IndisputableMonolith.Foundation.UniversalForcing.Strict.CategoricalMathlib
This module implements the LogicNat primitive recursion principle via Lean's inductive pattern matching on zero and successor. It supplies the Mathlib-compatible recursor for the strict categorical NNO surface. Researchers in categorical logic and NNO foundations cite it for the universal property. The implementation uses direct pattern matching on the inductive type.
claimThe recursor on the natural numbers object $\LogicNat$ satisfies the universal property: for any type $X$ with $z : X$ and $s : X \to X$, there exists a unique $f : \LogicNat \to X$ such that $f(0) = z$ and $f(n+1) = s(f(n))$.
background
The module sits in the Strict/Categorical layer of UniversalForcing and imports the upstream Categorical module whose doc states: "Strict categorical/Lawvere-style realization hook. The carrier is the canonical LogicNat NNO surface from CategoricalLogicRealization." It supplies the recursor, recursor_zero, recursor_succ, nno_universal_existence and nno_universal_uniqueness siblings. The setting is the Lawvere-style NNO realization inside the Recognition forcing chain (T0-T8) with J-uniqueness and the eight-tick octave.
proof idea
This is a definition module, no proofs. It encodes the LogicNat primitive recursion principle directly via Lean's built-in pattern matching on the inductive type, exposing the universal property as the CategoricalMathlibCert.
why it matters in Recognition Science
The module supplies the already-proved recursor universal property that the downstream MathlibNNO module exposes as the Mathlib-facing bridge to CategoryTheory. Its doc states the content is kept here so that MathlibNNO can connect the strict categorical realization without duplicating theorems.
scope and limits
- Does not implement Mathlib's full CategoryTheory NNO API.
- Does not refine the carrier beyond the LogicNat surface.
- Does not address higher categorical limits or adjunctions.
- Does not depend on external NNO constructions outside the imported Categorical module.