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Bhavik Mehta b-mehta Cambridge PhD Mathematics student at Trinity College, Cambridge

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Lecture notes from Cambridge maths

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Pull request review commentleanprover-community/mathlib

feat(data/nat): a predicate for prime powers

+/-+Copyright (c) 2022 Bhavik Mehta. All rights reserved.+Released under Apache 2.0 license as described in the file LICENSE.+Authors: Bhavik Mehta+-/+import algebra.associated+import data.nat.factorization++/-!+# Prime powers++This file deals with prime powers: numbers which are positive integer powers of a single prime.+-/++variables {R : Type*} [comm_monoid_with_zero R] (n p : R) (k : ℕ)++/-- `n` is a prime power if there is a prime `p` and a positive natural `k` such that `n` can be+written as `p^k`. -/+def is_prime_pow : Prop :=+  ∃ (p : R) (k : ℕ), prime p ∧ 0 < k ∧ p ^ k = n++lemma is_prime_pow_def :+  is_prime_pow n ↔ ∃ (p : R) (k : ℕ), prime p ∧ 0 < k ∧ p ^ k = n := iff.rfl++/-- An equivalent definition for prime powers: `n` is a prime power iff there is a prime `p` and a+natural `k` such that `n` can be written as `p^(k+1)`. -/+lemma is_prime_pow_iff_pow_succ :+  is_prime_pow n ↔ ∃ (p : R) (k : ℕ), prime p ∧ p ^ (k + 1) = n :=+(is_prime_pow_def _).trans+⟨λ ⟨p, k, hp, hk, hn⟩, ⟨_, _, hp, by rwa [nat.sub_add_cancel hk]⟩,+  λ ⟨p, k, hp, hn⟩, ⟨_, _, hp, nat.succ_pos', hn⟩⟩++lemma not_is_prime_pow_zero [no_zero_divisors R] :+  ¬ is_prime_pow (0 : R) :=+begin+  simp only [is_prime_pow_def, not_exists, not_and', and_imp],+  intros x n hn hx,+  rw pow_eq_zero hx,+  simp,+end++lemma not_is_prime_pow_one : ¬ is_prime_pow (1 : R) :=+begin+  simp only [is_prime_pow_def, not_exists, not_and', and_imp],+  intros x n hn hx ht,+  exact ht.not_unit (is_unit_of_pow_eq_one x n hx hn),+end++lemma prime.is_prime_pow {p : R} (hp : prime p) : is_prime_pow p :=+⟨p, 1, hp, zero_lt_one, by simp⟩++lemma is_prime_pow.pow {n : R} (hn : is_prime_pow n)+  {k : ℕ} (hk : k ≠ 0) : is_prime_pow (n ^ k) :=+let ⟨p, k', hp, hk', hn⟩ := hn in ⟨p, k * k', hp, mul_pos hk.bot_lt hk', by rw [pow_mul', hn]⟩++lemma unique_is_prime_pow {R : Type*} [cancel_comm_monoid_with_zero R] {p₁ p₂ : R} {k₁ k₂ : ℕ}

Yup you're right, I moved this and cleaned it and forgot to remove the old version!

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Pull request review commentleanprover-community/mathlib

