toZ_one
toZ_one projects an element of the δ=1 subgroup of ℤ back to its underlying integer. Researchers building order isomorphisms for ledger units or scales cite it as the forward map when constructing equivalences between the subgroup and the base ring. The definition is a direct extraction of the subtype value field.
claimLet $p$ belong to the subgroup $Delta_1 = {x : ℤ | ∃ n : ℤ, x = n · 1}$. The projection $toZ_1(p)$ returns the underlying integer $p.val$.
background
DeltaSub δ is the subtype of ℤ consisting of all multiples of δ, i.e., {x : ℤ // ∃ n : ℤ, x = n * δ}. Specializing to δ = 1 produces the full ring of integers and yields a clean order isomorphism, as stated in the module documentation. The same projection appears in the RecogSpec.Scales module and is referenced from UnitMapping's abstract placeholder version of DeltaSub.
proof idea
One-line definition that returns the .val field of the subtype p : DeltaSub 1.
why it matters in Recognition Science
toZ_one supplies the forward map of equiv_delta_one : DeltaSub 1 ≃ ℤ, which sends n·1 to n. This equivalence is used in fromZ_toZ_one, toZ_fromZ_one, and mapDeltaOne_fromZ_one to handle δ=1 cases in ledger units and scale mappings. It realizes the module's specialization to δ=1 for a clean order isomorphism and supports downstream integer-subgroup constructions in the Recognition framework.
scope and limits
- Does not construct or prove properties of DeltaSub for general δ.
- Does not establish the equivalence or its inverses.
- Does not handle real-valued affine maps or time mappings.
- Does not interact with the phi-ladder or mass formulas.
formal statement (Lean)
13def toZ_one (p : DeltaSub 1) : ℤ := p.val
proof body
Definition body.
14