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[Merged by Bors] - feat(linear_algebra/trace): dual_tensor_hom is an equivalence + basis-free characterization of the trace #10372

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[Merged by Bors] - feat(linear_algebra/trace): dual_tensor_hom is an equivalence + basis-free characterization of the trace #10372

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@antoinelab01 antoinelab01 added the awaiting-review The author would like community review of the PR label Nov 17, 2021
@antoinelab01 @ocfnash
Update src/linear_algebra/tensor_product_basis.lean
8451a8b 
Co-authored-by: Oliver Nash <github@olivernash.org>
eric-wieser
eric-wieser reviewed Nov 19, 2021
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Using the explicit right inverse, I think you should refactor this to:

  • def dual_tensor_hom_equiv_of_basis which requires only a comm_ring, and an explicit basis
  • A proof that dual_tensor_hom_equiv_of_basis b1 = dual_tensor_hom_equiv_of_basis b2 (trivially follows from linear_equiv.ext)
  • def dual_tensor_hom_equiv := dual_tensor_hom_equiv_of_basis (fin_basis R M) which requires a field

@eric-wieser eric-wieser added awaiting-author A reviewer has asked the author a question or requested changes and removed awaiting-review The author would like community review of the PR labels Nov 19, 2021
riccardobrasca
riccardobrasca approved these changes Apr 14, 2022
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src/data/matrix/basis.lean Outdated
variable [fintype n]

lemma trace_zero (h : j ≠ i) : trace n α α (std_basis_matrix i j c) = 0 := by simp [h]

lemma trace_same : trace n α α (std_basis_matrix i i c) = c := by simp
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Since the proof is simp, do you really need it? What happens if you remove it?

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At any rate this lemma should include std_basis_matrix in its name

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You're right, I don't really need it.

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Thanks!

bors d+

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bors bot commented Apr 19, 2022

✌️ antoinelab01 can now approve this pull request. To approve and merge a pull request, simply reply with bors r+. More detailed instructions are available here.

@github-actions github-actions bot added delegated The PR author may merge after reviewing final suggestions. and removed awaiting-review The author would like community review of the PR labels Apr 19, 2022
eric-wieser
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src/data/finsupp/basic.lean Outdated
Comment on lines 1292 to 1298
@[simp] lemma fintype_sum_single [add_comm_monoid M] [fintype α] (i : α) (m : M) :
∑ (j : α), (single i m) j = m :=
by simp [single]

@[simp] lemma fintype_sum_single' [add_comm_monoid M] [fintype α] (i : α) (m : M) :
∑ (j : α), (single j m) i = m :=
by simp [single]
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I think these should instead be called sum_univ_single, since syntactically this is finset.univ.sum _ not fintype.sum _.

antoinelab01 and others added 4 commits April 19, 2022 14:11
@antoinelab01 @eric-wieser
Update src/linear_algebra/contraction.lean
ef1e848 
Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
@antoinelab01 @eric-wieser
Update src/linear_algebra/trace.lean
63cc7a6 
Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
@antoinelab01 @eric-wieser
Update src/linear_algebra/trace.lean
86cadfc 
Co-authored-by: Eric Wieser <wieser.eric@gmail.com>
@antoinelab01
changed name to sum_univ_single
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bors r+

bors bot pushed a commit that referenced this pull request Apr 19, 2022
@antoinelab01 @ocfnash
feat(linear_algebra/trace): dual_tensor_hom is an equivalence + basi…
046daa1 
…s-free characterization of the trace (#10372)




Co-authored-by: antoinelab01 <66086247+antoinelab01@users.noreply.github.com>
Co-authored-by: Oliver Nash <github@olivernash.org>
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Build failed (retrying...):

bors bot pushed a commit that referenced this pull request Apr 19, 2022
@antoinelab01 @ocfnash
feat(linear_algebra/trace): dual_tensor_hom is an equivalence + basi…
b40293f 
…s-free characterization of the trace (#10372)




Co-authored-by: antoinelab01 <66086247+antoinelab01@users.noreply.github.com>
Co-authored-by: Oliver Nash <github@olivernash.org>
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ocfnash commented Apr 20, 2022

@antoinelab01 I merged master because this was failing to build for bors. Do you think you can fix this?

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ocfnash commented Apr 20, 2022

bors merge

@github-actions github-actions bot added the ready-to-merge All that is left is for bors to build and merge this PR. (Remember you need to say `bors r+`.) label Apr 20, 2022
bors bot pushed a commit that referenced this pull request Apr 20, 2022
@antoinelab01 @ocfnash
feat(linear_algebra/trace): dual_tensor_hom is an equivalence + basi…
b0805a5 
…s-free characterization of the trace (#10372)




Co-authored-by: antoinelab01 <66086247+antoinelab01@users.noreply.github.com>
Co-authored-by: Oliver Nash <github@olivernash.org>
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Pull request successfully merged into master.

