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

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+/-+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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+/-+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

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Oops yeah you're right!

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

YaelDillies

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

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Nice golf!

bors merge

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

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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

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.

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

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

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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

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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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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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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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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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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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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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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partial proof of cor 1

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

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

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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.

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

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

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

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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`.

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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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.

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

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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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generalize

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LGTM! Thanks!

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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?

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+/-+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?

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