feat(data/nat): a predicate for prime powers

+/-+Copyright (c) 2022 Bhavik Mehta. All rights reserved.+Released under Apache 2.0 license as described in the file LICENSE.+Authors: Bhavik Mehta+-/+import algebra.associated+import data.nat.factorization++/-!+# Prime powers++This file deals with prime powers: numbers which are positive integer powers of a single prime.+-/++variables {R : Type*} [comm_monoid_with_zero R] (n p : R) (k : ℕ)++/-- `n` is a prime power if there is a prime `p` and a positive natural `k` such that `n` can be+written as `p^k`. -/+def is_prime_pow : Prop :=+  ∃ (p : R) (k : ℕ), prime p ∧ 0 < k ∧ p ^ k = n++lemma is_prime_pow_def :+  is_prime_pow n ↔ ∃ (p : R) (k : ℕ), prime p ∧ 0 < k ∧ p ^ k = n := iff.rfl++/-- An equivalent definition for prime powers: `n` is a prime power iff there is a prime `p` and a+natural `k` such that `n` can be written as `p^(k+1)`. -/+lemma is_prime_pow_iff_pow_succ :+  is_prime_pow n ↔ ∃ (p : R) (k : ℕ), prime p ∧ p ^ (k + 1) = n :=+(is_prime_pow_def _).trans+⟨λ ⟨p, k, hp, hk, hn⟩, ⟨_, _, hp, by rwa [nat.sub_add_cancel hk]⟩,+  λ ⟨p, k, hp, hn⟩, ⟨_, _, hp, nat.succ_pos', hn⟩⟩++lemma not_is_prime_pow_zero [no_zero_divisors R] :+  ¬ is_prime_pow (0 : R) :=+begin+  simp only [is_prime_pow_def, not_exists, not_and', and_imp],+  intros x n hn hx,+  rw pow_eq_zero hx,+  simp,+end++lemma not_is_prime_pow_one : ¬ is_prime_pow (1 : R) :=+begin+  simp only [is_prime_pow_def, not_exists, not_and', and_imp],+  intros x n hn hx ht,+  exact ht.not_unit (is_unit_of_pow_eq_one x n hx hn),+end++lemma prime.is_prime_pow {p : R} (hp : prime p) : is_prime_pow p :=+⟨p, 1, hp, zero_lt_one, by simp⟩++lemma is_prime_pow.pow {n : R} (hn : is_prime_pow n)+  {k : ℕ} (hk : k ≠ 0) : is_prime_pow (n ^ k) :=+let ⟨p, k', hp, hk', hn⟩ := hn in ⟨p, k * k', hp, mul_pos hk.bot_lt hk', by rw [pow_mul', hn]⟩++lemma unique_is_prime_pow {R : Type*} [cancel_comm_monoid_with_zero R] {p₁ p₂ : R} {k₁ k₂ : ℕ}+  (hp₁ : prime p₁) (hp₂ : prime p₂) (hk₁ : 0 < k₁) (h : associated (p₁ ^ k₁) (p₂ ^ k₂)):+  associated p₁ p₂ :=+begin+  have : p₁ ∣ p₂ ^ k₂,+  { rw ←h.dvd_iff_dvd_right,+    apply dvd_pow_self _ hk₁.ne' },+  rw ←hp₁.dvd_prime_iff_associated hp₂,+  exact hp₁.dvd_of_dvd_pow this,+end++-- lemma unique_is_prime_pow {p₁ p₂ k₁ k₂ : ℕ}

There's the associated version in algebra/associates now, I think this version can be recovered because ℕ has unique units? I'll add that version explicitly though

b-mehta

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Pull request review commentleanprover-community/mathlib

feat(data/nat): a predicate for prime powers

+/-

Oops yeah you're right!

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pull request commentleanprover-community/mathlib

feat(order/atoms): A set/finset/multiset is an atom iff it's a singleton

Even then, this special case is valuable. Can we try to get either that version, or this version in?

YaelDillies

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pull request commentleanprover-community/mathlib

feat(order/atoms): A set/finset/multiset is an atom iff it's a singleton

Provided you have the time, may I bump this? I think this is something we really ought to have in mathlib!

YaelDillies

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feat(data/finset/basic): random golf

Nice golf!

bors merge

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Bhavik Mehta

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fixup

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Stuart Presnell

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feat(data/nat/factorization): Evaluating a multiplicative function over prime power divisors (#11167) For any multiplicative function `f` with `f 1 = 1` and any `n > 0`, we can evaluate `f n` by evaluating `f` at `p ^ k` over the factorization of `n`. Also provides an alternative version that swaps the `0 < n` condition for an extra `f 0 = 1` condition, as suggested by @ericrbg. This allows a very simple proof that `n.factorization.prod pow = n`

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Gabriel Ebner

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feat(data/set/basic): add decidable instances for boolean operations (#11354) Add decidability instances for `a ∈ s ∩ t`, etc.