Build succeeded:

@bors bors bot changed the title feat(linear_algebra/trace): dual_tensor_hom is an equivalence + basis-free characterization of the trace [Merged by Bors] - feat(linear_algebra/trace): dual_tensor_hom is an equivalence + basis-free characterization of the trace Apr 20, 2022
@bors bors bot closed this Apr 20, 2022
@bors bors bot deleted the trace branch April 20, 2022 18:42
Julian added a commit that referenced this pull request Apr 21, 2022
@Julian
Merge remote-tracking branch 'origin/master' into torsion-free-groups
eb3119e 
* origin/master: (394 commits)
  feat(data/set/[basic|prod]): make `×ˢ` bind more strongly, and define `mem.out` (#13422)
  feat(order/basic): Simple shortcut lemmas (#13421)
  chore(number_theory/dioph): Cleanup (#13403)
  feat(analysis/normed_space/exponential): ring homomorphisms are preserved by the exponential (#13402)
  feat(algebraic_geometry/projective_spectrum): degree zero part of a localized ring (#13398)
  feat(set_theory/cardinal): A set of cardinals is small iff it's bounded (#13373)
  feat(data/polynomial/{derivative, iterated_deriv}): reduce assumptions (#13368)
  feat(algebra/monoid_algebra/grading): Use the new graded_algebra API (#13360)
  feat(algebra/group/to_additive): let @[to_additive] mimic alias’s docstrings (#13330)
  feat(set_theory/cofinality): Basic fundamental sequences (#13326)
  feat(special_functions/pow): continuity of real to complex power (#13244)
  feat(group_theory/torsion): extension closedness, and torsion scalars in modules (#13172)
  feat(category_theory/path_category): canonical quotient of a path category (#13159)
  refactor(number_theory/padics/padic_norm): Switch nat and rat definitions (#12454)
  feat(analysis/normed): more lemmas about the sup norm on pi types and matrices (#13536)
  fix(category_theory/monoidal): improve hygiene in coherence tactic (#13507)
  feat(src/number_theory/cyclotomic/discriminant): add discr_prime_pow_ne_two (#13465)
  chore(algebra/group/type_tags): missing simp lemmas (#13553)
  feat(measure_theory): allow measurability to prove ae_strongly_measurable (#13427)
  refactor(algebra/hom/group): generalize a few lemmas to `monoid_hom_class` (#13447)
  chore(data/list/cycle): Add basic `simp` lemmas + minor golfing (#13533)
  feat(algebra/hom/non_unital_alg): introduce notation for non-unital algebra homomorphisms (#13470)
  chore(algebra/group/defs): Declare `field_simps` attribute earlier (#13543)
  feat(analysis/normed/normed_field): add `one_le_(nn)norm_one` for nontrivial normed rings (#13556)
  refactor(analysis/calculus/cont_diff): reorder the file (#13468)
  move(set_theory/*): Organize in folders (#13530)
  chore(number_theory/zsqrtd/basic): simplify le_total proof (#13555)
  feat(group_theory/perm/basic): Iterating a permutation is the same as taking a power (#13554)
  feat(data/real/sqrt): `sqrt x < y ↔ x < y^2` (#13546)
  feat(algebra/hom/group and *): introduce `mul_hom M N` notation `M →ₙ* N` (#13526)
  feat(group_theory/schreier): Schreier's lemma in terms of `group.fg` and `group.rank` (#13361)
  feat(linear_algebra/trace): dual_tensor_hom is an equivalence + basis-free characterization of the trace (#10372)
  feat(order/filter/basic): allow functions between different types in lemmas about [co]map by a constant function (#13542)
  feat(data/finset/basic): simp `to_finset_eq_empty` (#13531)
  feat(topology/algebra/algebra): ℚ-scalar multiplication is continuous (#13458)
  chore(model_theory/encoding): Improve the encoding of terms  (#13532)
  feat(topology/separation): Finite sets in T2 spaces (#12845)
  feat(analysis/inner_product_space/gram_schmidt_ortho): Gram-Schmidt Orthogonalization and Orthonormalization (#12857)
  chore(algebra/big_operators/fin): golf finset.prod_range (#13535)
  chore(analysis/normed_space/star): make an argument explicit (#13523)
  feat(*): `op_op_op_comm` lemmas (#13528)
  chore(data/real/nnreal): add commuted version of `nnreal.mul_finset_sup` (#13512)
  chore(*/matrix): order `m` and `n` alphabetically (#13510)
  feat(analysis/calculus/specific_functions): trivial extra lemmas (#13516)
  feat(analysis): lemmas about nnnorm and nndist (#13498)
  feat(data/int/basic): add lemma `int.abs_le_one_iff` (#13513)
  feat(category_theory/limits): add characteristic predicate for zero objects (#13511)
  feat(order/filter/n_ary): Add lemma equating map₂ to map on the product (#13490)
  fix(analysis/locally_convex/balanced_hull_core): minimize import (#13450)
  feat(order/cover): define `wcovby` (#13424)
  ...
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