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[Merged by Bors] - feat(analysis/convex): add dual cone ready-to-merge

Add definition of the dual cone of a set in a real inner product space

Hello! I am not sure if this is the "right" definition of a dual cone. An alternative would be to define the dual cone as a cone in the dual vector space and to use y(x) instead of ⟪ x, y ⟫.

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feat(analysis/convex): add dual cone

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chore(algebra/covariant_and_contravariant): fix typos in module doc-strings

This PR changes slightly the doc-strings to make the autogenerated documentation more consistent. I also removed an unstylish double space.


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feat(data/setoid/partition): indexed partition

 _ (subtype (@is_partition α)) _ (partial_order.to_preorder _) $ partition.rel_i end partition  end setoid++/-- A partition of a type indexed by another type. -/+@[nolint has_inhabited_instance]+structure indexed_partition {ι α : Type*} (s : ι → set α) :=+(eq_of_mem : ∀ {x i j}, x ∈ s i → x ∈ s j → i = j)+(some : ι → α)+(some_mem : ∀ i, some i ∈ s i)+(index : α → ι)+(mem_index : ∀ x, x ∈ s (index x))++/-- The non-constructive constructor for `indexed_partition`. -/+noncomputable+def indexed_partition.mk' {ι α : Type*} (s : ι → set α) (dis : ∀ i j, i ≠ j → disjoint (s i) (s j))+(empty : ∀ i, (s i).nonempty) (ex : ∀ x, ∃ i, x ∈ s i) : indexed_partition s :=+{ eq_of_mem := begin+    classical,+    intros x i j hxi hxj,+    by_contra h,+    exact dis _ _ h ⟨hxi, hxj⟩+  end,+  some := λ i, (empty i).some,+  some_mem := λ i, (empty i).some_spec,+  index := λ x, (ex x).some,+  mem_index := λ x, (ex x).some_spec }++namespace indexed_partition++open set++variables {ι α : Type*} {s : ι → set α} (hs : indexed_partition s)+include hs++lemma exists_mem (x : α) : ∃ i, x ∈ s i := ⟨hs.index x, hs.mem_index x⟩++lemma Union : (⋃ i, s i) = univ :=+by { ext x, simp [hs.exists_mem x] }++lemma disjoint : ∀ {i j}, i ≠ j → disjoint (s i) (s j) :=+λ i j h x ⟨hxi, hxj⟩,h (hs.eq_of_mem hxi hxj)
λ i j h x ⟨hxi, hxj⟩, h (hs.eq_of_mem hxi hxj)
PatrickMassot

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feat(measure_theory): define `measure.trim`, restriction of a measure to a sub-sigma algebra

 attribute [integral_simps] integral_neg integral_smul L1.integral_add L1.integra  attribute [irreducible] integral L1.integral +section integral_trim++variables {H β γ : Type*} [normed_group H] [measurable_space H]+  {m m0 : measurable_space β} {μ : measure β}++/-- Simple function seen as simple function of a larger `measurable_space`. -/+def simple_func.to_larger_space (hm : m ≤ m0) (f : @simple_func β m γ) : simple_func β γ :=+⟨@simple_func.to_fun β m γ f, λ x, hm _ (@simple_func.measurable_set_fiber β γ m f x),+  @simple_func.finite_range β γ m f⟩++lemma simple_func.coe_to_larger_space_eq (hm : m ≤ m0) (f : @simple_func β m γ) :+  ⇑(f.to_larger_space hm) = f :=+rfl++lemma integral_simple_func_larger_space (hm : m ≤ m0) (f : @simple_func β m F)+  (hf_int : integrable f μ) :+  ∫ x, f x ∂μ = ∑ x in (@simple_func.range β F m f), (ennreal.to_real (μ (f ⁻¹' {x}))) • x :=+begin+  simp_rw ← f.coe_to_larger_space_eq hm,+  have hf_int : integrable (f.to_larger_space hm) μ, by rwa simple_func.coe_to_larger_space_eq,+  rw simple_func.integral_eq_sum _ hf_int,+  congr,+end++lemma integral_trim_simple_func (hm : m ≤ m0) (f : @simple_func β m F) (hf_int : integrable f μ) :+  ∫ x, f x ∂μ = @integral β F m _ _ _ _ _ _ (μ.trim hm) f :=+begin+  have hf : @measurable _ _ m _ f, from @simple_func.measurable β F m _ f,+  have hf_int_m := hf_int.trim hm hf,+  rw [integral_simple_func_larger_space le_rfl f hf_int_m,+    integral_simple_func_larger_space hm f hf_int],+  congr,+  ext1 x,+  congr,+  exact (trim_measurable_set_eq hm (@simple_func.measurable_set_fiber β F m f x)).symm,+end++lemma integral_trim (hm : m ≤ m0) {f : β → F} (hf : @measurable β F m _ f)+  (hf_int : integrable f μ) :