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Alex J Best

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feat(data/finsupp/pointwise): add a definition of the pointwise action of functions on finsupps (#10933) I couldn't find this, and it seems like quite a natural way to talk about multiplying functions with finsupps. I'm not sure what additional lemmas would be useful yet, as I don't have a particular application in mind at present so suggestions/additions are welcome

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Paul Lezeau

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feat(algebra/associated): add lemmas to split #9345 (#10941) This PR contains lemmas from PR [#9345](https://github.com/leanprover-community/mathlib/pull/9345), which was starting to get quite lengthy. Co-authored-by: Anne Baanen.

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Violeta Hernández

commit sha 4e7e5a6e756d15c07e9eda778f0b2699f6e262ad

feat(set_theory/ordinal_arithmetic): Enumerating unbounded sets of ordinals with ordinals (#10979) This PR introduces `enum_ord`, which enumerates an unbounded set of ordinals using ordinals. This is used to build an explicit order isomorphism `enum_ord.order_iso`.

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Joseph Myers

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feat(analysis/inner_product_space/orientation): orientations of real inner product spaces (#11269) Add definitions and lemmas relating to orientations of real inner product spaces, in particular constructing an orthonormal basis with a given orientation in finite positive dimension. This is in a new file since nothing else about inner product spaces needs to depend on orientations.

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tb65536

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feat(group_theory/abelianization): Add fintype instance (#11302) Adds `fintype` instance for `abelianization`.

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Joseph Myers

commit sha 3b55b94857df2d8909e2c1f256eb9318b7a2c84d

feat(analysis/inner_product_space/basic): inner products of linear combinations of orthonormal vectors (#11323) There are some lemmas about the inner product of a linear combination of orthonormal vectors with one vector from that orthonormal family. Add similar lemmas where both sides of the inner product are linear combinations.

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Joseph Myers

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feat(analysis/normed_space/linear_isometry): basis ext lemmas (#11331) Add lemmas that two linear isometries / linear isometric equivalences are equal if they are equal on basis vectors, similar to such lemmas for equality on basis vectors of other kinds of maps.

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Riccardo Brasca

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feat(ring_theory/discriminant): remove an assumption (#11359) We remove a `nonempty` assumption.

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Joachim Breitner

commit sha d8a75bd0cfc5c6f13ad8638e9dba100cf6fa2d44

chore(simple_graph/basic): Fix typo in docstring: adjacent vertices, not edges (#11356)

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Alex J Best

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feat(algebra/big_operators): add product versions of some sum lemmas (#11358) and to_additive to get the old ones back

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Violeta Hernández

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refactor(logic/small, *): Infer `f : α → β` when followed by a simple condition on `f` (#11037) Co-authored-by: Yury G. Kudryashov <urkud@urkud.name> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

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tb65536

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refactor(analysis/complex/circle): The circle group is commutative (#11368) This PR upgrades the `group circle` instance to a `comm_group circle` instance.

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Violeta Hernández

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feat(set_theory/ordinal_arithmetic): Extra lemmas about suprema (#11178) Proved lemmas pertaining to when suprema or least strict upper bounds are zero.

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Yuma Mizuno

commit sha b7f8f725434b04f0ab663020eef2d91acba265f1

feat(category_theory/bicategory/functor): define oplax functors and their composition (#11277) This PR defines oplax functors between bicategories and their composition.

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ccn

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feat(linear_algebra/finite_dimensional): Define rank of set of vectors (#11290) Added in the definition of "rank of a set of vectors" and a useful lemma about the rank when one set is a subset of the other. Read the zulip stream here: https://leanprover.zulipchat.com/#narrow/stream/113489-new-members/topic/First.20Time.20Contributing Co-authored-by: ccn <callum.cassidynolan@mail.utoronto.ca>

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Yakov Pechersky

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feat(ring_theory/laurent): coe from R[[x]] to R((x)) (#11318) And actually the changes reported in #11295 Generalize `power_series.coeff_smul` Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

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Alex J Best

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feat(data/mv_polynomial): assorted mv_polynomial and finsupp lemmas (#11319) Mostly around total degree, supports and homogeneous components. From flt-regular.

