boolCost, h, h']

theorem boolCost_symm (p q : Bool) : boolCost p q = boolCost q p := by
  by_cases h : p = q
  · subst h
    simp [boolCost]
  · have h' : q ≠ p := by intro hqp; exact h hqp.symm
    simp [boolCost, h, h']

What this page does not claim

The Boolean cost is not claimed to be unique among costs on two values. The Boolean cost is not claimed to satisfy any composition law. The discrete realization is not claimed to be preferred over continuous realizations.

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

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