[Merged by Bors] - refactor(linear_algebra/pi): add linear_map.single to match add_monoid_hom.single
#6315
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…noid_hom.single` This changes the definition of `std_basis` to be exactly `linear_map.single`, but proves equality with the old definition. In future, it might make sense to remove `std_basis` entirely.
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Feb 19, 2021
| @[simps] | ||
| def single [decidable_eq ι] (i : ι) : φ i →ₗ[R] (Πi, φ i) := | ||
| { to_fun := pi.single i, | ||
| map_smul' := λ r x, begin |
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Since you simplify linear_map.single i x to pi.single i x, it makes sense to add a lemma pi.single_smul somewhere near the definition of pi.semimodule. BTW, function.update_eq_iff gives a slightly shorter proof.
Otherwise LGTM.
bors d+
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We don't have pi.single_add yet either, but I think we should - I'll leave adding all of those lemmas to a follow-up PR.
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But I will have a go at golfing this proof.
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I came up with
map_smul' := λ r x, begin
ext j,
have := (function.apply_update (λ i, (•) r) 0 i x j).symm,
simp only [pi.zero_apply, smul_zero] at this,
exact this,
end,Do you have a better proof in mind?
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bryangingechen
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Let's golf the proofs in the follow-up, since adding |
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bors r+ |
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…noid_hom.single` (#6315) This changes the definition of `std_basis` to be exactly `linear_map.single`, but proves equality with the old definition. In future, it might make sense to remove `std_basis` entirely.
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bors r+ Not sure what happened there, bors appears to have discarded that batch. |
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The log says that bors crashed, hopefully just a transient thing: {%DBConnection.ConnectionError{
message: "connection not available because of disconnection"
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…noid_hom.single` (#6315) This changes the definition of `std_basis` to be exactly `linear_map.single`, but proves equality with the old definition. In future, it might make sense to remove `std_basis` entirely.
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bors
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linear_map.single to match add_monoid_hom.singlelinear_map.single to match add_monoid_hom.single
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…noid_hom.single` (#6315) This changes the definition of `std_basis` to be exactly `linear_map.single`, but proves equality with the old definition. In future, it might make sense to remove `std_basis` entirely.
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This changes the definition of
std_basisto be exactlylinear_map.single, but proves equality with the old definition.In future, it might make sense to remove
std_basisentirely.