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Frédéric Dupuis

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feat(analysis/normed_space/star): add lemmas about continuity and norm of identity (#11324)

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Bhavik Mehta

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fill in some sorry

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partial proof of cor 1

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fix up

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restore arithmetic function

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Eric

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chore(*): update to lean 3.36.0 (#11253) The main breaking change is the change in elaboration of double membership binders into x hx y hy, from x y hx hy. Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>

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Yury G. Kudryashov

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chore(analysis/normed_space/banach): move more to the `continuous_linear_map` NS (#11263) ## Rename * `open_mapping` → `continuous_linear_map.is_open_map`; * `open_mapping_affine` → `affine_map.is_open_map`; ### New lemmas * `continuous_linear_map.quotient_map`, * `continuous_linear_map.interior_preimage`, * `continuous_linear_map.closure_preimage`, * `continuous_linear_map.frontier_preimage`.

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Rob Lewis

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docs(algebra/covariant_and_contravariant): minor typos (#11293)

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Floris van Doorn

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feat(topology/basic): add explicit definition of continuous_at (#11296) This was convenient in a demo.

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Eric Wieser

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chore(set_theory/game/domineering): extract repeated goal into lemma and golf (#11298) `fst_pred_mem_erase_of_mem_right` and `snd_pred_mem_erase_of_mem_left` were common subgoals that appeared in two lemmas each.

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Jireh Loreaux

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feat(linear_algebra/eigenspace): prove eigenvalues are exactly elements of the spectrum when the space is finite dimensional (#10961) This adds `has_eigenvalue_iff_mem_spectrum` and then uses it to golf `exists_eigenvalue` - [x] depends on: #10912 - [x] depends on: #10919 Co-authored-by: Vierkantor <vierkantor@vierkantor.com>

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Arthur Paulino

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feat(combinatorics/simple_graph/matchings): even_card_vertices_of_perfect_matching (#11083) Co-authored-by: YaelDillies <yael.dillies@gmail.com>

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Eric

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chore(ring_theory/fractional_ideal): fix typo (#11311)

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Stuart Presnell

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feat(data/nat/factorization): Add lemmas on factorizations of pairs of coprime numbers (#10850)

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Joseph Myers

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feat(linear_algebra/basis): `basis.ext`, `basis.ext'` for semilinear maps (#11317) Extend `basis.ext` and `basis.ext'` to apply to the general (semilinear) case of `linear_map` and `linear_equiv`.

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Alex J Best

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chore(*): miscellaneous to_additive related cleanup (#11316) A few cleanup changes related to to_additive: * After https://github.com/leanprover-community/lean/pull/618 was merged, we no longer need to add namespaces manually in filtered_colimits and open subgroup * to_additive can now generate some more lemmas in big_operators/fin * to_additive now handles a proof in measure/haar better than it used to so remove a workaround there

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Yaël Dillies

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split(data/set/functor): Split off `data.set.lattice` (#11327) This moves the functor structure of `set` in a new file `data.set.functor`. Also adds `alternative set` because it's quick and easy.

