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refactor(LinearAlgebra/BilinearMap): Congruences and bilinear maps - #8255

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refactor(LinearAlgebra/BilinearMap): Congruences and bilinear maps#8255
mans0954 wants to merge 13 commits into
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mans0954/LinearMap-congr

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@mans0954 mans0954 commented Nov 7, 2023

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Currently Mathlib has BilinForm.congr which defines a congruence between bilinear forms on the domain and range of an equivalence. This PR generalises this result to bilinear maps. The bilinear form version is recovered as a special case of the more general result.

Required for #8256


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Comment thread Mathlib/LinearAlgebra/BilinearMap.lean Outdated
Comment on lines +386 to +388
/-- Apply linear equivalences on the arguments of a bilinear map. -/
def congrl₁₂ (e₁ : Mₗ ≃ₗ[R] Nₗ) (e₂ : Pₗ ≃ₗ[R] Qₗ) :
(Mₗ →ₗ[R] Pₗ →ₗ[R] M) ≃ₗ[R] (Nₗ →ₗ[R] Qₗ →ₗ[R] M) where

@j-loreaux j-loreaux Dec 4, 2023

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I'm not convinced we need this at all, as it's just two applications of LinearEquiv.arrowCongr. And even if we did want it, it should take a third equivalence as an argument and be called LinearEquiv.arrowCongr₂

Suggested change
/-- Apply linear equivalences on the arguments of a bilinear map. -/
def congrl₁₂ (e₁ : Mₗ ≃ₗ[R] Nₗ) (e₂ : Pₗ ≃ₗ[R] Qₗ) :
(Mₗ →ₗ[R] Pₗ →ₗ[R] M) ≃ₗ[R] (Nₗ →ₗ[R] Qₗ →ₗ[R] M) where
/-- Apply linear equivalences on the arguments of a bilinear map. -/
def congrl₁₂ (e₁ : Mₗ ≃ₗ[R] Nₗ) (e₂ : Pₗ ≃ₗ[R] Qₗ) :
(Mₗ →ₗ[R] Pₗ →ₗ[R] M) ≃ₗ[R] (Nₗ →ₗ[R] Qₗ →ₗ[R] M) :=
e₁.arrowCongr <| e₂.arrowCongr <| .refl R M

(Note this is definitionally equal to what you wrote.) If you want, you can update the definition of BilinForm.congr to use LinearEquiv.arrowCongr

@j-loreaux j-loreaux added awaiting-author A reviewer has asked the author a question or requested changes. t-algebra Algebra (groups, rings, fields, etc) and removed awaiting-review labels Dec 4, 2023
@ghost ghost added the merge-conflict The PR has a merge conflict with master, and needs manual merging. (this label is managed by a bot) label Dec 8, 2023
@ghost ghost removed the merge-conflict The PR has a merge conflict with master, and needs manual merging. (this label is managed by a bot) label Dec 23, 2023
@mans0954

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Not needed.

@mans0954 mans0954 closed this Dec 23, 2023
@YaelDillies
YaelDillies deleted the mans0954/LinearMap-congr branch August 12, 2025 05:39
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