Thanks for spotting this! Indeed I could remove the integrability assumption, here and in the two other results.

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RemyDegenne

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remove integrability assumptions

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feat(measure_theory): define `measure.trim`, restriction of a measure to a sub-sigma algebra

 attribute [integral_simps] integral_neg integral_smul L1.integral_add L1.integra  attribute [irreducible] integral L1.integral +section integral_trim++variables {H β γ : Type*} [normed_group H] [measurable_space H]+  {m m0 : measurable_space β} {μ : measure β}++/-- Simple function seen as simple function of a larger `measurable_space`. -/+def simple_func.to_larger_space (hm : m ≤ m0) (f : @simple_func β m γ) : simple_func β γ :=+⟨@simple_func.to_fun β m γ f, λ x, hm _ (@simple_func.measurable_set_fiber β γ m f x),+  @simple_func.finite_range β γ m f⟩++lemma simple_func.coe_to_larger_space_eq (hm : m ≤ m0) (f : @simple_func β m γ) :+  ⇑(f.to_larger_space hm) = f :=+rfl++lemma integral_simple_func_larger_space (hm : m ≤ m0) (f : @simple_func β m F)+  (hf_int : integrable f μ) :+  ∫ x, f x ∂μ = ∑ x in (@simple_func.range β F m f), (ennreal.to_real (μ (f ⁻¹' {x}))) • x :=+begin+  simp_rw ← f.coe_to_larger_space_eq hm,+  have hf_int : integrable (f.to_larger_space hm) μ, by rwa simple_func.coe_to_larger_space_eq,+  rw simple_func.integral_eq_sum _ hf_int,+  congr,+end++lemma integral_trim_simple_func (hm : m ≤ m0) (f : @simple_func β m F) (hf_int : integrable f μ) :+  ∫ x, f x ∂μ = @integral β F m _ _ _ _ _ _ (μ.trim hm) f :=+begin+  have hf : @measurable _ _ m _ f, from @simple_func.measurable β F m _ f,+  have hf_int_m := hf_int.trim hm hf,+  rw [integral_simple_func_larger_space le_rfl f hf_int_m,+    integral_simple_func_larger_space hm f hf_int],+  congr,+  ext1 x,+  congr,+  exact (trim_measurable_set_eq hm (@simple_func.measurable_set_fiber β F m f x)).symm,+end++lemma integral_trim (hm : m ≤ m0) {f : β → F} (hf : @measurable β F m _ f)+  (hf_int : integrable f μ) :

Is the integrability assumption really necessary here? If f is not integrable with respect to the big sigma-algebra, it won't either with respect to the small sigma-algebra, and both integrals will vanish, right?