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Yakov Pechersky

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feat(field_theory/ratfunc): ratfunc.lift_on without is_domain (#11227) We might want to state results about rational functions without assuming that the base ring is an integral domain. Cf. Misconceptions about $K_X$, Kleiman, Steven; Stacks01X1 Co-authored-by: Yakov Pechersky <ypechersky@treeline.bio>

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Alex J Best

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feat(vscode): add a snippet for inserting a module docstring template (#11312) We already have a vscode snippet for adding copyright headers, this PR adds a similar one to generate a default module docstring with many of the common sections stubbed out. By default it takes the filename, converts underscores to spaces and capitalizes each word to create the title, as this seems a sensible default. But otherwise all text is a static default example following the documentation style page to make it easier to remember the various recommended secitons. To test do `ctrl+shift+p` to open the command pallette, type insert snippet, enter, and type module and it should show up. See also #3186

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Joseph Myers

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feat(algebra/periodic): lifting to function on quotient group (#11321) I want to make more use of the type `real.angle` in `analysis.special_functions.trigonometric.angle`, including defining functions from this type in terms of periodic functions from `ℝ`. To support defining such functions, add a definition `periodic.lift` that lifts a periodic function from `α` to a function from `α ⧸ (add_subgroup.zmultiples c)`, along with a lemma `periodic.lift_coe` about the values of the resulting function.

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Yaël Dillies

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refactor(data/{sigma,psigma}/order): Use `lex` synonym and new notation (#11235) This introduces notations `Σₗ i, α i` and `Σₗ' i, α i` for `lex (Σ i, α i)` and `lex (Σ' i, α i)` and use them instead of the instance switch with locale `lex`.

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Eric

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chore(*): update to 3.37.0c (#11325) the major breaking change this version is making `default`'s parameters implicit, as opposed to explicit. there was also some slight "free" golfing due to the better `out_param` simp support.

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Alex J Best

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feat(topology/algebra/monoid): to_additivize some lemmas (#11310) Uncomment a commented out to additive line that looks like its been there for 3 years (since https://github.com/leanprover-community/mathlib/commit/581cf19bf1885ef874c39c9902a93f579bc8c22d) The changes to to_additive in the past few years now make the generated lemma useful. Also to_additivize a bunch of other lemmas in this file.

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Rob Lewis

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fix(docs/references.bib): syntax error (#11342) This broke the docs build.

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Eric Wieser

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chore(data/set/function): move inv_fun_on out of `logic/function/basic` (#11330) This removes `function.inv_fun_on_eq'` as it is a duplicate of `inj_on.left_inv_on_inv_fun_on`.

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stuart suggestions

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generalize

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pull request commentleanprover-community/mathlib

feat(data/sym/sym2) eq in sym2 from distinct common members

Looks sensible to me! Let's see if Kyle has anything to add about this

bors d=kmill

vbeffara

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pull request commentleanprover-community/mathlib

feat(order/antichain): Strong antichains

LGTM! Thanks!

bors merge

YaelDillies

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pull request commentleanprover-community/mathlib

feat(order/antichain): Strong antichains

Looks sensible! I think there's a lemma you can prove about the relation of order_bot, maybe that if it's order_bot then there are no strong antichains or they're all trivial?

YaelDillies

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Pull request review commentleanprover-community/mathlib

feat(data/sum/interval): The disjoint sum of two locally finite orders is locally finite

+/-+Copyright (c) 2022 Yaël Dillies. All rights reserved.+Released under Apache 2.0 license as described in the file LICENSE.+Authors: Yaël Dillies+-/+import data.sum.order+import order.locally_finite++/-!+# Finite intervals in a disjoint union++This file provides the `locally_finite_order` instance for the disjoint sum of two orders.++## TODO++Do the same for the lexicographic sum of orders.+-/++open function sum++namespace finset+variables {α₁ α₂ β₁ β₂ γ₁ γ₂ : Type*}++section sum_lift₂+variables (f f₁ g₁ : α₁ → β₁ → finset γ₁) (g f₂ g₂ : α₂ → β₂ → finset γ₂)++/-- Lifts maps `α₁ → β₁ → finset γ₁` and `α₂ → β₂ → finset γ₂` to a map+`α₁ ⊕ α₂ → β₁ ⊕ β₂ → finset (γ₁ ⊕ γ₂)`. Could be generalized to alternative monads if we can make

I don't think this resolves the issue, the ambiguity is still present. How about writing "alternative functors" to make it clear that you mean alternative as a technical term?

YaelDillies

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