[Merged by Bors] - feat(linear_algebra/multilinear): Add dom_dom_congr #5136
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eric-wieser
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eric-wieser
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eric-wieser
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src/data/equiv/basic.lean
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| { simp only [h, dif_pos], | ||
| have A : e.symm ⟨i, h⟩ = j ↔ i = e j, | ||
| by { rw equiv.symm_apply_eq, exact subtype.ext_iff_val }, | ||
| by_cases h' : i = e j, | ||
| { rw [A.2 h', h'], simp }, | ||
| { have : ¬ e.symm ⟨i, h⟩ = j, by simpa [← A] using h', | ||
| simp [h, h', this] } }, | ||
| { rw dif_pos h, | ||
| rw [function.update_apply_equiv_apply, equiv.symm_symm, function.comp], | ||
| rw [function.update_apply, function.update_apply], | ||
| rw dif_pos h, | ||
| have h_coe : (⟨i, h⟩ : s) = e j ↔ i = e j := subtype.ext_iff.trans (by rw subtype.coe_mk), | ||
| simp_rw h_coe, | ||
| congr, }, |
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Happy to revert these, but I figured there was value to showing the reader that this is an extension of the update_apply_equiv_apply case.
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Co-authored-by: Bhavik Mehta <bhavikmehta8@gmail.com>
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| (m : multilinear_map R (λ i : ι, M₂) M₃) (σ : equiv.perm ι) : | ||
| multilinear_map R (λ i : ι, M₂) M₃ := |
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In hindsight, this should have had signature
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| (m : multilinear_map R (λ i : ι, M₂) M₃) (σ : equiv.perm ι) : | |
| multilinear_map R (λ i : ι, M₂) M₃ := | |
| (m : multilinear_map R (λ i : ι1, M₂) M₃) (σ : ι1 ≃ ι2) : | |
| multilinear_map R (λ i : ι2, M₂) M₃ := |
I'll make a follow-up PR to generalize this, since I don't want to have to build all the way from function.basic again! Let's let bors finish merging this one.
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I think this should perhaps be something like:
@[simps apply]
private def dom_dom_congr_aux {M₂ M₃ : Type*} {ι₁ ι₂ : Type*} [decidable_eq ι₁] [decidable_eq ι₂]
[add_comm_monoid M₂] [add_comm_monoid M₃] [semimodule R M₂] [semimodule R M₃]
(σ : ι₁ ≃ ι₂) (m : multilinear_map R (λ i : ι₁, M₂) M₃) : multilinear_map R (λ i : ι₂, M₂) M₃ :=
{ to_fun := λ v, m (λ i, v (σ i)),
map_add' := λ v i a b, by { simp_rw function.update_apply_equiv_apply v, rw m.map_add, },
map_smul' := λ v i a b, by { simp_rw function.update_apply_equiv_apply v, rw m.map_smul, }, }
/-- Transfer the equivalence between argument indices to an equivalence between maps
The naming is derived from `finsupp.dom_congr`, noting that here the permutation applies to the
domain of the domain. -/
@[simps apply symm_apply]
def dom_dom_congr {M₂ M₃ : Type*} {ι₁ ι₂ : Type*} [decidable_eq ι₁] [decidable_eq ι₂]
[add_comm_monoid M₂] [add_comm_monoid M₃] [semimodule R M₂] [semimodule R M₃]
(σ : ι₁ ≃ ι₂) :
multilinear_map R (λ i : ι₁, M₂) M₃ ≃+ multilinear_map R (λ i : ι₂, M₂) M₃ :=
{ to_fun := dom_dom_congr_aux σ,
inv_fun := dom_dom_congr_aux σ.symm,
left_inv := λ m, by {ext, simp},
right_inv := λ m, by {ext, simp},
map_add' := λ a b, by {ext, simp} }
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Pull request successfully merged into master. Build succeeded: |
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https://leanprover.zulipchat.com/#narrow/stream/217875-Is-there.20code.20for.20X.3F/topic/If.20.60f.60.20is.20a.20multilinear.20map.20then.20.60.CE.BB.20v.2C.20f.20%28v.20.E2.88.98.20.CF.83%29.60.20is.20too/near/217986991