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feat(measure_theory): define `measure.trim`, restriction of a measure to a sub-sigma algebra

 end  end is_complete +namespace measure_theory++lemma outer_measure.to_measure_zero [measurable_space α] : (0 : outer_measure α).to_measure+  ((le_top).trans outer_measure.zero_caratheodory.symm.le) = 0 :=+by rw [← measure.measure_univ_eq_zero, to_measure_apply _ _ measurable_set.univ,+  outer_measure.coe_zero, pi.zero_apply]++section trim++/-- Restriction of a measure to a sub-sigma algebra.+It is common to see a measure `μ` on a measurable space structure `m0` as being also a measure on+any `m ≤ m0`. Since measures in mathlib have to be trimmed to the measurable space, `μ` itself+cannot be a measure on `m`, hence the definition of `μ.trim hm`.++This notion is related to `outer_measure.trim`, see the lemma+`to_outer_measure_trim_eq_trim_to_outer_measure`. -/+def measure.trim {m m0 : measurable_space α} (μ : @measure α m0) (hm : m ≤ m0) : @measure α m :=+@outer_measure.to_measure α m μ.to_outer_measure (hm.trans (le_to_outer_measure_caratheodory μ))++@[simp] lemma trim_eq_self [measurable_space α] {μ : measure α} : μ.trim le_rfl = μ :=+by simp [measure.trim]++variables {m m0 : measurable_space α} {μ : measure α} {s : set α}++lemma to_outer_measure_trim_eq_trim_to_outer_measure (μ : measure α) (hm : m ≤ m0) :+  @measure.to_outer_measure _ m (μ.trim hm) = @outer_measure.trim _ m μ.to_outer_measure :=+by rw [measure.trim, to_measure_to_outer_measure]++@[simp] lemma zero_trim (hm : m ≤ m0) : (0 : measure α).trim hm = (0 : @measure α m) :=+by simp [measure.trim, outer_measure.to_measure_zero]++lemma trim_measurable_set_eq (hm : m ≤ m0) (hs : @measurable_set α m s) : μ.trim hm s = μ s :=+by simp [measure.trim, hs]++lemma le_trim (hm : m ≤ m0) : μ s ≤ μ.trim hm s :=+by {simp_rw [measure.trim], exact (@le_to_measure_apply _ m _ _ _), }
by { simp_rw [measure.trim], exact (@le_to_measure_apply _ m _ _ _), }
RemyDegenne

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feat(measure_theory): define `measure.trim`, restriction of a measure to a sub-sigma algebra

 begin   exact hfi.ae_measurable.null_measurable_set (measurable_set_singleton 0).compl end +lemma set_integral_trim {α} {m m0 : measurable_space α} {μ : measure α} (hm : m ≤ m0) {f : α → E}

same comment as above: I am not sure that you need the integrability assumption.

RemyDegenne

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feat(measure_theory): define `measure.trim`, restriction of a measure to a sub-sigma algebra

