le_refl
plain-language theorem explainer
Reflexivity of the order on LogicNat follows directly from the definition of ≤ via addition and the identity n + zero = n. Researchers building path interpolation lemmas or non-negativity arguments in archaeology and complexity cite this fact to fix endpoints or bound quantities. The proof is a one-line constructor that applies the add_zero theorem to supply the witness.
Claim. Let LogicNat be the inductive type generated by the identity constructor (zero-cost element) and the step constructor. For every $n$ in LogicNat the relation $n ≤ n$ holds, witnessed by the zero element together with the identity $n + 0 = n$.
background
LogicNat is the inductive type forced by the Law of Logic, with identity as the zero-cost multiplicative identity and step as one further iteration of the generator; its structure mirrors the orbit {1, γ, γ², …} inside the positive reals. The order ≤ on LogicNat is witnessed by an element k such that the sum of the left argument with k recovers the right argument. The upstream add_zero theorem states that n + zero = n for any n in LogicNat and is proved by reflexivity of equality.
proof idea
The proof is a one-line wrapper that applies add_zero: the zero element is supplied as the witness for the existential in the definition of ≤, and add_zero immediately gives the required equality n + zero = n.
why it matters
This reflexivity result is invoked in downstream theorems such as interp_zero and interp_one (PathSpace), popularity_nonneg (Archaeology), sat_recognition_time_bound (Complexity), and geodesic_minimizes_unconditional (FunctionalConvexity). It supplies the basic arithmetic needed to support the J-cost functional and the forcing chain from T0 to T8 inside the Recognition Science framework.
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