English
A BumpCovering with smooth bump functions yields a SmoothPartitionOfUnity by pairing each bump with the corresponding index.
Русский
Покрытие бампами с гладкими функциями даёт SmoothPartitionOfUnity, если каждой контура bump присвоить соответствующий индекс.
LaTeX
$$Let f be a BumpCovering with hf i smooth; then toSmoothPartitionOfUnity f hf is a SmoothPartitionOfUnity.$$
Lean4
/-- A `BumpCovering` such that all functions in this covering are smooth generates a smooth
partition of unity.
In our formalization, not every `f : BumpCovering ι M s` with smooth functions `f i` is a
`SmoothBumpCovering`; instead, a `SmoothBumpCovering` is a covering by supports of
`SmoothBumpFunction`s. So, we define `BumpCovering.toSmoothPartitionOfUnity`, then reuse it
in `SmoothBumpCovering.toSmoothPartitionOfUnity`. -/
def toSmoothPartitionOfUnity (f : BumpCovering ι M s) (hf : ∀ i, ContMDiff I 𝓘(ℝ) ∞ (f i)) :
SmoothPartitionOfUnity ι I M s :=
{ f.toPartitionOfUnity with toFun := fun i => ⟨f.toPartitionOfUnity i, f.contMDiff_toPartitionOfUnity hf i⟩ }