 attribute [integral_simps] integral_neg integral_smul L1.integral_add L1.integra  attribute [irreducible] integral L1.integral +section integral_trim++variables {H β γ : Type*} [normed_group H] [measurable_space H]+  {m m0 : measurable_space β} {μ : measure β}++/-- Simple function seen as simple function of a larger `measurable_space`. -/+def simple_func.to_larger_space (hm : m ≤ m0) (f : @simple_func β m γ) : simple_func β γ :=+⟨@simple_func.to_fun β m γ f, λ x, hm _ (@simple_func.measurable_set_fiber β γ m f x),+  @simple_func.finite_range β γ m f⟩++lemma simple_func.coe_to_larger_space_eq (hm : m ≤ m0) (f : @simple_func β m γ) :+  ⇑(f.to_larger_space hm) = f :=+rfl++lemma integral_simple_func_larger_space (hm : m ≤ m0) (f : @simple_func β m F)+  (hf_int : integrable f μ) :+  ∫ x, f x ∂μ = ∑ x in (@simple_func.range β F m f), (ennreal.to_real (μ (f ⁻¹' {x}))) • x :=+begin+  simp_rw ← f.coe_to_larger_space_eq hm,+  have hf_int : integrable (f.to_larger_space hm) μ, by rwa simple_func.coe_to_larger_space_eq,+  rw simple_func.integral_eq_sum _ hf_int,+  congr,+end++lemma integral_trim_simple_func (hm : m ≤ m0) (f : @simple_func β m F) (hf_int : integrable f μ) :+  ∫ x, f x ∂μ = @integral β F m _ _ _ _ _ _ (μ.trim hm) f :=+begin+  have hf : @measurable _ _ m _ f, from @simple_func.measurable β F m _ f,+  have hf_int_m := hf_int.trim hm hf,+  rw [integral_simple_func_larger_space le_rfl f hf_int_m,+    integral_simple_func_larger_space hm f hf_int],+  congr,+  ext1 x,+  congr,+  exact (trim_measurable_set_eq hm (@simple_func.measurable_set_fiber β F m f x)).symm,+end++lemma integral_trim (hm : m ≤ m0) {f : β → F} (hf : @measurable β F m _ f)+  (hf_int : integrable f μ) :+  ∫ x, f x ∂μ = @integral β F m _ _ _ _ _ _ (μ.trim hm) f :=+begin+  let f_seq := @simple_func.approx_on F β _ _ _ m _ hf set.univ 0 (set.mem_univ 0) _,+  have hf_seq_meas : ∀ n, @measurable _ _ m _ (f_seq n),+    from λ n, @simple_func.measurable β F m _ (f_seq n),+  have hf_seq_int : ∀ n, integrable (f_seq n) μ,+    from simple_func.integrable_approx_on_univ (hf.mono hm le_rfl) hf_int,+  have hf_seq_int_m : ∀ n, @integrable β F m _ _ (f_seq n) (μ.trim hm),+    from λ n, (hf_seq_int n).trim hm (hf_seq_meas n) ,+  have hf_seq_eq : ∀ n, ∫ x, f_seq n x ∂μ = @integral β F m _ _ _ _ _ _ (μ.trim hm) (f_seq n),+    from λ n, integral_trim_simple_func hm (f_seq n) (hf_seq_int n),+  have h_lim_1 : at_top.tendsto (λ n, ∫ x, f_seq n x ∂μ) (𝓝 (∫ x, f x ∂μ)),+  { refine tendsto_integral_of_L1 f hf_int (eventually_of_forall hf_seq_int) _,+    exact simple_func.tendsto_approx_on_univ_L1_edist (hf.mono hm le_rfl) hf_int, },+  have h_lim_2 :  at_top.tendsto (λ n, ∫ x, f_seq n x ∂μ)+    (𝓝 (@integral β F m _ _ _ _ _ _ (μ.trim hm) f)),+  { simp_rw hf_seq_eq,+    refine @tendsto_integral_of_L1 β F m _ _ _ _ _ _ (μ.trim hm) _ f+      (hf_int.trim hm hf) _ _ (eventually_of_forall hf_seq_int_m) _,+    exact @simple_func.tendsto_approx_on_univ_L1_edist β F m _ _ _ _ f _ hf (hf_int.trim hm hf), },+  exact tendsto_nhds_unique h_lim_1 h_lim_2,+end++lemma integral_trim' (hm : m ≤ m0) {f : β → F} (hf : @integrable β F m _ _ f (μ.trim hm)) :

If my comment above is true, ae_measurability should be enough for this one.

RemyDegenne

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Nicolò Cavalleri

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feat(geometry/manifold): Some lemmas for smooth functions (#7752)

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Riccardo Brasca

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feat(linear_algebra/free_module): add class module.free (#7801) We introduce here a new class `module.free`. Co-authored-by: Johan Commelin <johan@commelin.net>

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Adam Topaz

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feat(topology/category/Profinite/cofiltered_clopen): Theorem about clopen sets in cofiltered limits of profinite sets (#7837) This PR proves the theorem that any clopen set in a cofiltered limit of profinite sets arises from a clopen set in one of the factors of the limit. This generalizes a theorem used in LTE.

