intCost_self
The equality cost on integers vanishes exactly on identical arguments. Researchers constructing order realizations for the Universal Forcing chain cite this fact to confirm reflexivity of the zero-cost relation. The proof is a one-line simplification that unfolds the cost definition.
claimFor every integer $a$, if the cost function returns 0 on equal arguments and 1 otherwise, then the cost of $a$ with itself equals 0.
background
The OrderRealization module supplies a lightweight order-theoretic realization on the integers. Its carrier is the integers with cost defined to be zero on equal pairs and one otherwise; the forced arithmetic travels through a certified internal orbit while the carrier interpretation is the standard embedding of LogicNat into the integers.
proof idea
One-line wrapper that applies the simplifier to the definition of the cost function.
why it matters in Recognition Science
The result is invoked inside the orderRealization definition that builds the LogicRealization with integer carrier and the given cost; it likewise supports the strict ordered realization. In the Recognition Science setting it secures the zero element of the cost structure needed for the forcing chain steps T0-T8.
scope and limits
- Does not address cost functions on carriers other than the integers.
- Does not supply bounds or estimates beyond the exact diagonal value.
- Does not interact with the J-cost or phi-ladder constructions.
formal statement (Lean)
23@[simp] theorem intCost_self (a : ℤ) : intCost a a = 0 := by
proof body
One-line wrapper that applies simp.
24 simp [intCost]
25