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feat(Topology/PartitionOfUnity): add pointwise_decomposition_finsum + companions#39162

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feat(Topology/PartitionOfUnity): add pointwise_decomposition_finsum + companions#39162
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@anovickis anovickis commented May 11, 2026

Add three short lemmas to Mathlib/Topology/PartitionOfUnity.lean extending the existing PartitionOfUnity API:

  • pointwise_decomposition_finsum — for f : X → ℝ and x ∈ 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_nonneg0 ≤ 1 - ∑ᶠ i, ρ i x. Direct rearrangement of the existing sum_le_one field; useful as a complement-mass remainder bound in chart-by-chart estimates.
  • abs_le_one|ρ i x| ≤ 1. Combines the existing nonneg and le_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 f into 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

… 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>
@github-actions github-actions Bot added the new-contributor This PR was made by a contributor with at most 5 merged PRs. Welcome to the community! label May 11, 2026
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Welcome new contributor!

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@github-actions github-actions Bot added the t-topology Topological spaces, uniform spaces, metric spaces, filters label May 11, 2026
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PR summary aafc9e6204

Import changes for modified files

No significant changes to the import graph

Import changes for all files
Files Import difference

Declarations diff

+ abs_le_one
+ one_minus_sum_nonneg
+ pointwise_decomposition_finsum

You can run this locally as follows
## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.


No changes to strong technical debt.
No changes to weak technical debt.

@grunweg grunweg added the LLM-generated PRs with substantial input from LLMs - review accordingly label May 11, 2026
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