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feat(algebra/pointwise): Dynamics of powers of a subset (#7836) If `S` is a subset of a group `G`, then the powers of `S` eventually stabilize in size. Co-authored-by: tb65536 <tb65536@users.noreply.github.com>

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fix(tactic/lint): punctuation of messages (#7869) Previously, the linter framework would append punctuation (`.` or `:`) to the message provided by the linter, but this was confusing and lead to some double punctuation. Now all linters specify their own punctuation. Co-authored-by: Floris van Doorn <fpv@andrew.cmu.edu>

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chore(linear_algebra/quadratic_form): fix typo (#7907)

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chore(category_theory/filtered): Adds missing instances (#7909) Co-authored-by: Patrick Massot <patrickmassot@free.fr>

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chore(algebra/{ordered_group, linear_ordered_comm_group_with_zero.lean}): rename one lemma, remove more @s (#7895) The more substantial part of this PR is changing the name of a lemma from `div_lt_div_iff'` to `mul_inv_lt_mul_inv_iff'`: the lemma proves `a * b⁻¹ ≤ c * d⁻¹ ↔ a * d ≤ c * b`. Furthermore, in the same spirit as a couple of my recent short PRs, I am removing a few more `@`, in order to sweep under the rug, later on, a change in typeclass assumptions. This PR only changes a name, which was used only once, and a few proofs, but no statement. On the path towards PR #7645.

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chore(scripts): update nolints.txt (#7914) I am happy to remove some nolints for you!

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feat(algebra/covariant_and_contravariant): API for covariant_and_contravariant (#7889) This PR introduces more API for `covariant` and `contravariant` stuff . Besides the API, I have not actually made further use of the typeclasses or of the API. This happens in subsequent PRs. This is a step towards PR #7645.

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Patrick Massot

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feat(measure_theory): continuity of primitives (#7864) From the sphere eversion project This proves some continuity of interval integrals with respect to parameters and continuity of primitives of measurable functions. The statements are a bit abstract, but they allow to have: ```lean example {f : ℝ → E} (h_int : integrable f) (a : ℝ) : continuous (λ b, ∫ x in a .. b, f x ∂ volume) := h_int.continuous_primitive a ``` under the usual assumptions on `E`: `normed_group E`, `second_countable_topology E`, `normed_space ℝ E` `complete_space E`, `measurable_space E`, `borel_space E`, say `E = ℝ` for instance. Of course global integrability is not needed, assuming integrability on all finite length intervals is enough: ```lean example {f : ℝ → E} (h_int : ∀ a b : ℝ, interval_integrable f volume a b) (a : ℝ) : continuous (λ b, ∫ x in a .. b, f x ∂ volume) := continuous_primitive h_int a ```

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Ruben Van de Velde

commit sha 5c114586544998ea34d18cfe63700b5fe7507eb4

chore(analysis/normed_space/normed_group_hom): golf proof of normed_group_hom.bounded (#7896)

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sgouezel

commit sha a359bd989d3bc0917c282b59ba86dfafd5f26c4a

chore(measure_theory): measurability statements for coercions, coherent naming (#7854) Also add a few lemmas on measure theory

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Yaël Dillies

commit sha e971eae617baf2c358b32fe3dff33798e00e446f

docs(data/nat/totient): add module docstring (#7899)

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Yaël Dillies

commit sha b3244884a52c0dfa2ca48825fb45bcf85f9551e3

docs(set_theory/schroeder_bernstein): add module docstring (#7900)

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damiano

commit sha 4fe77812bd05841b05b7dd5ac43cb5faf4917255

chore(algebra/lie/basic + classical): golf some proofs (#7903) Another PR with some golfing, to get acquainted with the files! Oliver, I really like how you set this up! Also, feel free to say that you do not like the golfing: there is a subtle tension between proving stuff fast and making it accessible!

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Rémy Degenne

commit sha 300451336dc75ef64934b14050c93c3f1084797e

feat(measure_theory/ess_sup): monotonicity of ess_sup/ess_inf w.r.t. the measure (#7917)

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leanprover-community-bot

commit sha fec6c8a4fdf630d30cc103095a69a2fc4bc6091e

chore(scripts): update nolints.txt (#7922) I am happy to remove some nolints for you!

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feat(analysis/normed_space): nnreal.coe_nat_abs

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PatrickMassot

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docs(data/rat/denumerable): add module docstring

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YaelDillies

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Benjamin Davidson

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name changes

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