[Merged by Bors] - feat(LinearIndependent): LinearIndepOn predicate #21799
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PR summary 62b8136604Import changes for modified filesNo significant changes to the import graph Import changes for all files
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b-mehta
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Feb 13, 2025
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I'm on board, but I'd like to hear from @eric-wieser before merging.
eric-wieser
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Feb 13, 2025
Co-authored-by: Bhavik Mehta <[email protected]>
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b-mehta
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Feb 14, 2025
b-mehta
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Co-authored-by: Bhavik Mehta <[email protected]>
Co-authored-by: Bhavik Mehta <[email protected]>
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Added a predicate stating that the vectors in a family indexed by a set are linearly independent. As per [discussion on zulip](https://leanprover.zulipchat.com/#narrow/channel/287929-mathlib4/topic/.60Set.2Eincl.60)
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Currently, if `s` is a set of vectors in a module `W`, the way to state that the vectors in `s` are linearly independent is `LinearIndependent R (Subtype.val : ↥s → W)` (or similar), and if `v : ι -> W` and `s : Set ι` is a set of indices, the way to state that `s` indexes a linearly independent subcollection is `LinearIndependent R (fun x : ↥s ↦ (f x : W))` or similar. #21799 introduced a definition `LinearIndepOn`, which gives an alternative spelling `LinearIndepOn R f s` instead of `LinearIndependent R (fun x : ↥s ↦ (f x : W))` for the indexed version, and (therefore) `LinearIndepOn R id s` for the set version. This PR updates spellings throughout mathlib, changing terms of type `LinearIndependent R (Subtype.val : ↥s → W)` to `LinearIndepOn R id s` and `LinearIndependent R (fun x : ↥s ↦ (f x : W))` to `LinearIndepOn R f s`. The API using the old spellings has been deprecated, and replaced with new one - the diff is designed to be as small as possible subject to deprecating the old spellings. In some cases, where no extra work was needed, we generalized lemmas that previously apply to `LinearIndepOn R id s` to `LinearIndepOn R v s` for arbitrary `v`. But there is room to do this further in other places, some of which have been marked TODO. Most changes happen in `LinearAlgebra/LinearIndependent.lean`, but only about 100 lines have been added to the length, many of which are deprecated alias lines. Nearly all other changes are knock-on effects, where statements, proofs and (in two cases) definitions that use the old spellings are updated. 22 files are affected, but the majority only in the form of slight modifications of proofs to avoid deprecated lemmas. The two files other than `LinearIndependent.lean` with changed definitions are: - `Mathlib/LinearAlgebra/Dimension/Basic.lean` (definition of Module.rank) - `Mathlib/LinearAlgebra/Dimension/RankNullity.lean` (changed structure field of `HasRankNullity`) The files with changed theorem statements are: - `Mathlib/FieldTheory/Fixed.lean` - `Mathlib/LinearAlgebra/Basis/VectorSpace.lean` - `Mathlib/LinearAlgebra/Dimension/Constructions.lean` - `Mathlib/LinearAlgebra/Dimension/Finite.lean` - `Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean` - `Mathlib/LinearAlgebra/Dimension/LinearMap.lean` - `Mathlib/LinearAlgebra/Dimension/Localization.lean` - `Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean` [Zulip discussion](https://leanprover.zulipchat.com/#narrow/channel/144837-PR-reviews/topic/.2321886.20LinearIndepOn.20refactor/near/500527995)
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Added a predicate stating that the vectors in a family indexed by a set are linearly independent.
As per discussion on zulip