feat(Topology/PartitionOfUnity): add pointwise_decomposition_finsum + companions#39162
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… companions Adds three short lemmas extending the existing PartitionOfUnity API: - pointwise_decomposition_finsum: f x = ∑ᶠ i, ρ i x · f x for x ∈ s. This is the pointwise step that lifts to integral linearity in measure-theoretic PoU integration: ∫_s f dμ = ∑ᶠ i, ∫_s (ρ i · f) dμ. - one_minus_sum_nonneg: 0 ≤ 1 - ∑ᶠ i, ρ i x. Direct rearrangement of sum_le_one; useful as complement-mass remainder. - abs_le_one: |ρ i x| ≤ 1. Combines nonneg + le_one; convenience. All three are short (≤ 2-line proofs) using existing structure fields. Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
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Add three short lemmas to
Mathlib/Topology/PartitionOfUnity.leanextending the existingPartitionOfUnityAPI:pointwise_decomposition_finsum— forf : X → ℝandx ∈ s,f x = ∑ᶠ i, ρ i x · f x. This is the pointwise step that lifts to integral linearity in measure-theoretic PoU integration:∫_s f dμ = ∑ᶠ i, ∫_s (ρ i · f) dμ.one_minus_sum_nonneg—0 ≤ 1 - ∑ᶠ i, ρ i x. Direct rearrangement of the existingsum_le_onefield; useful as a complement-mass remainder bound in chart-by-chart estimates.abs_le_one—|ρ i x| ≤ 1. Combines the existingnonnegandle_one; convenience for absolute-value bounds.All three are short proofs using existing structure fields (
sum_eq_one,sum_le_one,nonneg,le_one).These came up while writing chart-by-chart Stokes-on-manifold estimates where one wants to decompose
∫_M finto chart-supported pieces using a partition of unity. The pointwise identity is the obvious first step; the other two are complementary algebraic bounds that show up in remainder estimates.🤖 Generated with Claude Code