mapDelta_fromZ
plain-language theorem explainer
The lemma shows that the mapDelta operation applied to a δ-subgroup element built by fromZ recovers the affine evaluation f.slope · n + f.offset. Researchers reducing explicit time or action maps on integer multiples of δ cite this result. The proof is a one-line term simplification that unfolds mapDelta and invokes toZ_fromZ.
Claim. Let $δ ∈ ℤ$ with $δ ≠ 0$, let $f$ be an affine map $ℤ → ℝ$ with slope $m$ and offset $b$, and let $n ∈ ℤ$. Then mapDelta(δ, f, fromZ(δ, n)) = m · n + b, where fromZ embeds the coefficient n into the subgroup generated by δ.
background
The UnitMapping module treats the subgroup generated by a fixed nonzero integer δ as an abstract placeholder for scaling units. AffineMapZ is the structure (slope : ℝ, offset : ℝ) representing the map n ↦ slope · n + offset. The function mapDelta composes an affine map with the integer projection toZ that extracts the coefficient from an element of the δ-subgroup.
proof idea
Term-mode proof consisting of a single simplification step. It unfolds the definition of mapDelta and applies the upstream lemma toZ_fromZ δ hδ, after which the expression reduces directly to the affine evaluation on n.
why it matters
The result is invoked by mapDeltaTime_fromZ and mapDeltaAction_fromZ to obtain the explicit linear forms for time and action. It supplies the concrete reduction step that links the general mapDelta definition to the fromZ construction inside the unit-mapping layer. No open questions are touched.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.