boolCost_self
plain-language theorem explainer
boolCost_self establishes that the comparison cost between any Boolean value and itself is zero under the discrete propositional carrier. Researchers constructing Law-of-Logic realizations in the Recognition Science framework cite it when verifying the identity property for the first non-continuous test case of Universal Forcing. The result follows from a one-line simplification that unfolds the cost definition directly.
Claim. For every Boolean value $p$, the comparison cost satisfies $C(p,p)=0$, where the cost function is defined by $C(p,q)=0$ if $p=q$ and $C(p,q)=1$ otherwise.
background
The DiscreteLogicRealization module supplies the second Law-of-Logic realization, using a discrete Boolean/propositional carrier as the first non-continuous test case for Universal Forcing. The central definition is the Boolean comparison cost: zero when the arguments are equal and one when they differ, expressed as if p = q then 0 else 1. This module imports UniversalForcing.Strict.DiscreteBoolean, which supplies the identical cost definition and its self-property.
proof idea
The proof is a one-line wrapper that applies the simp tactic to the definition of boolCost, which immediately reduces the self-comparison case to zero.
why it matters
The result feeds the discrete propositional Law-of-Logic realization (boolRealization) that sets Carrier to Bool and compare to the cost function, and likewise supports the strict Boolean realization. It verifies the zero-cost identity step required for the discrete carrier in the forcing chain, consistent with the Recognition Composition Law and the emphasis on self-similar fixed points.
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